Trigonometrik özdeşlikler listesi

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vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Giriş</div> </a> </li> <li id="toc-Pisagor_özdeşlikleri" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Pisagor_özdeşlikleri"> <div class="vector-toc-text"> 1 Pisagor özdeşlikleri </div> </a> <ul id="toc-Pisagor_özdeşlikleri-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Yansımalar,_kaymalar_ve_periyodiklik" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Yansımalar,_kaymalar_ve_periyodiklik"> <div class="vector-toc-text"> 2 Yansımalar, kaymalar ve periyodiklik </div> </a> <button aria-controls="toc-Yansımalar,_kaymalar_ve_periyodiklik-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Yansımalar, kaymalar ve periyodiklik alt bölümünü aç/kapa </button> <ul id="toc-Yansımalar,_kaymalar_ve_periyodiklik-sublist" class="vector-toc-list"> <li id="toc-Yansımalar" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Yansımalar"> <div class="vector-toc-text"> 2.1 Yansımalar </div> </a> <ul id="toc-Yansımalar-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kaymalar_ve_periyodiklik" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Kaymalar_ve_periyodiklik"> <div class="vector-toc-text"> 2.2 Kaymalar ve periyodiklik </div> </a> <ul id="toc-Kaymalar_ve_periyodiklik-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-İşaretler" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#İşaretler"> <div class="vector-toc-text"> 2.3 İşaretler </div> </a> <ul id="toc-İşaretler-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Açı_toplam_ve_fark_özdeşlikleri" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Açı_toplam_ve_fark_özdeşlikleri"> <div class="vector-toc-text"> 3 Açı toplam ve fark özdeşlikleri </div> </a> <button aria-controls="toc-Açı_toplam_ve_fark_özdeşlikleri-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Açı toplam ve fark özdeşlikleri alt bölümünü aç/kapa </button> <ul id="toc-Açı_toplam_ve_fark_özdeşlikleri-sublist" class="vector-toc-list"> <li id="toc-Sonsuz_sayıda_açının_toplamlarının_sinüs_ve_kosinüsleri" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sonsuz_sayıda_açının_toplamlarının_sinüs_ve_kosinüsleri"> <div class="vector-toc-text"> 3.1 Sonsuz sayıda açının toplamlarının sinüs ve kosinüsleri </div> </a> <ul id="toc-Sonsuz_sayıda_açının_toplamlarının_sinüs_ve_kosinüsleri-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Toplamların_tanjantları_ve_kotanjantları" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Toplamların_tanjantları_ve_kotanjantları"> <div class="vector-toc-text"> 3.2 Toplamların tanjantları ve kotanjantları </div> </a> <ul id="toc-Toplamların_tanjantları_ve_kotanjantları-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Toplamların_sekantları_ve_kosekantları" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Toplamların_sekantları_ve_kosekantları"> <div class="vector-toc-text"> 3.3 Toplamların sekantları ve kosekantları </div> </a> <ul id="toc-Toplamların_sekantları_ve_kosekantları-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Batlamyus_teoremi" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Batlamyus_teoremi"> <div class="vector-toc-text"> 3.4 Batlamyus teoremi </div> </a> <ul id="toc-Batlamyus_teoremi-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Açının_katları_ve_yarım_açı_formülleri" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Açının_katları_ve_yarım_açı_formülleri"> <div class="vector-toc-text"> 4 Açının katları ve yarım açı formülleri </div> </a> <button aria-controls="toc-Açının_katları_ve_yarım_açı_formülleri-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Açının katları ve yarım açı formülleri alt bölümünü aç/kapa </button> <ul id="toc-Açının_katları_ve_yarım_açı_formülleri-sublist" class="vector-toc-list"> <li id="toc-Açının_katları_formülleri" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Açının_katları_formülleri"> <div class="vector-toc-text"> 4.1 Açının katları formülleri </div> </a> <ul id="toc-Açının_katları_formülleri-sublist" class="vector-toc-list"> <li id="toc-Çift_açı_formülleri" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Çift_açı_formülleri"> <div class="vector-toc-text"> 4.1.1 Çift açı formülleri </div> </a> <ul id="toc-Çift_açı_formülleri-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Üç_kat_açı_formülleri" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Üç_kat_açı_formülleri"> <div class="vector-toc-text"> 4.1.2 Üç kat açı formülleri </div> </a> <ul id="toc-Üç_kat_açı_formülleri-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Çok_kat_açı_formülleri" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Çok_kat_açı_formülleri"> <div class="vector-toc-text"> 4.1.3 Çok kat açı formülleri </div> </a> <ul id="toc-Çok_kat_açı_formülleri-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Chebyshev_yöntemi" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Chebyshev_yöntemi"> <div class="vector-toc-text"> 4.1.4 Chebyshev yöntemi </div> </a> <ul id="toc-Chebyshev_yöntemi-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Yarım_açı_formülleri" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Yarım_açı_formülleri"> <div class="vector-toc-text"> 4.2 Yarım açı formülleri </div> </a> <ul id="toc-Yarım_açı_formülleri-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tablo" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tablo"> <div class="vector-toc-text"> 4.3 Tablo </div> </a> <ul id="toc-Tablo-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Kuvvet_indirgeme_formülleri" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Kuvvet_indirgeme_formülleri"> <div class="vector-toc-text"> 5 Kuvvet indirgeme formülleri </div> </a> <ul id="toc-Kuvvet_indirgeme_formülleri-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Çarpım-toplam_ve_toplam-çarpım_özdeşlikleri" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Çarpım-toplam_ve_toplam-çarpım_özdeşlikleri"> <div class="vector-toc-text"> 6 Çarpım-toplam ve toplam-çarpım özdeşlikleri </div> </a> <button aria-controls="toc-Çarpım-toplam_ve_toplam-çarpım_özdeşlikleri-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Çarpım-toplam ve toplam-çarpım özdeşlikleri alt bölümünü aç/kapa </button> <ul id="toc-Çarpım-toplam_ve_toplam-çarpım_özdeşlikleri-sublist" class="vector-toc-list"> <li id="toc-Çarpım-toplam_özdeşlikleri" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Çarpım-toplam_özdeşlikleri"> <div class="vector-toc-text"> 6.1 Çarpım-toplam özdeşlikleri </div> </a> <ul id="toc-Çarpım-toplam_özdeşlikleri-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Toplam-çarpım_özdeşlikleri" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Toplam-çarpım_özdeşlikleri"> <div class="vector-toc-text"> 6.2 Toplam-çarpım özdeşlikleri </div> </a> <ul id="toc-Toplam-çarpım_özdeşlikleri-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hermite_kotanjant_özdeşliği" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hermite_kotanjant_özdeşliği"> <div class="vector-toc-text"> 6.3 Hermite kotanjant özdeşliği </div> </a> <ul id="toc-Hermite_kotanjant_özdeşliği-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Trigonometrik_fonksiyonların_sonlu_çarpımları" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Trigonometrik_fonksiyonların_sonlu_çarpımları"> <div class="vector-toc-text"> 6.4 Trigonometrik fonksiyonların sonlu çarpımları </div> </a> <ul id="toc-Trigonometrik_fonksiyonların_sonlu_çarpımları-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Doğrusal_kombinasyonlar" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Doğrusal_kombinasyonlar"> <div class="vector-toc-text"> 7 Doğrusal kombinasyonlar </div> </a> <button aria-controls="toc-Doğrusal_kombinasyonlar-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Doğrusal kombinasyonlar alt bölümünü aç/kapa </button> <ul id="toc-Doğrusal_kombinasyonlar-sublist" class="vector-toc-list"> <li id="toc-Sinüs_ve_kosinüs" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sinüs_ve_kosinüs"> <div class="vector-toc-text"> 7.1 Sinüs ve kosinüs </div> </a> <ul id="toc-Sinüs_ve_kosinüs-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Keyfi_faz_kayması" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Keyfi_faz_kayması"> <div class="vector-toc-text"> 7.2 Keyfi faz kayması </div> </a> <ul id="toc-Keyfi_faz_kayması-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-İkiden_fazla_sinüzoid" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#İkiden_fazla_sinüzoid"> <div class="vector-toc-text"> 7.3 İkiden fazla sinüzoid </div> </a> <ul id="toc-İkiden_fazla_sinüzoid-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Lagrange_trigonometrik_özdeşlikleri" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Lagrange_trigonometrik_özdeşlikleri"> <div class="vector-toc-text"> 8 Lagrange trigonometrik özdeşlikleri </div> </a> <ul id="toc-Lagrange_trigonometrik_özdeşlikleri-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Belirli_doğrusal_kesirli_dönüşümler" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Belirli_doğrusal_kesirli_dönüşümler"> <div class="vector-toc-text"> 9 Belirli doğrusal kesirli dönüşümler </div> </a> <ul id="toc-Belirli_doğrusal_kesirli_dönüşümler-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Karmaşık_üstel_fonksiyon_ile_ilişkisi" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Karmaşık_üstel_fonksiyon_ile_ilişkisi"> <div class="vector-toc-text"> 10 Karmaşık üstel fonksiyon ile ilişkisi </div> </a> <ul id="toc-Karmaşık_üstel_fonksiyon_ile_ilişkisi-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Seri_açılımları" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Seri_açılımları"> <div class="vector-toc-text"> 11 Seri açılımları </div> </a> <ul id="toc-Seri_açılımları-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sonsuz_çarpım_formülleri" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Sonsuz_çarpım_formülleri"> <div class="vector-toc-text"> 12 Sonsuz çarpım formülleri </div> </a> <ul id="toc-Sonsuz_çarpım_formülleri-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ters_trigonometrik_fonksiyonlar" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Ters_trigonometrik_fonksiyonlar"> <div class="vector-toc-text"> 13 Ters trigonometrik fonksiyonlar </div> </a> <ul id="toc-Ters_trigonometrik_fonksiyonlar-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Değişken_içermeyen_özdeşlikler" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Değişken_içermeyen_özdeşlikler"> <div class="vector-toc-text"> 14 Değişken içermeyen özdeşlikler </div> </a> <button aria-controls="toc-Değişken_içermeyen_özdeşlikler-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Değişken içermeyen özdeşlikler alt bölümünü aç/kapa </button> <ul id="toc-Değişken_içermeyen_özdeşlikler-sublist" class="vector-toc-list"> <li id="toc-π'nin_hesaplanması" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#π'nin_hesaplanması"> <div class="vector-toc-text"> 14.1 π'nin hesaplanması </div> </a> <ul id="toc-π'nin_hesaplanması-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Öklid'in_bir_özdeşliği" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Öklid'in_bir_özdeşliği"> <div class="vector-toc-text"> 14.2 Öklid'in bir özdeşliği </div> </a> <ul id="toc-Öklid'in_bir_özdeşliği-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Trigonometrik_fonksiyonların_bileşimi" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Trigonometrik_fonksiyonların_bileşimi"> <div class="vector-toc-text"> 15 Trigonometrik fonksiyonların bileşimi </div> </a> <ul id="toc-Trigonometrik_fonksiyonların_bileşimi-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-α_+_β_+_γ_=_180°_durumu_için_diğer_"koşullu"_özdeşlikler" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#α_+_β_+_γ_=_180°_durumu_için_diğer_"koşullu"_özdeşlikler"> <div class="vector-toc-text"> 16 α + β + γ = 180° durumu için diğer "koşullu" özdeşlikler </div> </a> <ul id="toc-α_+_β_+_γ_=_180°_durumu_için_diğer_"koşullu"_özdeşlikler-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tarihsel_stenolar" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Tarihsel_stenolar"> <div class="vector-toc-text"> 17 Tarihsel stenolar </div> </a> <ul id="toc-Tarihsel_stenolar-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Diğer" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Diğer"> <div class="vector-toc-text"> 18 Diğer </div> </a> <button aria-controls="toc-Diğer-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Diğer alt bölümünü aç/kapa </button> <ul id="toc-Diğer-sublist" class="vector-toc-list"> <li id="toc-Dirichlet_çekirdeği" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dirichlet_çekirdeği"> <div class="vector-toc-text"> 18.1 Dirichlet çekirdeği </div> </a> <ul id="toc-Dirichlet_çekirdeği-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tanjant_yarım_açı_ikamesi" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tanjant_yarım_açı_ikamesi"> <div class="vector-toc-text"> 18.2 Tanjant yarım açı ikamesi </div> </a> <ul id="toc-Tanjant_yarım_açı_ikamesi-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Viète_sonsuz_çarpımı" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Viète_sonsuz_çarpımı"> <div class="vector-toc-text"> 18.3 Viète sonsuz çarpımı </div> </a> <ul id="toc-Viète_sonsuz_çarpımı-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Ayrıca_bakınız" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Ayrıca_bakınız"> <div class="vector-toc-text"> 19 Ayrıca bakınız </div> </a> <ul id="toc-Ayrıca_bakınız-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kaynakça" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Kaynakça"> <div class="vector-toc-text"> 20 Kaynakça </div> </a> <ul id="toc-Kaynakça-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliyografya" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Bibliyografya"> <div class="vector-toc-text"> 21 Bibliyografya </div> </a> <ul id="toc-Bibliyografya-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dış_bağlantılar" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Dış_bağlantılar"> <div class="vector-toc-text"> 22 Dış bağlantılar </div> </a> <ul id="toc-Dış_bağlantılar-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="İçindekiler" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="İçindekiler tablosunu değiştir" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > İçindekiler tablosunu değiştir </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">Trigonometrik özdeşlikler listesi</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Başka bir dildeki sayfaya gidin. 41 dilde mevcut" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-41" aria-hidden="true" > 41 dil </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%82%D8%A7%D8%A6%D9%85%D8%A9_%D8%A7%D9%84%D9%85%D8%AA%D8%B7%D8%A7%D8%A8%D9%82%D8%A7%D8%AA_%D8%A7%D9%84%D9%85%D8%AB%D9%84%D8%AB%D9%8A%D8%A9" title="قائمة المتطابقات المثلثية - Arapça" lang="ar" hreflang="ar" data-title="قائمة المتطابقات المثلثية" data-language-autonym="العربية" data-language-local-name="Arapça" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Triqonometriyan%C4%B1n_%C9%99sas_d%C3%BCsturlar%C4%B1" title="Triqonometriyanın əsas düsturları - Azerbaycan dili" lang="az" hreflang="az" data-title="Triqonometriyanın əsas düsturları" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaycan dili" class="interlanguage-link-target">Azərbaycanca</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A2%D1%80%D1%8B%D0%B3%D0%B0%D0%BD%D0%B0%D0%BC%D0%B5%D1%82%D1%80%D1%8B%D1%87%D0%BD%D1%8B%D1%8F_%D1%84%D0%BE%D1%80%D0%BC%D1%83%D0%BB%D1%8B" title="Трыганаметрычныя формулы - Belarusça" lang="be" hreflang="be" data-title="Трыганаметрычныя формулы" data-language-autonym="Беларуская" data-language-local-name="Belarusça" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A2%D1%80%D0%B8%D0%B3%D0%BE%D0%BD%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%87%D0%BD%D0%B8_%D1%82%D1%8A%D0%B6%D0%B4%D0%B5%D1%81%D1%82%D0%B2%D0%B0" title="Тригонометрични тъждества - Bulgarca" lang="bg" hreflang="bg" data-title="Тригонометрични тъждества" data-language-autonym="Български" data-language-local-name="Bulgarca" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%A4%E0%A7%8D%E0%A6%B0%E0%A6%BF%E0%A6%95%E0%A7%8B%E0%A6%A3%E0%A6%AE%E0%A6%BF%E0%A6%A4%E0%A6%BF%E0%A6%95_%E0%A6%85%E0%A6%AD%E0%A7%87%E0%A6%A6%E0%A6%B8%E0%A6%AE%E0%A7%82%E0%A6%B9%E0%A7%87%E0%A6%B0_%E0%A6%A4%E0%A6%BE%E0%A6%B2%E0%A6%BF%E0%A6%95%E0%A6%BE" title="ত্রিকোণমিতিক অভেদসমূহের তালিকা - Bengalce" lang="bn" hreflang="bn" data-title="ত্রিকোণমিতিক অভেদসমূহের তালিকা" data-language-autonym="বাংলা" data-language-local-name="Bengalce" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Llista_d%27identitats_trigonom%C3%A8triques" title="Llista d'identitats trigonomètriques - Katalanca" lang="ca" hreflang="ca" data-title="Llista d'identitats trigonomètriques" data-language-autonym="Català" data-language-local-name="Katalanca" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%BE%DB%8E%DA%95%D8%B3%D8%AA%DB%8C_%DA%BE%D8%A7%D9%88%D8%A6%DB%95%D9%86%D8%AC%D8%A7%D9%85%DB%95_%D8%B3%DB%8E%DA%AF%DB%86%D8%B4%DB%95%DB%8C%DB%8C%DB%8C%DB%95%DA%A9%D8%A7%D9%86" title="پێڕستی ھاوئەنجامە سێگۆشەیییەکان - Orta Kürtçe" lang="ckb" hreflang="ckb" data-title="پێڕستی ھاوئەنجامە سێگۆشەیییەکان" data-language-autonym="کوردی" data-language-local-name="Orta Kürtçe" class="interlanguage-link-target">کوردی</a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A2%D1%80%D0%B8%D0%B3%D0%BE%D0%BD%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%80%D0%B8_%C3%A7%D0%B0%D0%B2%D0%B0%D1%85%D0%BB%C4%83%D1%85%D1%81%D0%B5%D0%BC" title="Тригонометрири çавахлăхсем - Çuvaşça" lang="cv" hreflang="cv" data-title="Тригонометрири çавахлăхсем" data-language-autonym="Чӑвашла" data-language-local-name="Çuvaşça" class="interlanguage-link-target">Чӑвашла</a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Rhestr_unfathiannau_trigonometrig" title="Rhestr unfathiannau trigonometrig - Galce" lang="cy" hreflang="cy" data-title="Rhestr unfathiannau trigonometrig" data-language-autonym="Cymraeg" data-language-local-name="Galce" class="interlanguage-link-target">Cymraeg</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Formelsammlung_Trigonometrie" title="Formelsammlung Trigonometrie - Almanca" lang="de" hreflang="de" data-title="Formelsammlung Trigonometrie" data-language-autonym="Deutsch" data-language-local-name="Almanca" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/List_of_trigonometric_identities" title="List of trigonometric identities - İngilizce" lang="en" hreflang="en" data-title="List of trigonometric identities" data-language-autonym="English" data-language-local-name="İngilizce" class="interlanguage-link-target">English</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Anexo:Identidades_trigonom%C3%A9tricas" title="Anexo:Identidades trigonométricas - İspanyolca" lang="es" hreflang="es" data-title="Anexo:Identidades trigonométricas" data-language-autonym="Español" data-language-local-name="İspanyolca" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%81%D9%87%D8%B1%D8%B3%D8%AA_%D8%A7%D8%AA%D8%AD%D8%A7%D8%AF%D9%87%D8%A7%DB%8C_%D9%85%D8%AB%D9%84%D8%AB%D8%A7%D8%AA%DB%8C" title="فهرست اتحادهای مثلثاتی - Farsça" lang="fa" hreflang="fa" data-title="فهرست اتحادهای مثلثاتی" data-language-autonym="فارسی" data-language-local-name="Farsça" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Formule_de_trigonom%C3%A9trie" title="Formule de trigonométrie - Fransızca" lang="fr" hreflang="fr" data-title="Formule de trigonométrie" data-language-autonym="Français" data-language-local-name="Fransızca" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%96%D7%94%D7%95%D7%99%D7%95%D7%AA_%D7%98%D7%A8%D7%99%D7%92%D7%95%D7%A0%D7%95%D7%9E%D7%98%D7%A8%D7%99%D7%95%D7%AA" title="זהויות טריגונומטריות - İbranice" lang="he" hreflang="he" data-title="זהויות טריגונומטריות" data-language-autonym="עברית" data-language-local-name="İbranice" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%95%E0%A5%8B%E0%A4%A3%E0%A4%AE%E0%A4%BF%E0%A4%A4%E0%A5%80%E0%A4%AF_%E0%A4%B8%E0%A4%B0%E0%A5%8D%E0%A4%B5%E0%A4%B8%E0%A4%AE%E0%A4%BF%E0%A4%95%E0%A4%BE%E0%A4%93%E0%A4%82_%E0%A4%95%E0%A5%80_%E0%A4%B8%E0%A5%82%E0%A4%9A%E0%A5%80" title="त्रिकोणमितीय सर्वसमिकाओं की सूची - Hintçe" lang="hi" hreflang="hi" data-title="त्रिकोणमितीय सर्वसमिकाओं की सूची" data-language-autonym="हिन्दी" data-language-local-name="Hintçe" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Popis_trigonometrijskih_jednakosti" title="Popis trigonometrijskih jednakosti - Hırvatça" lang="hr" hreflang="hr" data-title="Popis trigonometrijskih jednakosti" data-language-autonym="Hrvatski" data-language-local-name="Hırvatça" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Trigonometrikus_azonoss%C3%A1gok" title="Trigonometrikus azonosságok - Macarca" lang="hu" hreflang="hu" data-title="Trigonometrikus azonosságok" data-language-autonym="Magyar" data-language-local-name="Macarca" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B5%D5%BC%D5%A1%D5%B6%D5%AF%D5%B5%D5%B8%D6%82%D5%B6%D5%A1%D5%B9%D5%A1%D6%83%D5%A1%D5%AF%D5%A1%D5%B6_%D5%B6%D5%B8%D6%82%D5%B5%D5%B6%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6%D5%B6%D5%A5%D6%80" title="Եռանկյունաչափական նույնություններ - Ermenice" lang="hy" hreflang="hy" data-title="Եռանկյունաչափական նույնություններ" data-language-autonym="Հայերեն" data-language-local-name="Ermenice" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Daftar_identitas_trigonometri" title="Daftar identitas trigonometri - Endonezce" lang="id" hreflang="id" data-title="Daftar identitas trigonometri" data-language-autonym="Bahasa Indonesia" data-language-local-name="Endonezce" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Identit%C3%A0_trigonometrica" title="Identità trigonometrica - İtalyanca" lang="it" hreflang="it" data-title="Identità trigonometrica" data-language-autonym="İtaliano" data-language-local-name="İtalyanca" class="interlanguage-link-target">İtaliano</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E4%B8%89%E8%A7%92%E9%96%A2%E6%95%B0%E3%81%AE%E5%85%AC%E5%BC%8F%E3%81%AE%E4%B8%80%E8%A6%A7" title="三角関数の公式の一覧 - Japonca" lang="ja" hreflang="ja" data-title="三角関数の公式の一覧" data-language-autonym="日本語" data-language-local-name="Japonca" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A2%D1%80%D0%B8%D0%B3%D0%BE%D0%BD%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F%D0%BB%D1%8B%D2%9B_%D2%AF%D0%B9%D0%BB%D0%B5%D1%81%D1%96%D0%BC%D0%B4%D1%96%D0%BA%D1%82%D0%B5%D1%80" title="Тригонометриялық үйлесімдіктер - Kazakça" lang="kk" hreflang="kk" data-title="Тригонометриялық үйлесімдіктер" data-language-autonym="Қазақша" data-language-local-name="Kazakça" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%82%BC%EA%B0%81_%ED%95%A8%EC%88%98_%ED%95%AD%EB%93%B1%EC%8B%9D" title="삼각 함수 항등식 - Korece" lang="ko" hreflang="ko" data-title="삼각 함수 항등식" data-language-autonym="한국어" data-language-local-name="Korece" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A1%D0%BF%D0%B8%D1%81%D0%BE%D0%BA_%D0%BD%D0%B0_%D1%82%D1%80%D0%B8%D0%B3%D0%BE%D0%BD%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%81%D0%BA%D0%B8_%D0%B5%D0%B4%D0%BD%D0%B0%D0%BA%D0%B2%D0%BE%D1%81%D1%82%D0%B8" title="Список на тригонометриски еднаквости - Makedonca" lang="mk" hreflang="mk" data-title="Список на тригонометриски еднаквости" data-language-autonym="Македонски" data-language-local-name="Makedonca" class="interlanguage-link-target">Македонски</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Lijst_van_goniometrische_gelijkheden" title="Lijst van goniometrische gelijkheden - Felemenkçe" lang="nl" hreflang="nl" data-title="Lijst van goniometrische gelijkheden" data-language-autonym="Nederlands" data-language-local-name="Felemenkçe" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Liste_over_trigonometriske_identiteter" title="Liste over trigonometriske identiteter - Norveççe Bokmål" lang="nb" hreflang="nb" data-title="Liste over trigonometriske identiteter" data-language-autonym="Norsk bokmål" data-language-local-name="Norveççe Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/To%C5%BCsamo%C5%9Bci_trygonometryczne" title="Tożsamości trygonometryczne - Lehçe" lang="pl" hreflang="pl" data-title="Tożsamości trygonometryczne" data-language-autonym="Polski" data-language-local-name="Lehçe" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Identidade_trigonom%C3%A9trica" title="Identidade trigonométrica - Portekizce" lang="pt" hreflang="pt" data-title="Identidade trigonométrica" data-language-autonym="Português" data-language-local-name="Portekizce" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Identit%C4%83%C8%9Bi_trigonometrice" title="Identități trigonometrice - Rumence" lang="ro" hreflang="ro" data-title="Identități trigonometrice" data-language-autonym="Română" data-language-local-name="Rumence" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A2%D1%80%D0%B8%D0%B3%D0%BE%D0%BD%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B8%D0%B5_%D1%82%D0%BE%D0%B6%D0%B4%D0%B5%D1%81%D1%82%D0%B2%D0%B0" title="Тригонометрические тождества - Rusça" lang="ru" hreflang="ru" data-title="Тригонометрические тождества" data-language-autonym="Русский" data-language-local-name="Rusça" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%A1%D2%AF%D1%80%D2%AF%D0%BD_%D1%82%D1%80%D0%B8%D0%B3%D0%BE%D0%BD%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D0%B9_%D1%84%D0%BE%D1%80%D0%BC%D1%83%D0%BB%D0%B0%D0%BB%D0%B0%D1%80" title="Сүрүн тригонометрическай формулалар - Yakutça" lang="sah" hreflang="sah" data-title="Сүрүн тригонометрическай формулалар" data-language-autonym="Саха тыла" data-language-local-name="Yakutça" class="interlanguage-link-target">Саха тыла</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A2%D1%80%D0%B8%D0%B3%D0%BE%D0%BD%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%98%D1%81%D0%BA%D0%B8_%D0%B8%D0%B4%D0%B5%D0%BD%D1%82%D0%B8%D1%82%D0%B5%D1%82%D0%B8" title="Тригонометријски идентитети - Sırpça" lang="sr" hreflang="sr" data-title="Тригонометријски идентитети" data-language-autonym="Српски / srpski" data-language-local-name="Sırpça" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Lista_%C3%B6ver_trigonometriska_identiteter" title="Lista över trigonometriska identiteter - İsveççe" lang="sv" hreflang="sv" data-title="Lista över trigonometriska identiteter" data-language-autonym="Svenska" data-language-local-name="İsveççe" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AE%E0%AF%81%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AF%8B%E0%AE%A3%E0%AE%B5%E0%AE%BF%E0%AE%AF%E0%AE%B2%E0%AF%8D_%E0%AE%AE%E0%AF%81%E0%AE%B1%E0%AF%8D%E0%AE%B1%E0%AF%8A%E0%AE%B0%E0%AF%81%E0%AE%AE%E0%AF%88%E0%AE%95%E0%AE%B3%E0%AE%BF%E0%AE%A9%E0%AF%8D_%E0%AE%AA%E0%AE%9F%E0%AF%8D%E0%AE%9F%E0%AE%BF%E0%AE%AF%E0%AE%B2%E0%AF%8D" title="முக்கோணவியல் முற்றொருமைகளின் பட்டியல் - Tamilce" lang="ta" hreflang="ta" data-title="முக்கோணவியல் முற்றொருமைகளின் பட்டியல்" data-language-autonym="தமிழ்" data-language-local-name="Tamilce" 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class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">kenar çubuğuna taşı</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">gizle</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Vikipedi, özgür ansiklopedi</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="tr" dir="ltr">Trigonometride, trigonometrik özdeşlikler <a href="/wiki/Trigonometrik_fonksiyonlar" title="Trigonometrik fonksiyonlar">trigonometrik fonksiyonları</a> içeren ve eşitliğin her iki tarafının da tanımlandığı <a href="/wiki/De%C4%9Fi%C5%9Fken_(matematik)" title="Değişken (matematik)">değişkenlerin</a> her değeri için doğru olan <a href="/w/index.php?title=E%C5%9Fitlik_(matematik)&action=edit&redlink=1" class="new" title="Eşitlik (matematik) (sayfa mevcut değil)">eşitliklerdir</a>. Geometrik olarak, bunlar bir veya daha fazla <a href="/wiki/A%C3%A7%C4%B1" title="Açı">açının</a> belirli fonksiyonlarını içeren <a href="/wiki/%C3%96zde%C5%9Flik" title="Özdeşlik">özdeşliklerdir</a>. Bunlar <a href="/wiki/Trigonometri#Üçgen_özdeşlikleri" title="Trigonometri">üçgen özdeşliklerinden</a> farklıdır, bunlar potansiyel olarak açıları içeren ama aynı zamanda kenar uzunluklarını veya bir <a href="/wiki/%C3%9C%C3%A7gen" title="Üçgen">üçgenin</a> diğer uzunluklarını da içeren özdeşliklerdir. Bu özdeşlikler, trigonometrik fonksiyonları içeren ifadelerin basitleştirilmesi gerektiğinde kullanışlıdır. Önemli bir uygulama, trigonometrik olmayan fonksiyonların <a href="/wiki/%C4%B0ntegral" title="İntegral">integrasyonudur</a>: yaygın bir teknik, önce <a href="/wiki/Trigonometrik_yerine_koyma" title="Trigonometrik yerine koyma">bir trigonometrik fonksiyonla ikame kuralı</a> kullanmayı ve ardından ortaya çıkan integrali bir trigonometrik özdeşlikle basitleştirmeyi içerir. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Pisagor_özdeşlikleri">Pisagor özdeşlikleri</h2>[<a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&veaction=edit&section=1" title="Değiştirilen bölüm: Pisagor özdeşlikleri" class="mw-editsection-visualeditor">değiştir</a> | <a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&action=edit&section=1" title="Bölümün kaynak kodunu değiştir: Pisagor özdeşlikleri">kaynağı değiştir</a>]</div> <div role="note" class="hatnote navigation-not-searchable">Ana madde: <a href="/wiki/Pisagor_trigonometrik_%C3%B6zde%C5%9Fli%C4%9Fi" title="Pisagor trigonometrik özdeşliği">Pisagor trigonometrik özdeşliği</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Dosya:Trigonometric_functions_and_their_reciprocals_on_the_unit_circle.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Trigonometric_functions_and_their_reciprocals_on_the_unit_circle.svg/400px-Trigonometric_functions_and_their_reciprocals_on_the_unit_circle.svg.png" decoding="async" width="400" height="251" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Trigonometric_functions_and_their_reciprocals_on_the_unit_circle.svg/600px-Trigonometric_functions_and_their_reciprocals_on_the_unit_circle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Trigonometric_functions_and_their_reciprocals_on_the_unit_circle.svg/800px-Trigonometric_functions_and_their_reciprocals_on_the_unit_circle.svg.png 2x" data-file-width="750" data-file-height="470" /></a><figcaption><div class="center" style="width:auto; margin-left:auto; margin-right:auto;">Trigonometrik fonksiyonlar ve bunların birim çember üzerindeki karşılıkları. Tüm dik açılı üçgenler benzerdir, yani karşılık gelen kenarları arasındaki oranlar aynıdır. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Sin}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mi>i</mi> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Sin}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cf6f19fe7950ae68369991469282500c2890b04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.696ex; height:2.176ex;" alt="{\displaystyle Sin}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle cos}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mi>o</mi> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle cos}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2b7d4d815b5433116804c1a216f5f7d6a39ad26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.225ex; height:1.676ex;" alt="{\displaystyle cos}"> ve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle tan}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mi>a</mi> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle tan}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39f5df8d960a86c3df96729399ae560b2b7614d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.464ex; height:2.009ex;" alt="{\displaystyle tan}"> için birim uzunluktaki yarıçap, onları tanımlayan üçgenin hipotenüsünü oluşturur. Karşılıklı özdeşlikler, bu birim çizginin artık hipotenüs olmadığı üçgenlerde kenarların oranları olarak ortaya çıkar. <style data-mw-deduplicate="TemplateStyles:r33633371">@media screen{html.skin-theme-clientpref-night .mw-parser-output div:not(.notheme)>.tmp-color,html.skin-theme-clientpref-night .mw-parser-output p>.tmp-color,html.skin-theme-clientpref-night .mw-parser-output table:not(.notheme) .tmp-color{color:inherit!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output div:not(.notheme)>.tmp-color,html.skin-theme-clientpref-os .mw-parser-output p>.tmp-color,html.skin-theme-clientpref-os .mw-parser-output table:not(.notheme) .tmp-color{color:inherit!important}}</style>Mavi gölgeli üçgen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+\cot ^{2}\theta =\csc ^{2}\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <msup> <mi>cot</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <msup> <mi>csc</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+\cot ^{2}\theta =\csc ^{2}\theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/569181e0b7d46ae94b93ef730ec2f4374dca2ced" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.245ex; height:2.843ex;" alt="{\displaystyle 1+\cot ^{2}\theta =\csc ^{2}\theta }"> özdeşliğini, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r33633371">kırmızı gölgeli üçgen ise <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan ^{2}\theta +1=\sec ^{2}\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <mn>1</mn> <mo>=</mo> <msup> <mi>sec</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan ^{2}\theta +1=\sec ^{2}\theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66ad7564019222853545df76d2d51d9f29aed4b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.506ex; height:2.843ex;" alt="{\displaystyle \tan ^{2}\theta +1=\sec ^{2}\theta }"> özdeşliğini göstermektedir.</div></figcaption></figure> <a href="/w/index.php?title=Sin%C3%BCs_ve_kosin%C3%BCs&action=edit&redlink=1" class="new" title="Sinüs ve kosinüs (sayfa mevcut değil)">Sinüs ve kosinüs</a> arasındaki temel ilişki Pisagor özdeşliği ile verilir: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin ^{2}\theta +\cos ^{2}\theta =1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin ^{2}\theta +\cos ^{2}\theta =1,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e622f12d9b1a5305455a5dc57663ebfdfaaa3b6d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.778ex; height:3.009ex;" alt="{\displaystyle \sin ^{2}\theta +\cos ^{2}\theta =1,}"> burada <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin ^{2}\theta =(\sin \theta )^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin ^{2}\theta =(\sin \theta )^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3dd38c7ce98c3304ea7fb0febeede335bd1c4e56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.683ex; height:3.176ex;" alt="{\displaystyle \sin ^{2}\theta =(\sin \theta )^{2}}"> ve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos ^{2}\theta =(\cos \theta )^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos ^{2}\theta =(\cos \theta )^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f0394e39aa0f0b83e609df10de3d19874ee989d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.194ex; height:3.176ex;" alt="{\displaystyle \cos ^{2}\theta =(\cos \theta )^{2}}"> anlamına gelir. Bu <a href="/wiki/Pisagor_teoremi" title="Pisagor teoremi">Pisagor teoreminin</a> bir versiyonu olarak görülebilir ve <a href="/wiki/Birim_%C3%A7ember" title="Birim çember">birim çember</a> için <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+y^{2}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+y^{2}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec84b90236512e8d27ff1a8f7707b60b63327de1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.7ex; height:3.009ex;" alt="{\displaystyle x^{2}+y^{2}=1}"> denkleminden çıkar. Bu denklem sinüs ya da kosinüs için çözülebilir: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\sin \theta &=\pm {\sqrt {1-\cos ^{2}\theta }},\\\cos \theta &=\pm {\sqrt {1-\sin ^{2}\theta }}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </msqrt> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </msqrt> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sin \theta &=\pm {\sqrt {1-\cos ^{2}\theta }},\\\cos \theta &=\pm {\sqrt {1-\sin ^{2}\theta }}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98b3b78e432f783954c8eaace140ea68d6304190" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:22.863ex; height:7.176ex;" alt="{\displaystyle {\begin{aligned}\sin \theta &=\pm {\sqrt {1-\cos ^{2}\theta }},\\\cos \theta &=\pm {\sqrt {1-\sin ^{2}\theta }}.\end{aligned}}}"> Burada işaret <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }">'nın <a href="/w/index.php?title=%C3%87eyrek_(d%C3%BCzlem_geometri)&action=edit&redlink=1" class="new" title="Çeyrek (düzlem geometri) (sayfa mevcut değil)">çeyreğine</a> (kuadrantına) bağlıdır. Bu özdeşliği <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin ^{2}\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin ^{2}\theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02635acae84e17f9bae3ce887b425276ffeaa1a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.387ex; height:2.676ex;" alt="{\displaystyle \sin ^{2}\theta }">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos ^{2}\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos ^{2}\theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/311894fdc65ae89f6b2e40edac3d3281a0727680" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.643ex; height:2.676ex;" alt="{\displaystyle \cos ^{2}\theta }"> veya her ikisine böldüğünüzde aşağıdaki özdeşlikler elde edilir: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&1+\cot ^{2}\theta =\csc ^{2}\theta \\&1+\tan ^{2}\theta =\sec ^{2}\theta \\&\sec ^{2}\theta +\csc ^{2}\theta =\sec ^{2}\theta \csc ^{2}\theta \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mn>1</mn> <mo>+</mo> <msup> <mi>cot</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <msup> <mi>csc</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mn>1</mn> <mo>+</mo> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <msup> <mi>sec</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <msup> <mi>sec</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <msup> <mi>csc</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <msup> <mi>sec</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <msup> <mi>csc</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&1+\cot ^{2}\theta =\csc ^{2}\theta \\&1+\tan ^{2}\theta =\sec ^{2}\theta \\&\sec ^{2}\theta +\csc ^{2}\theta =\sec ^{2}\theta \csc ^{2}\theta \end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dde24bbb255f2555dfe651d6dd748458a72efc09" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:29.129ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}&1+\cot ^{2}\theta =\csc ^{2}\theta \\&1+\tan ^{2}\theta =\sec ^{2}\theta \\&\sec ^{2}\theta +\csc ^{2}\theta =\sec ^{2}\theta \csc ^{2}\theta \end{aligned}}}"> Bu özdeşlikleri kullanarak, herhangi bir trigonometrik fonksiyonu diğer herhangi bir fonksiyon cinsinden (artı veya eksi işaretin<a href="/w/index.php?title=E_kadar&action=edit&redlink=1" class="new" title="E kadar (sayfa mevcut değil)">e kadar</a>) ifade etmek mümkündür: <table class="wikitable" style="text-align:center"> <caption>Her bir trigonometrik fonksiyon diğer beş fonksiyonun her biri cinsinden<a href="#cite_note-AS4345-1">[1]</a> </caption> <tbody><tr> <th scope="row">cinsinden </th> <th scope="col"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efa733f6703578b0c3af870a3170b4ab0dd99c00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.333ex; height:2.176ex;" alt="{\displaystyle \sin \theta }"> </th> <th scope="col"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \csc \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>csc</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \csc \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a16f3e26e8000fd11f80dc2a4f080644ec0df49e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.459ex; height:2.176ex;" alt="{\displaystyle \csc \theta }"> </th> <th scope="col"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/611e5c70de1d1cf4ebc3b70d2b5467f45d17a483" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.589ex; height:2.176ex;" alt="{\displaystyle \cos \theta }"> </th> <th scope="col"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sec \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sec</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sec \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79faddf64a4cc2eaa8ab61445eabd5956a4c408c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.459ex; height:2.176ex;" alt="{\displaystyle \sec \theta }"> </th> <th scope="col"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08e7b66c250a5021a4deddf3b614c24db6c86ab3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.837ex; height:2.176ex;" alt="{\displaystyle \tan \theta }"> </th> <th scope="col"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cot \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cot</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cot \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72c44b29a1e7cb32527ab0a805691cb2211c8e5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.577ex; height:2.176ex;" alt="{\displaystyle \cot \theta }"> </th></tr> <tr> <th scope="row"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \theta =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \theta =}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85a567c5248097d657724e34cb8666208c3085ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.786ex; height:2.176ex;" alt="{\displaystyle \sin \theta =}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efa733f6703578b0c3af870a3170b4ab0dd99c00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.333ex; height:2.176ex;" alt="{\displaystyle \sin \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\csc \theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>csc</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\csc \theta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fc4edd8913183d4072ff4afc02cae272602d49d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:5.295ex; height:5.343ex;" alt="{\displaystyle {\frac {1}{\csc \theta }}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm {\sqrt {1-\cos ^{2}\theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm {\sqrt {1-\cos ^{2}\theta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5dcfc53f7ec6d4230f96e92b37e3b84726695a47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.778ex; height:3.509ex;" alt="{\displaystyle \pm {\sqrt {1-\cos ^{2}\theta }}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm {\frac {\sqrt {\sec ^{2}\theta -1}}{\sec \theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <msup> <mi>sec</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <mn>1</mn> </msqrt> <mrow> <mi>sec</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm {\frac {\sqrt {\sec ^{2}\theta -1}}{\sec \theta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4683f0e0ce2cc062d921c7aff7a29dbd267c2b7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.484ex; height:6.343ex;" alt="{\displaystyle \pm {\frac {\sqrt {\sec ^{2}\theta -1}}{\sec \theta }}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm {\frac {\tan \theta }{\sqrt {1+\tan ^{2}\theta }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm {\frac {\tan \theta }{\sqrt {1+\tan ^{2}\theta }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c11edb2977dd45ec68250c10977da1943bb73f55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:14.862ex; height:6.676ex;" alt="{\displaystyle \pm {\frac {\tan \theta }{\sqrt {1+\tan ^{2}\theta }}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm {\frac {1}{\sqrt {1+\cot ^{2}\theta }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>cot</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm {\frac {1}{\sqrt {1+\cot ^{2}\theta }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b4ceb9a5427a4d7c331791ef4e541b861f1be60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:14.602ex; height:6.509ex;" alt="{\displaystyle \pm {\frac {1}{\sqrt {1+\cot ^{2}\theta }}}}"> </td></tr> <tr> <th scope="row"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \csc \theta =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>csc</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \csc \theta =}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5563ddf95d132c38ef38814ab3da18f5f1031cea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.912ex; height:2.176ex;" alt="{\displaystyle \csc \theta =}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\sin \theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sin \theta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f1674827817d644a15badfaf28a1856e55196e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:5.169ex; height:5.343ex;" alt="{\displaystyle {\frac {1}{\sin \theta }}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \csc \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>csc</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \csc \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a16f3e26e8000fd11f80dc2a4f080644ec0df49e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.459ex; height:2.176ex;" alt="{\displaystyle \csc \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm {\frac {1}{\sqrt {1-\cos ^{2}\theta }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm {\frac {1}{\sqrt {1-\cos ^{2}\theta }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbee2311321fac7ec47e7c5b6a9ac7db43045614" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:14.614ex; height:6.509ex;" alt="{\displaystyle \pm {\frac {1}{\sqrt {1-\cos ^{2}\theta }}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm {\frac {\sec \theta }{\sqrt {\sec ^{2}\theta -1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sec</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <msqrt> <msup> <mi>sec</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <mn>1</mn> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm {\frac {\sec \theta }{\sqrt {\sec ^{2}\theta -1}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf592094b7280616edc681d61b860bb3b65238d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:14.484ex; height:6.676ex;" alt="{\displaystyle \pm {\frac {\sec \theta }{\sqrt {\sec ^{2}\theta -1}}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm {\frac {\sqrt {1+\tan ^{2}\theta }}{\tan \theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </msqrt> <mrow> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm {\frac {\sqrt {1+\tan ^{2}\theta }}{\tan \theta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d52e76aa506f01bff69089d44ffe9e9269a5aa4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.862ex; height:6.343ex;" alt="{\displaystyle \pm {\frac {\sqrt {1+\tan ^{2}\theta }}{\tan \theta }}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm {\sqrt {1+\cot ^{2}\theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>cot</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm {\sqrt {1+\cot ^{2}\theta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b3a1e8f33c1daf0f15851b590bc038d9a2e0f07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.766ex; height:3.509ex;" alt="{\displaystyle \pm {\sqrt {1+\cot ^{2}\theta }}}"> </td></tr> <tr> <th scope="row"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \theta =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \theta =}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6182f1f03d1c1c76dfb4c6943877c788511980d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.042ex; height:2.176ex;" alt="{\displaystyle \cos \theta =}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm {\sqrt {1-\sin ^{2}\theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm {\sqrt {1-\sin ^{2}\theta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/829520b8e3a7b74bcd37e28f4079482fa89af6cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.522ex; height:3.509ex;" alt="{\displaystyle \pm {\sqrt {1-\sin ^{2}\theta }}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm {\frac {\sqrt {\csc ^{2}\theta -1}}{\csc \theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <msup> <mi>csc</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <mn>1</mn> </msqrt> <mrow> <mi>csc</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm {\frac {\sqrt {\csc ^{2}\theta -1}}{\csc \theta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5126626818974a74ab53d01e97b325540c4678c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.484ex; height:6.343ex;" alt="{\displaystyle \pm {\frac {\sqrt {\csc ^{2}\theta -1}}{\csc \theta }}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/611e5c70de1d1cf4ebc3b70d2b5467f45d17a483" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.589ex; height:2.176ex;" alt="{\displaystyle \cos \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\sec \theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>sec</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sec \theta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68a698c7710dad53b8f5fb67d0ade8c702f675f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:5.295ex; height:5.343ex;" alt="{\displaystyle {\frac {1}{\sec \theta }}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm {\frac {1}{\sqrt {1+\tan ^{2}\theta }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm {\frac {1}{\sqrt {1+\tan ^{2}\theta }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab5d2a287d666cb2e9959257fc62a14f2266acd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:14.862ex; height:6.509ex;" alt="{\displaystyle \pm {\frac {1}{\sqrt {1+\tan ^{2}\theta }}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm {\frac {\cot \theta }{\sqrt {1+\cot ^{2}\theta }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>cot</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>cot</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm {\frac {\cot \theta }{\sqrt {1+\cot ^{2}\theta }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40cd59a06b5daa0d66c7953dcf62db95111720f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:14.602ex; height:6.676ex;" alt="{\displaystyle \pm {\frac {\cot \theta }{\sqrt {1+\cot ^{2}\theta }}}}"> </td></tr> <tr> <th scope="row"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sec \theta =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sec</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sec \theta =}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/542b3aa408c7cd3d04ab46f327ad1920db68335d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.912ex; height:2.176ex;" alt="{\displaystyle \sec \theta =}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm {\frac {1}{\sqrt {1-\sin ^{2}\theta }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm {\frac {1}{\sqrt {1-\sin ^{2}\theta }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46fc1f0fd9aa0649d38b93b5a5daf440570a8752" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:14.358ex; height:6.509ex;" alt="{\displaystyle \pm {\frac {1}{\sqrt {1-\sin ^{2}\theta }}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm {\frac {\csc \theta }{\sqrt {\csc ^{2}\theta -1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>csc</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <msqrt> <msup> <mi>csc</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <mn>1</mn> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm {\frac {\csc \theta }{\sqrt {\csc ^{2}\theta -1}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8076899848ec86145f7802b13fdecca936369b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:14.484ex; height:6.676ex;" alt="{\displaystyle \pm {\frac {\csc \theta }{\sqrt {\csc ^{2}\theta -1}}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\cos \theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\cos \theta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/500d8441b54264431f83aef66590aca0eee66383" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:5.425ex; height:5.343ex;" alt="{\displaystyle {\frac {1}{\cos \theta }}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sec \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sec</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sec \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79faddf64a4cc2eaa8ab61445eabd5956a4c408c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.459ex; height:2.176ex;" alt="{\displaystyle \sec \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm {\sqrt {1+\tan ^{2}\theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm {\sqrt {1+\tan ^{2}\theta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/660da09c139170d70e81f2d21a7c253327b57846" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.026ex; height:3.509ex;" alt="{\displaystyle \pm {\sqrt {1+\tan ^{2}\theta }}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm {\frac {\sqrt {1+\cot ^{2}\theta }}{\cot \theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>cot</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </msqrt> <mrow> <mi>cot</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm {\frac {\sqrt {1+\cot ^{2}\theta }}{\cot \theta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4634e0652f9ee8ba31889049de3b0da8a88e91a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.602ex; height:6.343ex;" alt="{\displaystyle \pm {\frac {\sqrt {1+\cot ^{2}\theta }}{\cot \theta }}}"> </td></tr> <tr> <th scope="row"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan \theta =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan \theta =}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/811edbd828a6027e90de36901ee7a868a3315b4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.29ex; height:2.176ex;" alt="{\displaystyle \tan \theta =}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm {\frac {\sin \theta }{\sqrt {1-\sin ^{2}\theta }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm {\frac {\sin \theta }{\sqrt {1-\sin ^{2}\theta }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc4b1be661d8edbcaacbccb7216b04963552f865" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:14.358ex; height:6.676ex;" alt="{\displaystyle \pm {\frac {\sin \theta }{\sqrt {1-\sin ^{2}\theta }}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm {\frac {1}{\sqrt {\csc ^{2}\theta -1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <msup> <mi>csc</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <mn>1</mn> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm {\frac {1}{\sqrt {\csc ^{2}\theta -1}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f928be3f12c1506151552863ffb548f0f1ccccf1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:14.484ex; height:6.509ex;" alt="{\displaystyle \pm {\frac {1}{\sqrt {\csc ^{2}\theta -1}}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm {\frac {\sqrt {1-\cos ^{2}\theta }}{\cos \theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </msqrt> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm {\frac {\sqrt {1-\cos ^{2}\theta }}{\cos \theta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3cfa6dd0fc1886aa6cd4031b5a3aed1dfb438b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.614ex; height:6.343ex;" alt="{\displaystyle \pm {\frac {\sqrt {1-\cos ^{2}\theta }}{\cos \theta }}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm {\sqrt {\sec ^{2}\theta -1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>sec</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm {\sqrt {\sec ^{2}\theta -1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b64a4b82e6c4ca046de6b6f58635a29b9892bc6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.648ex; height:3.509ex;" alt="{\displaystyle \pm {\sqrt {\sec ^{2}\theta -1}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08e7b66c250a5021a4deddf3b614c24db6c86ab3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.837ex; height:2.176ex;" alt="{\displaystyle \tan \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\cot \theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>cot</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\cot \theta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f572723cb9aeffbb394cbcd84e394344885a3ec1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:5.413ex; height:5.343ex;" alt="{\displaystyle {\frac {1}{\cot \theta }}}"> </td></tr> <tr> <th scope="row"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cot \theta =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cot</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cot \theta =}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/791414887be8a6e9b575d5915b886fdb98ffe0e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.03ex; height:2.176ex;" alt="{\displaystyle \cot \theta =}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm {\frac {\sqrt {1-\sin ^{2}\theta }}{\sin \theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </msqrt> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm {\frac {\sqrt {1-\sin ^{2}\theta }}{\sin \theta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0fb1330fd304c171a72a7ec113554d1fda24553" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.358ex; height:6.343ex;" alt="{\displaystyle \pm {\frac {\sqrt {1-\sin ^{2}\theta }}{\sin \theta }}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm {\sqrt {\csc ^{2}\theta -1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>csc</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm {\sqrt {\csc ^{2}\theta -1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96c820548070ce563d549e71aaeccf2be688d119" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.648ex; height:3.509ex;" alt="{\displaystyle \pm {\sqrt {\csc ^{2}\theta -1}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm {\frac {\cos \theta }{\sqrt {1-\cos ^{2}\theta }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm {\frac {\cos \theta }{\sqrt {1-\cos ^{2}\theta }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d8655a838e910f5e43a4d291d37b1923fd7fdb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:14.614ex; height:6.676ex;" alt="{\displaystyle \pm {\frac {\cos \theta }{\sqrt {1-\cos ^{2}\theta }}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm {\frac {1}{\sqrt {\sec ^{2}\theta -1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <msup> <mi>sec</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <mn>1</mn> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm {\frac {1}{\sqrt {\sec ^{2}\theta -1}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2749aeee23d40f14bbae55ace5e30ce1d019f6f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:14.484ex; height:6.509ex;" alt="{\displaystyle \pm {\frac {1}{\sqrt {\sec ^{2}\theta -1}}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\tan \theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\tan \theta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d5da158d9c04c7b673ee00ac23693aff8f8167e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:5.673ex; height:5.343ex;" alt="{\displaystyle {\frac {1}{\tan \theta }}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cot \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cot</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cot \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72c44b29a1e7cb32527ab0a805691cb2211c8e5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.577ex; height:2.176ex;" alt="{\displaystyle \cot \theta }"> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Yansımalar,_kaymalar_ve_periyodiklik">Yansımalar, kaymalar ve periyodiklik</h2>[<a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&veaction=edit&section=2" title="Değiştirilen bölüm: Yansımalar, kaymalar ve periyodiklik" class="mw-editsection-visualeditor">değiştir</a> | <a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&action=edit&section=2" title="Bölümün kaynak kodunu değiştir: Yansımalar, kaymalar ve periyodiklik">kaynağı değiştir</a>]</div> Birim çember incelenerek trigonometrik fonksiyonların aşağıdaki özellikleri belirlenebilir. <div class="mw-heading mw-heading3"><h3 id="Yansımalar">Yansımalar</h3>[<a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&veaction=edit&section=3" title="Değiştirilen bölüm: Yansımalar" class="mw-editsection-visualeditor">değiştir</a> | <a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&action=edit&section=3" title="Bölümün kaynak kodunu değiştir: Yansımalar">kaynağı değiştir</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Dosya:Unit_Circle_-_symmetry.svg" class="mw-file-description"><img alt="(a,b) koordinatlarında çizilmiş teta süpürme açısına sahip birim daire. Açı bir çeyrek pi (45 derece) artışlarla yansıtıldıkça, koordinatlar dönüştürülür. Bir çeyrek pi (45 derece veya 90 - teta) dönüşümü için koordinatlar (b,a)'ya dönüştürülür. Yansıma açısının bir çeyrek pi (toplam 90 derece veya 180 - teta) daha artırılması koordinatları (-a,b)'ye dönüştürür. Yansıma açısının bir çeyrek pi daha artırılması (toplam 135 derece veya 270 - teta) koordinatları (-b,-a)'ya dönüştürür. Son bir çeyrek pi'lik artış (toplam 180 derece veya 360 - teta) koordinatları (a,-b)'ye dönüştürür." src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Unit_Circle_-_symmetry.svg/330px-Unit_Circle_-_symmetry.svg.png" decoding="async" width="330" height="264" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Unit_Circle_-_symmetry.svg/495px-Unit_Circle_-_symmetry.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Unit_Circle_-_symmetry.svg/660px-Unit_Circle_-_symmetry.svg.png 2x" data-file-width="500" data-file-height="400" /></a><figcaption><div class="center" style="width:auto; margin-left:auto; margin-right:auto;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"> yansıma açısını <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\pi }{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\pi }{4}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f89d7c88c1c93dce69a46052a8e276e231063de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.168ex; height:4.676ex;" alt="{\displaystyle {\frac {\pi }{4}}}"> artışlarla kaydırırken (a, b) koordinatlarının dönüşümü.</div></figcaption></figure> Bir Öklid vektörünün yönü bir <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a45ee89f965ec5498b347cb54f45243ef88b218" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.737ex; height:2.509ex;" alt="{\displaystyle \theta ,}"> açısı ile temsil edildiğinde, bu açı serbest vektör (orijinden başlayan) ve pozitif <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}">-birim vektörü tarafından belirlenen açıdır. Aynı kavram, Öklid uzayında doğrulara da uygulanabilir; burada açı, verilen doğruya orijinden ve pozitif <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}">-ekseninden geçen bir paralel doğru tarafından belirlenen açıdır. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"> doğrultulu bir doğru (vektör) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b2cc8f6d373595f06dcd33f127dadf2b9d5727f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.134ex; height:2.009ex;" alt="{\displaystyle \alpha ,}"> doğrultulu bir doğru etrafında yansıtılırsa, bu yansıtılan doğrunun (vektörün) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta ^{\prime }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta ^{\prime }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ae4d8a422e6b538dce9bc6a2e8b28ee50f09458" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.775ex; height:2.509ex;" alt="{\displaystyle \theta ^{\prime }}"> doğrultu açısı <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta ^{\prime }=2\alpha -\theta ."> <semantics> <mrow> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>=</mo> <mn>2</mn> <mi>α</mi> <mo>−</mo> <mi>θ</mi> <mo>.</mo> </mrow> <annotation encoding="application/x-tex">\theta ^{\prime }=2\alpha -\theta .</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5270e0a931fbceb2e3f7e21e2b5bdd14f220c87f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.101ex; height:2.676ex;" alt="\theta ^{\prime }=2\alpha -\theta ."> değerine sahiptir. Bu açıların trigonometrik fonksiyonlarının değerleri <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta ,\;\theta ^{\prime }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>,</mo> <mspace width="thickmathspace" /> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta ,\;\theta ^{\prime }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af6e547f7a114a7fe9a0832f5ec359e359871320" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.545ex; height:2.843ex;" alt="{\displaystyle \theta ,\;\theta ^{\prime }}"> belirli açılar <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"> için basit özdeşlikleri karşılar: ya eşittirler, ya zıt işaretlidirler ya da tamamlayıcı trigonometrik fonksiyon kullanırlar. Bunlar indirgeme formülleri olarak da bilinir.<a href="#cite_note-2">[2]</a> <table class="wikitable"> <tbody><tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30cc00f65bbc630448311dd2dc82e7ce5e90985a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle \alpha =0}">'da yansıtılan <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"><a href="#cite_note-3">[3]</a> <a href="/wiki/Tek_ve_%C3%A7ift_fonksiyonlar" title="Tek ve çift fonksiyonlar">tek/çift</a> özdeşlikler </th> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ={\frac {\pi }{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ={\frac {\pi }{4}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8f7c5303bf3113490b1bfb240ce9dfd4165e0ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.754ex; height:4.676ex;" alt="{\displaystyle \alpha ={\frac {\pi }{4}}}">'te yansıtılan <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"> </th> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ={\frac {\pi }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ={\frac {\pi }{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cd12c3063518da1d406706bc669e61a15fe9c8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.754ex; height:4.676ex;" alt="{\displaystyle \alpha ={\frac {\pi }{2}}}">'de yansıtılan <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"> </th> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ={\frac {3\pi }{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ={\frac {3\pi }{4}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67eaa80b8a90a28e56090631f158ce15c291ddb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.917ex; height:5.176ex;" alt="{\displaystyle \alpha ={\frac {3\pi }{4}}}">'te yansıtılan <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"> </th> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56d4c38f1082fe503a8a69193cc3cd91466c656e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.918ex; height:1.676ex;" alt="{\displaystyle \alpha =\pi }">'de yansıtılan <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30cc00f65bbc630448311dd2dc82e7ce5e90985a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle \alpha =0}"> ile karşılaştıtma </th></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(-\theta )=-\sin \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(-\theta )=-\sin \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05280328a017fe88f5b1be9a4a3ad56c4b38e6cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.19ex; height:2.843ex;" alt="{\displaystyle \sin(-\theta )=-\sin \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \left({\tfrac {\pi }{2}}-\theta \right)=\cos \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>−</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \left({\tfrac {\pi }{2}}-\theta \right)=\cos \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c350885c85db49915f7d641a511c5ec7116c98e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:18.381ex; height:3.343ex;" alt="{\displaystyle \sin \left({\tfrac {\pi }{2}}-\theta \right)=\cos \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(\pi -\theta )=+\sin \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mo>−</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(\pi -\theta )=+\sin \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67285863e55f108c97427e206251949f79517757" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.555ex; height:2.843ex;" alt="{\displaystyle \sin(\pi -\theta )=+\sin \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \left({\tfrac {3\pi }{2}}-\theta \right)=-\cos \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>−</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \left({\tfrac {3\pi }{2}}-\theta \right)=-\cos \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/461be3dac49d99f476beb31ed3f9129bb8899321" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.044ex; height:4.843ex;" alt="{\displaystyle \sin \left({\tfrac {3\pi }{2}}-\theta \right)=-\cos \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(2\pi -\theta )=-\sin(\theta )=\sin(-\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mo>−</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(2\pi -\theta )=-\sin(\theta )=\sin(-\theta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5b4761d94912842d63029c535a7e0c7c7b680b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.801ex; height:2.843ex;" alt="{\displaystyle \sin(2\pi -\theta )=-\sin(\theta )=\sin(-\theta )}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(-\theta )=+\cos \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(-\theta )=+\cos \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f1299f605dca3cf1cc3a4f0efda1d6dc642e4d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.701ex; height:2.843ex;" alt="{\displaystyle \cos(-\theta )=+\cos \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \left({\tfrac {\pi }{2}}-\theta \right)=\sin \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>−</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \left({\tfrac {\pi }{2}}-\theta \right)=\sin \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a8f1ce404b120f990b9cab9d619365cf7dd170" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:18.381ex; height:3.343ex;" alt="{\displaystyle \cos \left({\tfrac {\pi }{2}}-\theta \right)=\sin \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(\pi -\theta )=-\cos \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mo>−</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(\pi -\theta )=-\cos \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcbcef6426672f28e34965669ec23f14197461e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.066ex; height:2.843ex;" alt="{\displaystyle \cos(\pi -\theta )=-\cos \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \left({\tfrac {3\pi }{2}}-\theta \right)=-\sin \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>−</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \left({\tfrac {3\pi }{2}}-\theta \right)=-\sin \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57d8e7b605172f0342b739c3f12770bb502a68cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.044ex; height:4.843ex;" alt="{\displaystyle \cos \left({\tfrac {3\pi }{2}}-\theta \right)=-\sin \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(2\pi -\theta )=+\cos(\theta )=\cos(-\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mo>−</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(2\pi -\theta )=+\cos(\theta )=\cos(-\theta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fecfdbf5707a1856b9831da92ff0455a5fc08600" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.568ex; height:2.843ex;" alt="{\displaystyle \cos(2\pi -\theta )=+\cos(\theta )=\cos(-\theta )}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan(-\theta )=-\tan \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan(-\theta )=-\tan \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44a9a642606a49ea00893d42db64d6b8638bdd3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.198ex; height:2.843ex;" alt="{\displaystyle \tan(-\theta )=-\tan \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan \left({\tfrac {\pi }{2}}-\theta \right)=\cot \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>−</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>cot</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan \left({\tfrac {\pi }{2}}-\theta \right)=\cot \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a27b89483b1a940ea0f4737dc6c7daab0e5bda05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:18.874ex; height:3.343ex;" alt="{\displaystyle \tan \left({\tfrac {\pi }{2}}-\theta \right)=\cot \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan(\pi -\theta )=-\tan \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mo>−</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan(\pi -\theta )=-\tan \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48ae7d252e3bd661b09b148c7ac097c2365a0bd5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.563ex; height:2.843ex;" alt="{\displaystyle \tan(\pi -\theta )=-\tan \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan \left({\tfrac {3\pi }{2}}-\theta \right)=+\cot \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>−</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>+</mo> <mi>cot</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan \left({\tfrac {3\pi }{2}}-\theta \right)=+\cot \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e360cc27fb05b28952dd08b1a17380e9b9645aea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.537ex; height:4.843ex;" alt="{\displaystyle \tan \left({\tfrac {3\pi }{2}}-\theta \right)=+\cot \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan(2\pi -\theta )=-\tan(\theta )=\tan(-\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mo>−</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan(2\pi -\theta )=-\tan(\theta )=\tan(-\theta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0d3112a652b7dd5454f92be3170a26441228ebe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.313ex; height:2.843ex;" alt="{\displaystyle \tan(2\pi -\theta )=-\tan(\theta )=\tan(-\theta )}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \csc(-\theta )=-\csc \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>csc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>csc</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \csc(-\theta )=-\csc \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50ec2b53a22eb3ad72b0c826652729caf49cee4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.441ex; height:2.843ex;" alt="{\displaystyle \csc(-\theta )=-\csc \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \csc \left({\tfrac {\pi }{2}}-\theta \right)=\sec \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>csc</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>−</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>sec</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \csc \left({\tfrac {\pi }{2}}-\theta \right)=\sec \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d35728de94a1931049181a16f07aced4bdcf696" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:18.377ex; height:3.343ex;" alt="{\displaystyle \csc \left({\tfrac {\pi }{2}}-\theta \right)=\sec \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \csc(\pi -\theta )=+\csc \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>csc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mo>−</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>+</mo> <mi>csc</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \csc(\pi -\theta )=+\csc \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d16113feedf9a19cde7ed704e1d049863ba9d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.805ex; height:2.843ex;" alt="{\displaystyle \csc(\pi -\theta )=+\csc \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \csc \left({\tfrac {3\pi }{2}}-\theta \right)=-\sec \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>csc</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>−</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mi>sec</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \csc \left({\tfrac {3\pi }{2}}-\theta \right)=-\sec \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/314bfa1b68758cf684ee270b2cf0be981154c3f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.04ex; height:4.843ex;" alt="{\displaystyle \csc \left({\tfrac {3\pi }{2}}-\theta \right)=-\sec \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \csc(2\pi -\theta )=-\csc(\theta )=\csc(-\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>csc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mo>−</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>csc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>csc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \csc(2\pi -\theta )=-\csc(\theta )=\csc(-\theta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b814b63281847bea022de29ce03944d6119730c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.178ex; height:2.843ex;" alt="{\displaystyle \csc(2\pi -\theta )=-\csc(\theta )=\csc(-\theta )}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sec(-\theta )=+\sec \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>+</mo> <mi>sec</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sec(-\theta )=+\sec \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fc2558c7e685a4fc5d0484b756e54a3334a8b2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.441ex; height:2.843ex;" alt="{\displaystyle \sec(-\theta )=+\sec \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sec \left({\tfrac {\pi }{2}}-\theta \right)=\csc \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sec</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>−</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>csc</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sec \left({\tfrac {\pi }{2}}-\theta \right)=\csc \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e096d56fa3b4452a4d9c03ed63b1cf347afd89c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:18.377ex; height:3.343ex;" alt="{\displaystyle \sec \left({\tfrac {\pi }{2}}-\theta \right)=\csc \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sec(\pi -\theta )=-\sec \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mo>−</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>sec</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sec(\pi -\theta )=-\sec \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/038d193be0e76650a4460389ca89b91e56a1bfba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.805ex; height:2.843ex;" alt="{\displaystyle \sec(\pi -\theta )=-\sec \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sec \left({\tfrac {3\pi }{2}}-\theta \right)=-\csc \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sec</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>−</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mi>csc</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sec \left({\tfrac {3\pi }{2}}-\theta \right)=-\csc \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c83359c72b3f0e928bf38940e23ce3d038d085f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.04ex; height:4.843ex;" alt="{\displaystyle \sec \left({\tfrac {3\pi }{2}}-\theta \right)=-\csc \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sec(2\pi -\theta )=+\sec(\theta )=\sec(-\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mo>−</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>+</mo> <mi>sec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>sec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sec(2\pi -\theta )=+\sec(\theta )=\sec(-\theta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a6fadca4e93d2c897027c73ad0570f18e1c0ef7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.178ex; height:2.843ex;" alt="{\displaystyle \sec(2\pi -\theta )=+\sec(\theta )=\sec(-\theta )}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cot(-\theta )=-\cot \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>cot</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cot(-\theta )=-\cot \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/171e81eb8b2b831b1a046ace7bb57af824cad982" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.678ex; height:2.843ex;" alt="{\displaystyle \cot(-\theta )=-\cot \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cot \left({\tfrac {\pi }{2}}-\theta \right)=\tan \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cot</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>−</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cot \left({\tfrac {\pi }{2}}-\theta \right)=\tan \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca71092e41028003cb12eba431338a7daacb1cc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:18.874ex; height:3.343ex;" alt="{\displaystyle \cot \left({\tfrac {\pi }{2}}-\theta \right)=\tan \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cot(\pi -\theta )=-\cot \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mo>−</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>cot</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cot(\pi -\theta )=-\cot \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/665e014254e39bbf65d5fec6abd62eb30801ff1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.042ex; height:2.843ex;" alt="{\displaystyle \cot(\pi -\theta )=-\cot \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cot \left({\tfrac {3\pi }{2}}-\theta \right)=+\tan \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cot</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>−</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>+</mo> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cot \left({\tfrac {3\pi }{2}}-\theta \right)=+\tan \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37024bdd21e9ebe8d0fd2374c072b8455b13b2d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.537ex; height:4.843ex;" alt="{\displaystyle \cot \left({\tfrac {3\pi }{2}}-\theta \right)=+\tan \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cot(2\pi -\theta )=-\cot(\theta )=\cot(-\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mo>−</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>cot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>cot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cot(2\pi -\theta )=-\cot(\theta )=\cot(-\theta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc4a5e19c74a4ecd60901a08fa737ee7bfd5ddef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.533ex; height:2.843ex;" alt="{\displaystyle \cot(2\pi -\theta )=-\cot(\theta )=\cot(-\theta )}"> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Kaymalar_ve_periyodiklik">Kaymalar ve periyodiklik</h3>[<a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&veaction=edit&section=4" title="Değiştirilen bölüm: Kaymalar ve periyodiklik" class="mw-editsection-visualeditor">değiştir</a> | <a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&action=edit&section=4" title="Bölümün kaynak kodunu değiştir: Kaymalar ve periyodiklik">kaynağı değiştir</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Dosya:Unit_Circle_-_shifts.svg" class="mw-file-description"><img alt="(a, b) koordinatlarında çizilen teta süpürme açısına sahip birim daire. Tarama açısı bir buçuk pi (90 derece) artırıldığında, koordinatlar (-b, a)'ya dönüşür. Bir başka yarım pi'lik artış (toplam 180 derece) koordinatları (-a,-b)'ye dönüştürür. Son bir yarım pi (toplam 270 derece) artış koordinatları (b, a)'ya dönüştürür." src="//upload.wikimedia.org/wikipedia/commons/thumb/8/85/Unit_Circle_-_shifts.svg/260px-Unit_Circle_-_shifts.svg.png" decoding="async" width="260" height="208" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/85/Unit_Circle_-_shifts.svg/390px-Unit_Circle_-_shifts.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/85/Unit_Circle_-_shifts.svg/520px-Unit_Circle_-_shifts.svg.png 2x" data-file-width="500" data-file-height="400" /></a><figcaption><div class="center" style="width:auto; margin-left:auto; margin-right:auto;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"> açısını <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\pi }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\pi }{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98f98bef5d4981ff6e2aa827d4699e347fb30db2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.168ex; height:4.676ex;" alt="{\displaystyle {\frac {\pi }{2}}}"> artışlarla kaydırırken (a, b) koordinatlarının dönüşümü.</div></figcaption></figure> <table class="wikitable"> <tbody><tr> <th>Bir çeyrek periyot kaydırma </th> <th>Bir yarım periyot kaydırma </th> <th>Tam periyotlarla kaydırma<a href="#cite_note-4">[4]</a> </th> <th>Periyot </th></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(\theta \pm {\tfrac {\pi }{2}})=\pm \cos \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo>±</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(\theta \pm {\tfrac {\pi }{2}})=\pm \cos \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/638e6444443ae83b660b457f3125fc03b24eb822" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:20.256ex; height:3.176ex;" alt="{\displaystyle \sin(\theta \pm {\tfrac {\pi }{2}})=\pm \cos \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(\theta +\pi )=-\sin \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>+</mo> <mi>π</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(\theta +\pi )=-\sin \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df6d4a2fee4ca45ffb3d979d7f7011b8934a407a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.555ex; height:2.843ex;" alt="{\displaystyle \sin(\theta +\pi )=-\sin \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(\theta +k\cdot 2\pi )=+\sin \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>+</mo> <mi>k</mi> <mo>⋅</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(\theta +k\cdot 2\pi )=+\sin \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a338e5dc21d9dd5ffd6a9d779d9f5666a5df4956" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.607ex; height:2.843ex;" alt="{\displaystyle \sin(\theta +k\cdot 2\pi )=+\sin \theta }"> </td> <td style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73efd1f6493490b058097060a572606d2c550a06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(\theta \pm {\tfrac {\pi }{2}})=\mp \sin \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo>∓</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(\theta \pm {\tfrac {\pi }{2}})=\mp \sin \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9275715bc7a6580c7a59c1664eeba69222de77d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:20.256ex; height:3.176ex;" alt="{\displaystyle \cos(\theta \pm {\tfrac {\pi }{2}})=\mp \sin \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(\theta +\pi )=-\cos \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>+</mo> <mi>π</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(\theta +\pi )=-\cos \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d099997d54de814027c19e5f98d90b42af52679" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.066ex; height:2.843ex;" alt="{\displaystyle \cos(\theta +\pi )=-\cos \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(\theta +k\cdot 2\pi )=+\cos \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>+</mo> <mi>k</mi> <mo>⋅</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(\theta +k\cdot 2\pi )=+\cos \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7fc35b521e8de74a741c23aa5d1e2c7aa6d8ae4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.118ex; height:2.843ex;" alt="{\displaystyle \cos(\theta +k\cdot 2\pi )=+\cos \theta }"> </td> <td style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73efd1f6493490b058097060a572606d2c550a06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \csc(\theta \pm {\tfrac {\pi }{2}})=\pm \sec \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>csc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo>±</mo> <mi>sec</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \csc(\theta \pm {\tfrac {\pi }{2}})=\pm \sec \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fde678959e71a3d07b0898b1db5494e5ef39bbce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:20.251ex; height:3.176ex;" alt="{\displaystyle \csc(\theta \pm {\tfrac {\pi }{2}})=\pm \sec \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \csc(\theta +\pi )=-\csc \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>csc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>+</mo> <mi>π</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>csc</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \csc(\theta +\pi )=-\csc \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545f264a0681da0142f2d089760c8362c241483d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.805ex; height:2.843ex;" alt="{\displaystyle \csc(\theta +\pi )=-\csc \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \csc(\theta +k\cdot 2\pi )=+\csc \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>csc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>+</mo> <mi>k</mi> <mo>⋅</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>+</mo> <mi>csc</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \csc(\theta +k\cdot 2\pi )=+\csc \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67c0ea64b64e16bbe1434fc77e5d6622a5d6be29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.858ex; height:2.843ex;" alt="{\displaystyle \csc(\theta +k\cdot 2\pi )=+\csc \theta }"> </td> <td style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73efd1f6493490b058097060a572606d2c550a06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sec(\theta \pm {\tfrac {\pi }{2}})=\mp \csc \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo>∓</mo> <mi>csc</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sec(\theta \pm {\tfrac {\pi }{2}})=\mp \csc \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07634225a3f9bbec3b9752c4edcbc924cb33d002" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:20.251ex; height:3.176ex;" alt="{\displaystyle \sec(\theta \pm {\tfrac {\pi }{2}})=\mp \csc \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sec(\theta +\pi )=-\sec \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>+</mo> <mi>π</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>sec</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sec(\theta +\pi )=-\sec \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dddb9ba4858c2461f58fca2dc5512e2d0f7218f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.805ex; height:2.843ex;" alt="{\displaystyle \sec(\theta +\pi )=-\sec \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sec(\theta +k\cdot 2\pi )=+\sec \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>+</mo> <mi>k</mi> <mo>⋅</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>+</mo> <mi>sec</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sec(\theta +k\cdot 2\pi )=+\sec \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48fbb490baeb75ac5fa3c6b3683ee902ace9aca9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.858ex; height:2.843ex;" alt="{\displaystyle \sec(\theta +k\cdot 2\pi )=+\sec \theta }"> </td> <td style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73efd1f6493490b058097060a572606d2c550a06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan(\theta \pm {\tfrac {\pi }{4}})={\tfrac {\tan \theta \pm 1}{1\mp \tan \theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> <mo>±</mo> <mn>1</mn> </mrow> <mrow> <mn>1</mn> <mo>∓</mo> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan(\theta \pm {\tfrac {\pi }{4}})={\tfrac {\tan \theta \pm 1}{1\mp \tan \theta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12dfafdd1684fbfd4dbccbc8e00d112a3af8ad0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:20.447ex; height:4.009ex;" alt="{\displaystyle \tan(\theta \pm {\tfrac {\pi }{4}})={\tfrac {\tan \theta \pm 1}{1\mp \tan \theta }}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan(\theta +{\tfrac {\pi }{2}})=-\cot \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>cot</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan(\theta +{\tfrac {\pi }{2}})=-\cot \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07aad352f11a2679802651eb8eed93b6153375c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:20.749ex; height:3.176ex;" alt="{\displaystyle \tan(\theta +{\tfrac {\pi }{2}})=-\cot \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan(\theta +k\cdot \pi )=+\tan \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>+</mo> <mi>k</mi> <mo>⋅</mo> <mi>π</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>+</mo> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan(\theta +k\cdot \pi )=+\tan \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66ea384b8f94dbb4169b0fce7f8f0b1f6e31d554" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.453ex; height:2.843ex;" alt="{\displaystyle \tan(\theta +k\cdot \pi )=+\tan \theta }"> </td> <td style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cot(\theta \pm {\tfrac {\pi }{4}})={\tfrac {\cot \theta \mp 1}{1\pm \cot \theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>cot</mi> <mo>⁡</mo> <mi>θ</mi> <mo>∓</mo> <mn>1</mn> </mrow> <mrow> <mn>1</mn> <mo>±</mo> <mi>cot</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cot(\theta \pm {\tfrac {\pi }{4}})={\tfrac {\cot \theta \mp 1}{1\pm \cot \theta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4b5f10f25939b1b01b28fa29f2e8ee6cc58111a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:20.003ex; height:4.009ex;" alt="{\displaystyle \cot(\theta \pm {\tfrac {\pi }{4}})={\tfrac {\cot \theta \mp 1}{1\pm \cot \theta }}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cot(\theta +{\tfrac {\pi }{2}})=-\tan \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cot(\theta +{\tfrac {\pi }{2}})=-\tan \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be3467e31a9042cf09b4f28387fdb7e701d62739" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:20.749ex; height:3.176ex;" alt="{\displaystyle \cot(\theta +{\tfrac {\pi }{2}})=-\tan \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cot(\theta +k\cdot \pi )=+\cot \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>+</mo> <mi>k</mi> <mo>⋅</mo> <mi>π</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>+</mo> <mi>cot</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cot(\theta +k\cdot \pi )=+\cot \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d490de2cf348e90f5ab328a35b393d4e6a2f314" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.933ex; height:2.843ex;" alt="{\displaystyle \cot(\theta +k\cdot \pi )=+\cot \theta }"> </td> <td style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="İşaretler">İşaretler</h3>[<a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&veaction=edit&section=5" title="Değiştirilen bölüm: İşaretler" class="mw-editsection-visualeditor">değiştir</a> | <a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&action=edit&section=5" title="Bölümün kaynak kodunu değiştir: İşaretler">kaynağı değiştir</a>]</div> Trigonometrik fonksiyonların işareti açının çeyreğine (kuadrantına) bağlıdır. Eğer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {-\pi }<\theta \leq \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>π</mi> </mrow> <mo><</mo> <mi>θ</mi> <mo>≤</mo> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {-\pi }<\theta \leq \pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37ba39ae213e688812a0fa68de996d2d4f4e90c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.76ex; height:2.343ex;" alt="{\displaystyle {-\pi }<\theta \leq \pi }"> ve sgn <a href="/wiki/%C4%B0%C5%9Faret_fonksiyonu" title="İşaret fonksiyonu">işaret fonksiyonu</a> ise, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\operatorname {sgn}(\sin \theta )=\operatorname {sgn}(\csc \theta )&={\begin{cases}+1&\ 0<\theta <\pi \ {\text{ ise}}\\-1&\ {-\pi }<\theta <0\ {\text{ ise}}\\0&\ \theta \in \{0,\pi \}\ {\text{ ise}}\end{cases}}\\[5mu]\operatorname {sgn}(\cos \theta )=\operatorname {sgn}(\sec \theta )&={\begin{cases}+1&\ {-{\tfrac {1}{2}}\pi }<\theta <{\tfrac {1}{2}}\pi \ {\text{ ise}}\\-1&\ {-\pi }<\theta <-{\tfrac {1}{2}}\pi \ \ {\text{veya}}\ \ {\tfrac {1}{2}}\pi <\theta <\pi \ {\text{ ise}}\\0&\ \theta \in {\bigl \{}{-{\tfrac {1}{2}}\pi },{\tfrac {1}{2}}\pi {\bigr \}}\ {\text{ ise}}\end{cases}}\\[5mu]\operatorname {sgn}(\tan \theta )=\operatorname {sgn}(\cot \theta )&={\begin{cases}+1&\ {-\pi }<\theta <-{\tfrac {1}{2}}\pi \ \ {\text{veya}}\ \ 0<\theta <{\tfrac {1}{2}}\pi \ {\text{ ise}}\\-1&\ {-{\tfrac {1}{2}}\pi }<\theta <0\ \ {\text{veya}}\ \ {\tfrac {1}{2}}\pi <\theta <\pi \ {\text{ ise}}\\0&\ \theta \in {\bigl \{}{-{\tfrac {1}{2}}\pi },0,{\tfrac {1}{2}}\pi ,\pi {\bigr \}}\ {\text{ ise}}\end{cases}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.578em 0.578em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>csc</mi> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mo>+</mo> <mn>1</mn> </mtd> <mtd> <mtext> </mtext> <mn>0</mn> <mo><</mo> <mi>θ</mi> <mo><</mo> <mi>π</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext> ise</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>π</mi> </mrow> <mo><</mo> <mi>θ</mi> <mo><</mo> <mn>0</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext> ise</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mtext> </mtext> <mi>θ</mi> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mi>π</mi> <mo fence="false" stretchy="false">}</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext> ise</mtext> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>sec</mi> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mo>+</mo> <mn>1</mn> </mtd> <mtd> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>π</mi> </mrow> <mo><</mo> <mi>θ</mi> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>π</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext> ise</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>π</mi> </mrow> <mo><</mo> <mi>θ</mi> <mo><</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>π</mi> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>veya</mtext> </mrow> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>π</mi> <mo><</mo> <mi>θ</mi> <mo><</mo> <mi>π</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext> ise</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mtext> </mtext> <mi>θ</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">{</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>π</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">}</mo> </mrow> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext> ise</mtext> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>cot</mi> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mo>+</mo> <mn>1</mn> </mtd> <mtd> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>π</mi> </mrow> <mo><</mo> <mi>θ</mi> <mo><</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>π</mi> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>veya</mtext> </mrow> <mtext> </mtext> <mtext> </mtext> <mn>0</mn> <mo><</mo> <mi>θ</mi> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>π</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext> ise</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>π</mi> </mrow> <mo><</mo> <mi>θ</mi> <mo><</mo> <mn>0</mn> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>veya</mtext> </mrow> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>π</mi> <mo><</mo> <mi>θ</mi> <mo><</mo> <mi>π</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext> ise</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mtext> </mtext> <mi>θ</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">{</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>π</mi> </mrow> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>π</mi> <mo>,</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">}</mo> </mrow> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext> ise</mtext> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {sgn}(\sin \theta )=\operatorname {sgn}(\csc \theta )&={\begin{cases}+1&\ 0<\theta <\pi \ {\text{ ise}}\\-1&\ {-\pi }<\theta <0\ {\text{ ise}}\\0&\ \theta \in \{0,\pi \}\ {\text{ ise}}\end{cases}}\\[5mu]\operatorname {sgn}(\cos \theta )=\operatorname {sgn}(\sec \theta )&={\begin{cases}+1&\ {-{\tfrac {1}{2}}\pi }<\theta <{\tfrac {1}{2}}\pi \ {\text{ ise}}\\-1&\ {-\pi }<\theta <-{\tfrac {1}{2}}\pi \ \ {\text{veya}}\ \ {\tfrac {1}{2}}\pi <\theta <\pi \ {\text{ ise}}\\0&\ \theta \in {\bigl \{}{-{\tfrac {1}{2}}\pi },{\tfrac {1}{2}}\pi {\bigr \}}\ {\text{ ise}}\end{cases}}\\[5mu]\operatorname {sgn}(\tan \theta )=\operatorname {sgn}(\cot \theta )&={\begin{cases}+1&\ {-\pi }<\theta <-{\tfrac {1}{2}}\pi \ \ {\text{veya}}\ \ 0<\theta <{\tfrac {1}{2}}\pi \ {\text{ ise}}\\-1&\ {-{\tfrac {1}{2}}\pi }<\theta <0\ \ {\text{veya}}\ \ {\tfrac {1}{2}}\pi <\theta <\pi \ {\text{ ise}}\\0&\ \theta \in {\bigl \{}{-{\tfrac {1}{2}}\pi },0,{\tfrac {1}{2}}\pi ,\pi {\bigr \}}\ {\text{ ise}}\end{cases}}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/100bf1f74f64b859146c7a9d5f1ad6ec558191df" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -15.171ex; width:72.979ex; height:31.509ex;" alt="{\displaystyle {\begin{aligned}\operatorname {sgn}(\sin \theta )=\operatorname {sgn}(\csc \theta )&={\begin{cases}+1&\ 0<\theta <\pi \ {\text{ ise}}\\-1&\ {-\pi }<\theta <0\ {\text{ ise}}\\0&\ \theta \in \{0,\pi \}\ {\text{ ise}}\end{cases}}\\[5mu]\operatorname {sgn}(\cos \theta )=\operatorname {sgn}(\sec \theta )&={\begin{cases}+1&\ {-{\tfrac {1}{2}}\pi }<\theta <{\tfrac {1}{2}}\pi \ {\text{ ise}}\\-1&\ {-\pi }<\theta <-{\tfrac {1}{2}}\pi \ \ {\text{veya}}\ \ {\tfrac {1}{2}}\pi <\theta <\pi \ {\text{ ise}}\\0&\ \theta \in {\bigl \{}{-{\tfrac {1}{2}}\pi },{\tfrac {1}{2}}\pi {\bigr \}}\ {\text{ ise}}\end{cases}}\\[5mu]\operatorname {sgn}(\tan \theta )=\operatorname {sgn}(\cot \theta )&={\begin{cases}+1&\ {-\pi }<\theta <-{\tfrac {1}{2}}\pi \ \ {\text{veya}}\ \ 0<\theta <{\tfrac {1}{2}}\pi \ {\text{ ise}}\\-1&\ {-{\tfrac {1}{2}}\pi }<\theta <0\ \ {\text{veya}}\ \ {\tfrac {1}{2}}\pi <\theta <\pi \ {\text{ ise}}\\0&\ \theta \in {\bigl \{}{-{\tfrac {1}{2}}\pi },0,{\tfrac {1}{2}}\pi ,\pi {\bigr \}}\ {\text{ ise}}\end{cases}}\end{aligned}}}"> Trigonometrik fonksiyonlar ortak periyot <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/461d52fa6fb5a16ef8a24e871488584db5398489" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.141ex; height:2.509ex;" alt="{\displaystyle 2\pi ,}"> ile periyodiktir, bu nedenle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-\pi ,\pi ],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mi>π</mi> <mo>,</mo> <mi>π</mi> <mo stretchy="false">]</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-\pi ,\pi ],}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c73d26a883cb3bac0800037cd3dd9dbd719f9334" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.704ex; height:2.843ex;" alt="{\displaystyle (-\pi ,\pi ],}"> aralığının dışındaki θ değerleri için tekrar eden değerler alırlar (yukarıdaki <a href="#Kaymalar_ve_periyodiklik">§ Kaymalar ve periyodiklik</a> bölümüne bakın). <div class="mw-heading mw-heading2"><h2 id="Açı_toplam_ve_fark_özdeşlikleri">Açı toplam ve fark özdeşlikleri</h2>[<a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&veaction=edit&section=6" title="Değiştirilen bölüm: Açı toplam ve fark özdeşlikleri" class="mw-editsection-visualeditor">değiştir</a> | <a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&action=edit&section=6" title="Bölümün kaynak kodunu değiştir: Açı toplam ve fark özdeşlikleri">kaynağı değiştir</a>]</div> <div role="note" class="hatnote navigation-not-searchable">Ayrıca bakınız: <a href="/wiki/Trigonometrik_%C3%B6zde%C5%9Fliklerin_ispatlar%C4%B1#Açı_toplam_özdeşlikleri" title="Trigonometrik özdeşliklerin ispatları">Trigonometrik özdeşliklerin ispatları § Açı toplam özdeşlikleri</a> ve <a href="/w/index.php?title=K%C3%BC%C3%A7%C3%BCk_a%C3%A7%C4%B1_yakla%C5%9F%C4%B1m%C4%B1&action=edit&redlink=1" class="new" title="Küçük açı yaklaşımı (sayfa mevcut değil)">Küçük açı yaklaşımı § Açı toplamı ve farkı</a></div> <div class="center"><div class="thumb tnone"> <div class="thumbinner" style="width:608px;"> <table style="background: transparent; border: 0; padding: 0; margin: 0;" cellspacing="0"> <tbody><tr> <td style="padding: 0; margin: 0;" class="thumbimage"><a href="/wiki/Dosya:AngleAdditionDiagramSine.svg" class="mw-file-description" title="Dar açıların sinüs ve kosinüsleri için açı toplam formüllerinin gösterimi. Vurgulanan parça birim uzunluktadır."><img alt="Dar açıların sinüs ve kosinüsleri için açı toplam formüllerinin gösterimi. Vurgulanan parça birim uzunluktadır." src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/AngleAdditionDiagramSine.svg/300px-AngleAdditionDiagramSine.svg.png" decoding="async" width="300" height="361" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/AngleAdditionDiagramSine.svg/450px-AngleAdditionDiagramSine.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f5/AngleAdditionDiagramSine.svg/600px-AngleAdditionDiagramSine.svg.png 2x" data-file-width="335" data-file-height="403" /></a> </td> <td style="padding: 0; margin: 0; border: 0; width: 2px"> </td> <td style="padding: 0; margin: 0;" class="thumbimage"><a href="/wiki/Dosya:Diagram_showing_the_angle_difference_trigonometry_identities_for_sin(a-b)_and_cos(a-b).svg" class="mw-file-description" title="Dar açıların sinüs ve kosinüsleri için açı toplam formüllerinin gösterimi. Vurgulanan parça birim uzunluktadır."><img alt="Dar açıların sinüs ve kosinüsleri için açı toplam formüllerinin gösterimi. Vurgulanan parça birim uzunluktadır." src="//upload.wikimedia.org/wikipedia/commons/thumb/4/43/Diagram_showing_the_angle_difference_trigonometry_identities_for_sin%28a-b%29_and_cos%28a-b%29.svg/300px-Diagram_showing_the_angle_difference_trigonometry_identities_for_sin%28a-b%29_and_cos%28a-b%29.svg.png" decoding="async" width="300" height="225" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/43/Diagram_showing_the_angle_difference_trigonometry_identities_for_sin%28a-b%29_and_cos%28a-b%29.svg/450px-Diagram_showing_the_angle_difference_trigonometry_identities_for_sin%28a-b%29_and_cos%28a-b%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/43/Diagram_showing_the_angle_difference_trigonometry_identities_for_sin%28a-b%29_and_cos%28a-b%29.svg/600px-Diagram_showing_the_angle_difference_trigonometry_identities_for_sin%28a-b%29_and_cos%28a-b%29.svg.png 2x" data-file-width="800" data-file-height="600" /></a> </td></tr> <tr> <td style="padding: 0; margin: 0; border: 0;"><div class="thumbcaption"><div class="center" style="width:auto; margin-left:auto; margin-right:auto;">Dar açıların sinüs ve kosinüsleri için açı toplam formüllerinin gösterimi. Vurgulanan parça birim uzunluktadır.</div></div> </td> <td style="padding: 0; margin: 0; border: 0;"> </td> <td style="padding: 0; margin: 0; border: 0;"><div class="thumbcaption"><div class="center" style="width:auto; margin-left:auto; margin-right:auto;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(\alpha -\beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>−</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(\alpha -\beta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/680bcef1cb476d51f78323083ab28fa9892056ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.325ex; height:2.843ex;" alt="{\displaystyle \sin(\alpha -\beta )}"> ve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(\alpha -\beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>−</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(\alpha -\beta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2497b20fdac2493cd2d8ce6474a77911c7f5c3eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.58ex; height:2.843ex;" alt="{\displaystyle \cos(\alpha -\beta )}"> için açı farkı özdeşliklerini gösteren şekil.</div></div> </td></tr></tbody></table> </div> </div></div> Bunlar aynı zamanda açı toplam ve fark teoremleri (veya formülleri) olarak da bilinir. <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\sin(\alpha +\beta )&=\sin \alpha \cos \beta +\cos \alpha \sin \beta \\\sin(\alpha -\beta )&=\sin \alpha \cos \beta -\cos \alpha \sin \beta \\\cos(\alpha +\beta )&=\cos \alpha \cos \beta -\sin \alpha \sin \beta \\\cos(\alpha -\beta )&=\cos \alpha \cos \beta +\sin \alpha \sin \beta \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mi>α</mi> <mi>cos</mi> <mo>⁡</mo> <mi>β</mi> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> <mi>sin</mi> <mo>⁡</mo> <mi>β</mi> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>−</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mi>α</mi> <mi>cos</mi> <mo>⁡</mo> <mi>β</mi> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> <mi>sin</mi> <mo>⁡</mo> <mi>β</mi> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> <mi>cos</mi> <mo>⁡</mo> <mi>β</mi> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>α</mi> <mi>sin</mi> <mo>⁡</mo> <mi>β</mi> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>−</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> <mi>cos</mi> <mo>⁡</mo> <mi>β</mi> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mi>α</mi> <mi>sin</mi> <mo>⁡</mo> <mi>β</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sin(\alpha +\beta )&=\sin \alpha \cos \beta +\cos \alpha \sin \beta \\\sin(\alpha -\beta )&=\sin \alpha \cos \beta -\cos \alpha \sin \beta \\\cos(\alpha +\beta )&=\cos \alpha \cos \beta -\sin \alpha \sin \beta \\\cos(\alpha -\beta )&=\cos \alpha \cos \beta +\sin \alpha \sin \beta \end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c94f036855a9492696b7ca3c2715ee9162124a50" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:37.166ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}\sin(\alpha +\beta )&=\sin \alpha \cos \beta +\cos \alpha \sin \beta \\\sin(\alpha -\beta )&=\sin \alpha \cos \beta -\cos \alpha \sin \beta \\\cos(\alpha +\beta )&=\cos \alpha \cos \beta -\sin \alpha \sin \beta \\\cos(\alpha -\beta )&=\cos \alpha \cos \beta +\sin \alpha \sin \beta \end{aligned}}}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(\alpha -\beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>−</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(\alpha -\beta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/680bcef1cb476d51f78323083ab28fa9892056ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.325ex; height:2.843ex;" alt="{\displaystyle \sin(\alpha -\beta )}"> ve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(\alpha -\beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>−</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(\alpha -\beta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2497b20fdac2493cd2d8ce6474a77911c7f5c3eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.58ex; height:2.843ex;" alt="{\displaystyle \cos(\alpha -\beta )}"> için açı farkı özdeşlikleri, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\beta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/789e8289e8f49d257d033611fbe19356b7747bdb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.14ex; height:2.509ex;" alt="{\displaystyle -\beta }"> yerine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"> koyarak ve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(-\beta )=-\sin(\beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(-\beta )=-\sin(\beta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a377a921c4be0e2e54868c90d779140b100a2a70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.096ex; height:2.843ex;" alt="{\displaystyle \sin(-\beta )=-\sin(\beta )}"> ile <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(-\beta )=\cos(\beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(-\beta )=\cos(\beta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/937b38c4b99ba19e898a2be1ae4dba351253f5cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.411ex; height:2.843ex;" alt="{\displaystyle \cos(-\beta )=\cos(\beta )}"> gerçeklerini kullanarak açı toplamı versiyonlarından türetilebilir. Açı toplamı özdeşlikleri için şeklin biraz değiştirilmiş bir versiyonu kullanılarak da elde edilebilirler, her ikisi de burada gösterilmektedir. Bu özdeşlikler, diğer trigonometrik fonksiyonlar için toplam ve fark özdeşliklerini de içeren aşağıdaki tablonun ilk iki satırında özetlenmiştir. <table class="wikitable" style="background-color:var(--background-color-base)"> <tbody><tr> <th>Sinüs </th> <td colspan="3" style="border-style: solid none solid solid; text-align: right;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(\alpha \pm \beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>±</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(\alpha \pm \beta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90e29100512e94f9d9104441fd51f7d799768d74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.325ex; height:2.843ex;" alt="{\displaystyle \sin(\alpha \pm \beta )}"> </td> <td style="border-style: solid none solid none; text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}"> </td> <td style="border-style: solid solid solid none; text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \alpha \cos \beta \pm \cos \alpha \sin \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>α</mi> <mi>cos</mi> <mo>⁡</mo> <mi>β</mi> <mo>±</mo> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> <mi>sin</mi> <mo>⁡</mo> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \alpha \cos \beta \pm \cos \alpha \sin \beta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c00bb77223cba5c2fed9ebde3c3c98008ec82470" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.736ex; height:2.509ex;" alt="{\displaystyle \sin \alpha \cos \beta \pm \cos \alpha \sin \beta }"><a href="#cite_note-5">[5]</a><a href="#cite_note-mathworld_addition-6">[6]</a> </td></tr> <tr> <th>Kosinüs </th> <td colspan="3" style="border-style: solid none solid solid; text-align: right;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(\alpha \pm \beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>±</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(\alpha \pm \beta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/654fc5fcd0236b2a746704421db12804d62acfd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.58ex; height:2.843ex;" alt="{\displaystyle \cos(\alpha \pm \beta )}"> </td> <td style="border-style: solid none solid none; text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}"> </td> <td style="border-style: solid solid solid none; text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \alpha \cos \beta \mp \sin \alpha \sin \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> <mi>cos</mi> <mo>⁡</mo> <mi>β</mi> <mo>∓</mo> <mi>sin</mi> <mo>⁡</mo> <mi>α</mi> <mi>sin</mi> <mo>⁡</mo> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \alpha \cos \beta \mp \sin \alpha \sin \beta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5237788d07304c9dc28712271ac893bbb2f4fd38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.736ex; height:2.509ex;" alt="{\displaystyle \cos \alpha \cos \beta \mp \sin \alpha \sin \beta }"><a href="#cite_note-mathworld_addition-6">[6]</a><a href="#cite_note-7">[7]</a> </td></tr> <tr> <th>Tanjant </th> <td colspan="3" style="border-style: solid none solid solid; text-align: right;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan(\alpha \pm \beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>±</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan(\alpha \pm \beta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35dd36e98ec5ff9a03a18cf7891d74d5f016c4f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.829ex; height:2.843ex;" alt="{\displaystyle \tan(\alpha \pm \beta )}"> </td> <td style="border-style: solid none solid none; text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}"> </td> <td style="border-style: solid solid solid none; text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\tan \alpha \pm \tan \beta }{1\mp \tan \alpha \tan \beta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>tan</mi> <mo>⁡</mo> <mi>α</mi> <mo>±</mo> <mi>tan</mi> <mo>⁡</mo> <mi>β</mi> </mrow> <mrow> <mn>1</mn> <mo>∓</mo> <mi>tan</mi> <mo>⁡</mo> <mi>α</mi> <mi>tan</mi> <mo>⁡</mo> <mi>β</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\tan \alpha \pm \tan \beta }{1\mp \tan \alpha \tan \beta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f4401dd220bb696e0e1ec141b8f2d586dc328d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.539ex; height:5.843ex;" alt="{\displaystyle {\frac {\tan \alpha \pm \tan \beta }{1\mp \tan \alpha \tan \beta }}}"><a href="#cite_note-mathworld_addition-6">[6]</a><a href="#cite_note-8">[8]</a> </td></tr> <tr> <th>Kosekant </th> <td colspan="3" style="border-style: solid none solid solid; text-align: right;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \csc(\alpha \pm \beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>csc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>±</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \csc(\alpha \pm \beta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0ec258f3d100d5b8f431eb3e4f340b717848160" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.45ex; height:2.843ex;" alt="{\displaystyle \csc(\alpha \pm \beta )}"> </td> <td style="border-style: solid none solid none; text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}"> </td> <td style="border-style: solid solid solid none; text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\sec \alpha \sec \beta \csc \alpha \csc \beta }{\sec \alpha \csc \beta \pm \csc \alpha \sec \beta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sec</mi> <mo>⁡</mo> <mi>α</mi> <mi>sec</mi> <mo>⁡</mo> <mi>β</mi> <mi>csc</mi> <mo>⁡</mo> <mi>α</mi> <mi>csc</mi> <mo>⁡</mo> <mi>β</mi> </mrow> <mrow> <mi>sec</mi> <mo>⁡</mo> <mi>α</mi> <mi>csc</mi> <mo>⁡</mo> <mi>β</mi> <mo>±</mo> <mi>csc</mi> <mo>⁡</mo> <mi>α</mi> <mi>sec</mi> <mo>⁡</mo> <mi>β</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\sec \alpha \sec \beta \csc \alpha \csc \beta }{\sec \alpha \csc \beta \pm \csc \alpha \sec \beta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efbbda7843e79d7add84d5c110b0ebc71d4604a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.563ex; height:5.843ex;" alt="{\displaystyle {\frac {\sec \alpha \sec \beta \csc \alpha \csc \beta }{\sec \alpha \csc \beta \pm \csc \alpha \sec \beta }}}"><a href="#cite_note-:0-9">[9]</a> </td></tr> <tr> <th>Sekant </th> <td colspan="3" style="border-style: solid none solid solid; text-align: right;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sec(\alpha \pm \beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>±</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sec(\alpha \pm \beta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f738bf697d1afe0dccab7e1d4673d520b520d27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.45ex; height:2.843ex;" alt="{\displaystyle \sec(\alpha \pm \beta )}"> </td> <td style="border-style: solid none solid none; text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}"> </td> <td style="border-style: solid solid solid none; text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\sec \alpha \sec \beta \csc \alpha \csc \beta }{\csc \alpha \csc \beta \mp \sec \alpha \sec \beta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sec</mi> <mo>⁡</mo> <mi>α</mi> <mi>sec</mi> <mo>⁡</mo> <mi>β</mi> <mi>csc</mi> <mo>⁡</mo> <mi>α</mi> <mi>csc</mi> <mo>⁡</mo> <mi>β</mi> </mrow> <mrow> <mi>csc</mi> <mo>⁡</mo> <mi>α</mi> <mi>csc</mi> <mo>⁡</mo> <mi>β</mi> <mo>∓</mo> <mi>sec</mi> <mo>⁡</mo> <mi>α</mi> <mi>sec</mi> <mo>⁡</mo> <mi>β</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\sec \alpha \sec \beta \csc \alpha \csc \beta }{\csc \alpha \csc \beta \mp \sec \alpha \sec \beta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5be7c6091c087488c157e2b113d9469afb5bc1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.563ex; height:5.843ex;" alt="{\displaystyle {\frac {\sec \alpha \sec \beta \csc \alpha \csc \beta }{\csc \alpha \csc \beta \mp \sec \alpha \sec \beta }}}"><a href="#cite_note-:0-9">[9]</a> </td></tr> <tr> <th>Kotanjant </th> <td colspan="3" style="border-style: solid none solid solid; text-align: right;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cot(\alpha \pm \beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>±</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cot(\alpha \pm \beta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a8fb2f5e0a8d0d75f94f58edfa7f0fd03d319e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.569ex; height:2.843ex;" alt="{\displaystyle \cot(\alpha \pm \beta )}"> </td> <td style="border-style: solid none solid none; text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}"> </td> <td style="border-style: solid solid solid none; text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\cot \alpha \cot \beta \mp 1}{\cot \beta \pm \cot \alpha }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>cot</mi> <mo>⁡</mo> <mi>α</mi> <mi>cot</mi> <mo>⁡</mo> <mi>β</mi> <mo>∓</mo> <mn>1</mn> </mrow> <mrow> <mi>cot</mi> <mo>⁡</mo> <mi>β</mi> <mo>±</mo> <mi>cot</mi> <mo>⁡</mo> <mi>α</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\cot \alpha \cot \beta \mp 1}{\cot \beta \pm \cot \alpha }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f4e81de903c71dde1c12f5f0e905e31a35faf47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.019ex; height:5.843ex;" alt="{\displaystyle {\frac {\cot \alpha \cot \beta \mp 1}{\cot \beta \pm \cot \alpha }}}"><a href="#cite_note-mathworld_addition-6">[6]</a><a href="#cite_note-10">[10]</a> </td></tr> <tr> <th>Arksinüs </th> <td colspan="3" style="border-style: solid none solid solid; text-align: right;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arcsin x\pm \arcsin y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arcsin</mi> <mo>⁡</mo> <mi>x</mi> <mo>±</mo> <mi>arcsin</mi> <mo>⁡</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arcsin x\pm \arcsin y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad629350c60a33db3bbc919b77473f40ddb65aab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.024ex; height:2.509ex;" alt="{\displaystyle \arcsin x\pm \arcsin y}"> </td> <td style="border-style: solid none solid none; text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}"> </td> <td style="border-style: solid solid solid none; text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arcsin \left(x{\sqrt {1-y^{2}}}\pm y{\sqrt {1-x^{2}{\vphantom {y}}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arcsin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>±</mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mpadded width="0"> <mphantom> <mi>y</mi> </mphantom> </mpadded> </mrow> </mrow> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arcsin \left(x{\sqrt {1-y^{2}}}\pm y{\sqrt {1-x^{2}{\vphantom {y}}}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ca946616db8ca8d1bb83ef3af04fdea80e7519c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.961ex; height:6.176ex;" alt="{\displaystyle \arcsin \left(x{\sqrt {1-y^{2}}}\pm y{\sqrt {1-x^{2}{\vphantom {y}}}}\right)}"><a href="#cite_note-11">[11]</a> </td></tr> <tr> <th>Arkkosinüs </th> <td colspan="3" style="border-style: solid none solid solid; text-align: right;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arccos x\pm \arccos y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccos</mi> <mo>⁡</mo> <mi>x</mi> <mo>±</mo> <mi>arccos</mi> <mo>⁡</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arccos x\pm \arccos y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eaeb31fef67f3cd13e6ea8b11708e30280c55853" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.535ex; height:2.509ex;" alt="{\displaystyle \arccos x\pm \arccos y}"> </td> <td style="border-style: solid none solid none; text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}"> </td> <td style="border-style: solid solid solid none; text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arccos \left(xy\mp {\sqrt {\left(1-x^{2}\right)\left(1-y^{2}\right)}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mi>y</mi> <mo>∓</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arccos \left(xy\mp {\sqrt {\left(1-x^{2}\right)\left(1-y^{2}\right)}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc5eed6756b40e66106f012846799f514f110e88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:33.893ex; height:4.843ex;" alt="{\displaystyle \arccos \left(xy\mp {\sqrt {\left(1-x^{2}\right)\left(1-y^{2}\right)}}\right)}"><a href="#cite_note-12">[12]</a> </td></tr> <tr> <th>Arktanjant </th> <td colspan="3" style="border-style: solid none solid solid; text-align: right;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arctan x\pm \arctan y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arctan</mi> <mo>⁡</mo> <mi>x</mi> <mo>±</mo> <mi>arctan</mi> <mo>⁡</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arctan x\pm \arctan y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7d13e7e755796d6de2212403e7ae28b143f2a04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.032ex; height:2.509ex;" alt="{\displaystyle \arctan x\pm \arctan y}"> </td> <td style="border-style: solid none solid none; text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}"> </td> <td style="border-style: solid solid solid none; text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arctan \left({\frac {x\pm y}{1\mp xy}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>±</mo> <mi>y</mi> </mrow> <mrow> <mn>1</mn> <mo>∓</mo> <mi>x</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arctan \left({\frac {x\pm y}{1\mp xy}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6866c9ee865ec1029449b0432c0f9c6d6b80e9cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.211ex; height:6.176ex;" alt="{\displaystyle \arctan \left({\frac {x\pm y}{1\mp xy}}\right)}"><a href="#cite_note-13">[13]</a> </td></tr> <tr> <th>Arkkotanjant </th> <td colspan="3" style="border-style: solid none solid solid; text-align: right;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {arccot} x\pm \operatorname {arccot} y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccot</mi> <mo>⁡</mo> <mi>x</mi> <mo>±</mo> <mi>arccot</mi> <mo>⁡</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {arccot} x\pm \operatorname {arccot} y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d0f5a0fa74449dc9361c74835622481cbbecbbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.512ex; height:2.509ex;" alt="{\displaystyle \operatorname {arccot} x\pm \operatorname {arccot} y}"> </td> <td style="border-style: solid none solid none; text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}"> </td> <td style="border-style: solid solid solid none; text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {arccot} \left({\frac {xy\mp 1}{y\pm x}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccot</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mi>y</mi> <mo>∓</mo> <mn>1</mn> </mrow> <mrow> <mi>y</mi> <mo>±</mo> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {arccot} \left({\frac {xy\mp 1}{y\pm x}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/261603c1b8cb5bb596b3870713a8612f36a90a9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.951ex; height:6.176ex;" alt="{\displaystyle \operatorname {arccot} \left({\frac {xy\mp 1}{y\pm x}}\right)}"> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Sonsuz_sayıda_açının_toplamlarının_sinüs_ve_kosinüsleri">Sonsuz sayıda açının toplamlarının sinüs ve kosinüsleri</h3>[<a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&veaction=edit&section=7" title="Değiştirilen bölüm: Sonsuz sayıda açının toplamlarının sinüs ve kosinüsleri" class="mw-editsection-visualeditor">değiştir</a> | <a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&action=edit&section=7" title="Bölümün kaynak kodunu değiştir: Sonsuz sayıda açının toplamlarının sinüs ve kosinüsleri">kaynağı değiştir</a>]</div> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sum _{i=1}^{\infty }\theta _{i}"> <semantics> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <annotation encoding="application/x-tex">\sum _{i=1}^{\infty }\theta _{i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/711f443e264cd22d5d7e7bab5f80c6b46ef9abb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:5.632ex; height:6.843ex;" alt="\sum _{i=1}^{\infty }\theta _{i}"> serisi, <a href="/wiki/Mutlak_yak%C4%B1nsama" title="Mutlak yakınsama">mutlak yakınsar</a> olduğunda; <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\sin }{\biggl (}\sum _{i=1}^{\infty }\theta _{i}{\biggl )}&=\sum _{{\text{tek}}\ k\geq 1}(-1)^{\frac {k-1}{2}}\!\!\sum _{\begin{smallmatrix}A\subseteq \{\,1,2,3,\dots \,\}\\\left|A\right|=k\end{smallmatrix}}{\biggl (}\prod _{i\in A}\sin \theta _{i}\prod _{i\not \in A}\cos \theta _{i}{\biggr )}\\{\cos }{\biggl (}\sum _{i=1}^{\infty }\theta _{i}{\biggr )}&=\sum _{{\text{çift}}\ k\geq 0}(-1)^{\frac {k}{2}}\,\sum _{\begin{smallmatrix}A\subseteq \{\,1,2,3,\dots \,\}\\\left|A\right|=k\end{smallmatrix}}{\biggl (}\prod _{i\in A}\sin \theta _{i}\prod _{i\not \in A}\cos \theta _{i}{\biggr )}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi>sin</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>tek</mtext> </mrow> <mtext> </mtext> <mi>k</mi> <mo>≥</mo> <mn>1</mn> </mrow> </munder> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </msup> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mi>A</mi> <mo>⊆</mo> <mo fence="false" stretchy="false">{</mo> <mspace width="thinmathspace" /> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>…</mo> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">}</mo> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>|</mo> <mi>A</mi> <mo>|</mo> </mrow> <mo>=</mo> <mi>k</mi> </mtd> </mtr> </mtable> </mstyle> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>A</mi> </mrow> </munder> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∉</mo> <mi>A</mi> </mrow> </munder> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi>cos</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>çift</mtext> </mrow> <mtext> </mtext> <mi>k</mi> <mo>≥</mo> <mn>0</mn> </mrow> </munder> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mn>2</mn> </mfrac> </mrow> </msup> <mspace width="thinmathspace" /> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mi>A</mi> <mo>⊆</mo> <mo fence="false" stretchy="false">{</mo> <mspace width="thinmathspace" /> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>…</mo> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">}</mo> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>|</mo> <mi>A</mi> <mo>|</mo> </mrow> <mo>=</mo> <mi>k</mi> </mtd> </mtr> </mtable> </mstyle> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>A</mi> </mrow> </munder> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∉</mo> <mi>A</mi> </mrow> </munder> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\sin }{\biggl (}\sum _{i=1}^{\infty }\theta _{i}{\biggl )}&=\sum _{{\text{tek}}\ k\geq 1}(-1)^{\frac {k-1}{2}}\!\!\sum _{\begin{smallmatrix}A\subseteq \{\,1,2,3,\dots \,\}\\\left|A\right|=k\end{smallmatrix}}{\biggl (}\prod _{i\in A}\sin \theta _{i}\prod _{i\not \in A}\cos \theta _{i}{\biggr )}\\{\cos }{\biggl (}\sum _{i=1}^{\infty }\theta _{i}{\biggr )}&=\sum _{{\text{çift}}\ k\geq 0}(-1)^{\frac {k}{2}}\,\sum _{\begin{smallmatrix}A\subseteq \{\,1,2,3,\dots \,\}\\\left|A\right|=k\end{smallmatrix}}{\biggl (}\prod _{i\in A}\sin \theta _{i}\prod _{i\not \in A}\cos \theta _{i}{\biggr )}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97a0e3ba0d96a66bb544de14b69085e52145d410" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.005ex; width:63.767ex; height:19.176ex;" alt="{\displaystyle {\begin{aligned}{\sin }{\biggl (}\sum _{i=1}^{\infty }\theta _{i}{\biggl )}&=\sum _{{\text{tek}}\ k\geq 1}(-1)^{\frac {k-1}{2}}\!\!\sum _{\begin{smallmatrix}A\subseteq \{\,1,2,3,\dots \,\}\\\left|A\right|=k\end{smallmatrix}}{\biggl (}\prod _{i\in A}\sin \theta _{i}\prod _{i\not \in A}\cos \theta _{i}{\biggr )}\\{\cos }{\biggl (}\sum _{i=1}^{\infty }\theta _{i}{\biggr )}&=\sum _{{\text{çift}}\ k\geq 0}(-1)^{\frac {k}{2}}\,\sum _{\begin{smallmatrix}A\subseteq \{\,1,2,3,\dots \,\}\\\left|A\right|=k\end{smallmatrix}}{\biggl (}\prod _{i\in A}\sin \theta _{i}\prod _{i\not \in A}\cos \theta _{i}{\biggr )}.\end{aligned}}}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sum _{i=1}^{\infty }\theta _{i}"> <semantics> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <annotation encoding="application/x-tex">\sum _{i=1}^{\infty }\theta _{i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/711f443e264cd22d5d7e7bab5f80c6b46ef9abb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:5.632ex; height:6.843ex;" alt="\sum _{i=1}^{\infty }\theta _{i}"> serisi mutlak yakınsadığı için, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lim _{i\to \infty }\theta _{i}=0,"> <semantics> <mrow> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mrow> <annotation encoding="application/x-tex">\lim _{i\to \infty }\theta _{i}=0,</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1643f1efe77defe0939eea0eeeeba36c06377b27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.039ex; height:3.843ex;" alt="\lim _{i\to \infty }\theta _{i}=0,"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lim _{i\to \infty }\sin \theta _{i}=0,"> <semantics> <mrow> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mrow> <annotation encoding="application/x-tex">\lim _{i\to \infty }\sin \theta _{i}=0,</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/866bfa4c2dda7bfe1c990bf18db68ff957059ca1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.281ex; height:3.843ex;" alt="\lim _{i\to \infty }\sin \theta _{i}=0,"> ve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lim _{i\to \infty }\cos \theta _{i}=1."> <semantics> <mrow> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>1.</mn> </mrow> <annotation encoding="application/x-tex">\lim _{i\to \infty }\cos \theta _{i}=1.</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2765c8ff2b146184f7a67c151d70db4d86bf225a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.537ex; height:3.843ex;" alt="\lim _{i\to \infty }\cos \theta _{i}=1."> Özellikle, bu iki özdeşlikte sonlu sayıda açının toplamları durumunda görülmeyen bir asimetri ortaya çıkar: her çarpımda yalnızca sonlu sayıda sinüs çarpanı vardır, ancak <a href="/w/index.php?title=Sonluluk&action=edit&redlink=1" class="new" title="Sonluluk (sayfa mevcut değil)">dual sonlu</a> çok sayıda kosinüs çarpanı vardır. Sonsuz sayıda sinüs çarpanı olan terimler zorunlu olarak sıfıra eşit olacaktır. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/302b19204ed378e99ff4575341a67eebdbe5a555" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.89ex; height:2.509ex;" alt="{\displaystyle \theta _{i}}"> açılarının yalnızca sonlu sayıda olanı sıfırdan farklı olduğunda, sağ taraftaki terimlerin yalnızca sonlu sayıda olanı sıfırdan farklıdır çünkü sonlu sayıda sinüs çarpanı hariç hepsi yok olur (sadeleşir). Ayrıca, her bir terimde sonlu sayıda kosinüs çarpanı hariç hepsi birimdir (tekildir). <div class="mw-heading mw-heading3"><h3 id="Toplamların_tanjantları_ve_kotanjantları">Toplamların tanjantları ve kotanjantları</h3>[<a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&veaction=edit&section=8" title="Değiştirilen bölüm: Toplamların tanjantları ve kotanjantları" class="mw-editsection-visualeditor">değiştir</a> | <a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&action=edit&section=8" title="Bölümün kaynak kodunu değiştir: Toplamların tanjantları ve kotanjantları">kaynağı değiştir</a>]</div> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a24f4c64db82ae2f1e89d9861304aec8598c7e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.172ex; height:2.009ex;" alt="{\displaystyle e_{k}}"> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=0,1,2,3,\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=0,1,2,3,\ldots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0682dbc430868c80201d88de10574907690a1e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.819ex; height:2.509ex;" alt="{\displaystyle k=0,1,2,3,\ldots }"> için) değişkenler içinde kinci derece <a href="/w/index.php?title=Temel_simetrik_polinom&action=edit&redlink=1" class="new" title="Temel simetrik polinom (sayfa mevcut değil)">temel simetrik polinom</a> olsun: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}=\tan \theta _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi>tan</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}=\tan \theta _{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d377f179e0327f81c3a3428a386436aec18ea12" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.865ex; height:2.509ex;" alt="{\displaystyle x_{i}=\tan \theta _{i}}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=0,1,2,3,\ldots ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=0,1,2,3,\ldots ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c380790b19b4c48945b0c7274e9ef9825966f37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.444ex; height:2.509ex;" alt="{\displaystyle i=0,1,2,3,\ldots ,}"> için yani, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}e_{0}&=1\\[6pt]e_{1}&=\sum _{i}x_{i}&&=\sum _{i}\tan \theta _{i}\\[6pt]e_{2}&=\sum _{i<j}x_{i}x_{j}&&=\sum _{i<j}\tan \theta _{i}\tan \theta _{j}\\[6pt]e_{3}&=\sum _{i<j<k}x_{i}x_{j}x_{k}&&=\sum _{i<j<k}\tan \theta _{i}\tan \theta _{j}\tan \theta _{k}\\&\ \ \vdots &&\ \ \vdots \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.9em 0.9em 0.9em 0.3em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mi>tan</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo><</mo> <mi>j</mi> </mrow> </munder> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mtd> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo><</mo> <mi>j</mi> </mrow> </munder> <mi>tan</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>tan</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo><</mo> <mi>j</mi> <mo><</mo> <mi>k</mi> </mrow> </munder> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mtd> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo><</mo> <mi>j</mi> <mo><</mo> <mi>k</mi> </mrow> </munder> <mi>tan</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>tan</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mi>tan</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd /> <mtd> <mtext> </mtext> <mtext> </mtext> <mo>⋮</mo> </mtd> <mtd /> <mtd> <mtext> </mtext> <mtext> </mtext> <mo>⋮</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}e_{0}&=1\\[6pt]e_{1}&=\sum _{i}x_{i}&&=\sum _{i}\tan \theta _{i}\\[6pt]e_{2}&=\sum _{i<j}x_{i}x_{j}&&=\sum _{i<j}\tan \theta _{i}\tan \theta _{j}\\[6pt]e_{3}&=\sum _{i<j<k}x_{i}x_{j}x_{k}&&=\sum _{i<j<k}\tan \theta _{i}\tan \theta _{j}\tan \theta _{k}\\&\ \ \vdots &&\ \ \vdots \end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb1927ae2be96f5156e18813a20486fdb0ff72f1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.838ex; width:48.693ex; height:28.843ex;" alt="{\displaystyle {\begin{aligned}e_{0}&=1\\[6pt]e_{1}&=\sum _{i}x_{i}&&=\sum _{i}\tan \theta _{i}\\[6pt]e_{2}&=\sum _{i<j}x_{i}x_{j}&&=\sum _{i<j}\tan \theta _{i}\tan \theta _{j}\\[6pt]e_{3}&=\sum _{i<j<k}x_{i}x_{j}x_{k}&&=\sum _{i<j<k}\tan \theta _{i}\tan \theta _{j}\tan \theta _{k}\\&\ \ \vdots &&\ \ \vdots \end{aligned}}}"> Öyleyse yukarıdaki sinüs ve kosinüs toplam formüllerini kullanarak, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\tan }{\Bigl (}\sum _{i}\theta _{i}{\Bigr )}&={\frac {{\sin }{\bigl (}\sum _{i}\theta _{i}{\bigr )}/\prod _{i}\cos \theta _{i}}{{\cos }{\bigl (}\sum _{i}\theta _{i}{\bigr )}/\prod _{i}\cos \theta _{i}}}\\[10pt]&={\frac {\displaystyle \sum _{{\text{tek}}\ k\geq 1}(-1)^{\frac {k-1}{2}}\sum _{\begin{smallmatrix}A\subseteq \{1,2,3,\dots \}\\\left|A\right|=k\end{smallmatrix}}\prod _{i\in A}\tan \theta _{i}}{\displaystyle \sum _{{\text{çift}}\ k\geq 0}~(-1)^{\frac {k}{2}}~~\sum _{\begin{smallmatrix}A\subseteq \{1,2,3,\dots \}\\\left|A\right|=k\end{smallmatrix}}\prod _{i\in A}\tan \theta _{i}}}={\frac {e_{1}-e_{3}+e_{5}-\cdots }{e_{0}-e_{2}+e_{4}-\cdots }}\\[10pt]{\cot }{\Bigl (}\sum _{i}\theta _{i}{\Bigr )}&={\frac {e_{0}-e_{2}+e_{4}-\cdots }{e_{1}-e_{3}+e_{5}-\cdots }}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="1.3em 1.3em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi>tan</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>sin</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>cos</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>tek</mtext> </mrow> <mtext> </mtext> <mi>k</mi> <mo>≥</mo> <mn>1</mn> </mrow> </munder> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </msup> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mi>A</mi> <mo>⊆</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>…</mo> <mo fence="false" stretchy="false">}</mo> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>|</mo> <mi>A</mi> <mo>|</mo> </mrow> <mo>=</mo> <mi>k</mi> </mtd> </mtr> </mtable> </mstyle> </mrow> </munder> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>A</mi> </mrow> </munder> <mi>tan</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>çift</mtext> </mrow> <mtext> </mtext> <mi>k</mi> <mo>≥</mo> <mn>0</mn> </mrow> </munder> <mtext> </mtext> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mn>2</mn> </mfrac> </mrow> </msup> <mtext> </mtext> <mtext> </mtext> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mi>A</mi> <mo>⊆</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>…</mo> <mo fence="false" stretchy="false">}</mo> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>|</mo> <mi>A</mi> <mo>|</mo> </mrow> <mo>=</mo> <mi>k</mi> </mtd> </mtr> </mtable> </mstyle> </mrow> </munder> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>A</mi> </mrow> </munder> <mi>tan</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>−</mo> <mo>⋯</mo> </mrow> <mrow> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>−</mo> <mo>⋯</mo> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi>cot</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>−</mo> <mo>⋯</mo> </mrow> <mrow> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>−</mo> <mo>⋯</mo> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\tan }{\Bigl (}\sum _{i}\theta _{i}{\Bigr )}&={\frac {{\sin }{\bigl (}\sum _{i}\theta _{i}{\bigr )}/\prod _{i}\cos \theta _{i}}{{\cos }{\bigl (}\sum _{i}\theta _{i}{\bigr )}/\prod _{i}\cos \theta _{i}}}\\[10pt]&={\frac {\displaystyle \sum _{{\text{tek}}\ k\geq 1}(-1)^{\frac {k-1}{2}}\sum _{\begin{smallmatrix}A\subseteq \{1,2,3,\dots \}\\\left|A\right|=k\end{smallmatrix}}\prod _{i\in A}\tan \theta _{i}}{\displaystyle \sum _{{\text{çift}}\ k\geq 0}~(-1)^{\frac {k}{2}}~~\sum _{\begin{smallmatrix}A\subseteq \{1,2,3,\dots \}\\\left|A\right|=k\end{smallmatrix}}\prod _{i\in A}\tan \theta _{i}}}={\frac {e_{1}-e_{3}+e_{5}-\cdots }{e_{0}-e_{2}+e_{4}-\cdots }}\\[10pt]{\cot }{\Bigl (}\sum _{i}\theta _{i}{\Bigr )}&={\frac {e_{0}-e_{2}+e_{4}-\cdots }{e_{1}-e_{3}+e_{5}-\cdots }}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/705065cd52a7e0a11c792f2cb1a8de1f78984e29" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -18.005ex; width:73.683ex; height:37.176ex;" alt="{\displaystyle {\begin{aligned}{\tan }{\Bigl (}\sum _{i}\theta _{i}{\Bigr )}&={\frac {{\sin }{\bigl (}\sum _{i}\theta _{i}{\bigr )}/\prod _{i}\cos \theta _{i}}{{\cos }{\bigl (}\sum _{i}\theta _{i}{\bigr )}/\prod _{i}\cos \theta _{i}}}\\[10pt]&={\frac {\displaystyle \sum _{{\text{tek}}\ k\geq 1}(-1)^{\frac {k-1}{2}}\sum _{\begin{smallmatrix}A\subseteq \{1,2,3,\dots \}\\\left|A\right|=k\end{smallmatrix}}\prod _{i\in A}\tan \theta _{i}}{\displaystyle \sum _{{\text{çift}}\ k\geq 0}~(-1)^{\frac {k}{2}}~~\sum _{\begin{smallmatrix}A\subseteq \{1,2,3,\dots \}\\\left|A\right|=k\end{smallmatrix}}\prod _{i\in A}\tan \theta _{i}}}={\frac {e_{1}-e_{3}+e_{5}-\cdots }{e_{0}-e_{2}+e_{4}-\cdots }}\\[10pt]{\cot }{\Bigl (}\sum _{i}\theta _{i}{\Bigr )}&={\frac {e_{0}-e_{2}+e_{4}-\cdots }{e_{1}-e_{3}+e_{5}-\cdots }}\end{aligned}}}"> Sağ taraftaki terim sayısı sol taraftaki terim sayısına bağlıdır. Örneğin: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\tan(\theta _{1}+\theta _{2})&={\frac {e_{1}}{e_{0}-e_{2}}}={\frac {x_{1}+x_{2}}{1\ -\ x_{1}x_{2}}}={\frac {\tan \theta _{1}+\tan \theta _{2}}{1\ -\ \tan \theta _{1}\tan \theta _{2}}},\\[8pt]\tan(\theta _{1}+\theta _{2}+\theta _{3})&={\frac {e_{1}-e_{3}}{e_{0}-e_{2}}}={\frac {(x_{1}+x_{2}+x_{3})\ -\ (x_{1}x_{2}x_{3})}{1\ -\ (x_{1}x_{2}+x_{1}x_{3}+x_{2}x_{3})}},\\[8pt]\tan(\theta _{1}+\theta _{2}+\theta _{3}+\theta _{4})&={\frac {e_{1}-e_{3}}{e_{0}-e_{2}+e_{4}}}\\[8pt]&={\frac {(x_{1}+x_{2}+x_{3}+x_{4})\ -\ (x_{1}x_{2}x_{3}+x_{1}x_{2}x_{4}+x_{1}x_{3}x_{4}+x_{2}x_{3}x_{4})}{1\ -\ (x_{1}x_{2}+x_{1}x_{3}+x_{1}x_{4}+x_{2}x_{3}+x_{2}x_{4}+x_{3}x_{4})\ +\ (x_{1}x_{2}x_{3}x_{4})}},\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="1.1em 1.1em 1.1em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mn>1</mn> <mtext> </mtext> <mo>−</mo> <mtext> </mtext> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>tan</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>tan</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mn>1</mn> <mtext> </mtext> <mo>−</mo> <mtext> </mtext> <mi>tan</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>tan</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>−</mo> <mtext> </mtext> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>1</mn> <mtext> </mtext> <mo>−</mo> <mtext> </mtext> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>−</mo> <mtext> </mtext> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>1</mn> <mtext> </mtext> <mo>−</mo> <mtext> </mtext> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>+</mo> <mtext> </mtext> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\tan(\theta _{1}+\theta _{2})&={\frac {e_{1}}{e_{0}-e_{2}}}={\frac {x_{1}+x_{2}}{1\ -\ x_{1}x_{2}}}={\frac {\tan \theta _{1}+\tan \theta _{2}}{1\ -\ \tan \theta _{1}\tan \theta _{2}}},\\[8pt]\tan(\theta _{1}+\theta _{2}+\theta _{3})&={\frac {e_{1}-e_{3}}{e_{0}-e_{2}}}={\frac {(x_{1}+x_{2}+x_{3})\ -\ (x_{1}x_{2}x_{3})}{1\ -\ (x_{1}x_{2}+x_{1}x_{3}+x_{2}x_{3})}},\\[8pt]\tan(\theta _{1}+\theta _{2}+\theta _{3}+\theta _{4})&={\frac {e_{1}-e_{3}}{e_{0}-e_{2}+e_{4}}}\\[8pt]&={\frac {(x_{1}+x_{2}+x_{3}+x_{4})\ -\ (x_{1}x_{2}x_{3}+x_{1}x_{2}x_{4}+x_{1}x_{3}x_{4}+x_{2}x_{3}x_{4})}{1\ -\ (x_{1}x_{2}+x_{1}x_{3}+x_{1}x_{4}+x_{2}x_{3}+x_{2}x_{4}+x_{3}x_{4})\ +\ (x_{1}x_{2}x_{3}x_{4})}},\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c9196b8ab5e0aef6cf7d7fec7d7546f2a6e7d2f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -14.505ex; width:92.731ex; height:30.176ex;" alt="{\displaystyle {\begin{aligned}\tan(\theta _{1}+\theta _{2})&={\frac {e_{1}}{e_{0}-e_{2}}}={\frac {x_{1}+x_{2}}{1\ -\ x_{1}x_{2}}}={\frac {\tan \theta _{1}+\tan \theta _{2}}{1\ -\ \tan \theta _{1}\tan \theta _{2}}},\\[8pt]\tan(\theta _{1}+\theta _{2}+\theta _{3})&={\frac {e_{1}-e_{3}}{e_{0}-e_{2}}}={\frac {(x_{1}+x_{2}+x_{3})\ -\ (x_{1}x_{2}x_{3})}{1\ -\ (x_{1}x_{2}+x_{1}x_{3}+x_{2}x_{3})}},\\[8pt]\tan(\theta _{1}+\theta _{2}+\theta _{3}+\theta _{4})&={\frac {e_{1}-e_{3}}{e_{0}-e_{2}+e_{4}}}\\[8pt]&={\frac {(x_{1}+x_{2}+x_{3}+x_{4})\ -\ (x_{1}x_{2}x_{3}+x_{1}x_{2}x_{4}+x_{1}x_{3}x_{4}+x_{2}x_{3}x_{4})}{1\ -\ (x_{1}x_{2}+x_{1}x_{3}+x_{1}x_{4}+x_{2}x_{3}+x_{2}x_{4}+x_{3}x_{4})\ +\ (x_{1}x_{2}x_{3}x_{4})}},\end{aligned}}}"> ve bunun gibi. Sadece sonlu sayıda terim olması durumu <a href="/wiki/Matematiksel_t%C3%BCmevar%C4%B1m" title="Matematiksel tümevarım">matematiksel tümevarım</a> ile kanıtlanabilir.<a href="#cite_note-14">[14]</a> Sonsuz sayıda terim olması durumu, bazı temel eşitsizlikler kullanılarak kanıtlanabilir.<a href="#cite_note-15">[15]</a> <div class="mw-heading mw-heading3"><h3 id="Toplamların_sekantları_ve_kosekantları">Toplamların sekantları ve kosekantları</h3>[<a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&veaction=edit&section=9" title="Değiştirilen bölüm: Toplamların sekantları ve kosekantları" class="mw-editsection-visualeditor">değiştir</a> | <a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&action=edit&section=9" title="Bölümün kaynak kodunu değiştir: Toplamların sekantları ve kosekantları">kaynağı değiştir</a>]</div> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\sec }{\Bigl (}\sum _{i}\theta _{i}{\Bigr )}&={\frac {\prod _{i}\sec \theta _{i}}{e_{0}-e_{2}+e_{4}-\cdots }}\\[8pt]{\csc }{\Bigl (}\sum _{i}\theta _{i}{\Bigr )}&={\frac {\prod _{i}\sec \theta _{i}}{e_{1}-e_{3}+e_{5}-\cdots }}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="1.1em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi>sec</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mi>sec</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>−</mo> <mo>⋯</mo> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi>csc</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mi>sec</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>−</mo> <mo>⋯</mo> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\sec }{\Bigl (}\sum _{i}\theta _{i}{\Bigr )}&={\frac {\prod _{i}\sec \theta _{i}}{e_{0}-e_{2}+e_{4}-\cdots }}\\[8pt]{\csc }{\Bigl (}\sum _{i}\theta _{i}{\Bigr )}&={\frac {\prod _{i}\sec \theta _{i}}{e_{1}-e_{3}+e_{5}-\cdots }}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b24f2b169fbe11608507c63f5bb728e03b2b4957" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.338ex; width:34.12ex; height:15.843ex;" alt="{\displaystyle {\begin{aligned}{\sec }{\Bigl (}\sum _{i}\theta _{i}{\Bigr )}&={\frac {\prod _{i}\sec \theta _{i}}{e_{0}-e_{2}+e_{4}-\cdots }}\\[8pt]{\csc }{\Bigl (}\sum _{i}\theta _{i}{\Bigr )}&={\frac {\prod _{i}\sec \theta _{i}}{e_{1}-e_{3}+e_{5}-\cdots }}\end{aligned}}}"> Burada <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a24f4c64db82ae2f1e89d9861304aec8598c7e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.172ex; height:2.009ex;" alt="{\displaystyle e_{k}}">, n değişkenlerinde kinci derece <a href="/w/index.php?title=Temel_simetrik_polinom&action=edit&redlink=1" class="new" title="Temel simetrik polinom (sayfa mevcut değil)">temel simetrik polinom</a> olup, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}=\tan \theta _{i},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi>tan</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}=\tan \theta _{i},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b935233927724db7dfe50d9c98179d9a408fb39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.512ex; height:2.509ex;" alt="{\displaystyle x_{i}=\tan \theta _{i},}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=1,\dots ,n,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=1,\dots ,n,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c1c6cab7b74aa04bca4aca382a0193c3f8ab432" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.283ex; height:2.509ex;" alt="{\displaystyle i=1,\dots ,n,}"> ve paydadaki terim sayısı ile paydaki çarpımdaki çarpan sayısı soldaki toplamdaki terim sayısına bağlıdır.<a href="#cite_note-16">[16]</a> Sadece sonlu sayıda terim olması durumu, bu tür terimlerin sayısı üzerine matematiksel tümevarım yoluyla kanıtlanabilir. Örneğin, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\sec(\alpha +\beta +\gamma )&={\frac {\sec \alpha \sec \beta \sec \gamma }{1-\tan \alpha \tan \beta -\tan \alpha \tan \gamma -\tan \beta \tan \gamma }}\\[8pt]\csc(\alpha +\beta +\gamma )&={\frac {\sec \alpha \sec \beta \sec \gamma }{\tan \alpha +\tan \beta +\tan \gamma -\tan \alpha \tan \beta \tan \gamma }}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="1.1em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>sec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo>+</mo> <mi>γ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sec</mi> <mo>⁡</mo> <mi>α</mi> <mi>sec</mi> <mo>⁡</mo> <mi>β</mi> <mi>sec</mi> <mo>⁡</mo> <mi>γ</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>tan</mi> <mo>⁡</mo> <mi>α</mi> <mi>tan</mi> <mo>⁡</mo> <mi>β</mi> <mo>−</mo> <mi>tan</mi> <mo>⁡</mo> <mi>α</mi> <mi>tan</mi> <mo>⁡</mo> <mi>γ</mi> <mo>−</mo> <mi>tan</mi> <mo>⁡</mo> <mi>β</mi> <mi>tan</mi> <mo>⁡</mo> <mi>γ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>csc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo>+</mo> <mi>γ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sec</mi> <mo>⁡</mo> <mi>α</mi> <mi>sec</mi> <mo>⁡</mo> <mi>β</mi> <mi>sec</mi> <mo>⁡</mo> <mi>γ</mi> </mrow> <mrow> <mi>tan</mi> <mo>⁡</mo> <mi>α</mi> <mo>+</mo> <mi>tan</mi> <mo>⁡</mo> <mi>β</mi> <mo>+</mo> <mi>tan</mi> <mo>⁡</mo> <mi>γ</mi> <mo>−</mo> <mi>tan</mi> <mo>⁡</mo> <mi>α</mi> <mi>tan</mi> <mo>⁡</mo> <mi>β</mi> <mi>tan</mi> <mo>⁡</mo> <mi>γ</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sec(\alpha +\beta +\gamma )&={\frac {\sec \alpha \sec \beta \sec \gamma }{1-\tan \alpha \tan \beta -\tan \alpha \tan \gamma -\tan \beta \tan \gamma }}\\[8pt]\csc(\alpha +\beta +\gamma )&={\frac {\sec \alpha \sec \beta \sec \gamma }{\tan \alpha +\tan \beta +\tan \gamma -\tan \alpha \tan \beta \tan \gamma }}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e082e69aab7a9c782f590f3a93262ba3101899b7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:60.728ex; height:14.176ex;" alt="{\displaystyle {\begin{aligned}\sec(\alpha +\beta +\gamma )&={\frac {\sec \alpha \sec \beta \sec \gamma }{1-\tan \alpha \tan \beta -\tan \alpha \tan \gamma -\tan \beta \tan \gamma }}\\[8pt]\csc(\alpha +\beta +\gamma )&={\frac {\sec \alpha \sec \beta \sec \gamma }{\tan \alpha +\tan \beta +\tan \gamma -\tan \alpha \tan \beta \tan \gamma }}.\end{aligned}}}"> <div class="mw-heading mw-heading3"><h3 id="Batlamyus_teoremi">Batlamyus teoremi</h3>[<a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&veaction=edit&section=10" title="Değiştirilen bölüm: Batlamyus teoremi" class="mw-editsection-visualeditor">değiştir</a> | <a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&action=edit&section=10" title="Bölümün kaynak kodunu değiştir: Batlamyus teoremi">kaynağı değiştir</a>]</div> <div role="note" class="hatnote navigation-not-searchable">Ana madde: <a href="/wiki/Batlamyus_teoremi" title="Batlamyus teoremi">Batlamyus teoremi</a></div> <div role="note" class="hatnote navigation-not-searchable">Ayrıca bakınız: <a href="/wiki/Trigonometri_tarihi#Klasik_antik_dönem" title="Trigonometri tarihi">Trigonometri tarihi § Klasik antik dönem</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Dosya:Diagram_illustrating_the_relation_between_Ptolemy%27s_theorem_and_the_angle_sum_trig_identity_for_sin.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Diagram_illustrating_the_relation_between_Ptolemy%27s_theorem_and_the_angle_sum_trig_identity_for_sin.svg/330px-Diagram_illustrating_the_relation_between_Ptolemy%27s_theorem_and_the_angle_sum_trig_identity_for_sin.svg.png" decoding="async" width="330" height="330" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Diagram_illustrating_the_relation_between_Ptolemy%27s_theorem_and_the_angle_sum_trig_identity_for_sin.svg/495px-Diagram_illustrating_the_relation_between_Ptolemy%27s_theorem_and_the_angle_sum_trig_identity_for_sin.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Diagram_illustrating_the_relation_between_Ptolemy%27s_theorem_and_the_angle_sum_trig_identity_for_sin.svg/660px-Diagram_illustrating_the_relation_between_Ptolemy%27s_theorem_and_the_angle_sum_trig_identity_for_sin.svg.png 2x" data-file-width="770" data-file-height="770" /></a><figcaption><div class="center" style="width:auto; margin-left:auto; margin-right:auto;">Batlamyus teoremi ile sinüs için açı toplamı trigonometri özdeşliği arasındaki ilişkiyi gösteren şekil. Batlamyus teoremi, karşılıklı kenarların uzunluklarının çarpımlarının toplamının köşegenlerin uzunluklarının çarpımına eşit olduğunu belirtir. Bu kenar uzunlukları yukarıdaki şekilde gösterilen sin ve cos değerleri cinsinden ifade edildiğinde, sinüs için açı toplamı trigonometrik özdeşliği elde edilir: sin(α + β) = sin α cos β + cos α sin β.</div></figcaption></figure> Batlamyus teoremi, trigonometrik özdeşlikler tarihinde önemlidir, çünkü sinüs ve kosinüs için toplam ve fark formüllerine eşdeğer sonuçlar ilk kez bu şekilde kanıtlanmıştır. Teorem, yandaki şekilde gösterildiği gibi <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ABCD}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> <mi>C</mi> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ABCD}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/412b7d8df4db6ca8093d971320c405598c49c339" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.198ex; height:2.176ex;" alt="{\displaystyle ABCD}"> çembersel dörtgeninde, karşılıklı kenarların uzunluklarının çarpımlarının toplamının köşegenlerin uzunluklarının çarpımına eşit olduğunu belirtir. Köşegenlerden veya kenarlardan birinin dairenin çapı olduğu özel durumlarda, bu teorem doğrudan açı toplamı ve fark trigonometrik özdeşliklerine yol açar.<a href="#cite_note-cut-the-knot.org-17">[17]</a> Bu ilişki en kolay şekilde, burada gösterildiği gibi daire bir çap uzunluğunda olacak şekilde inşa edildiğinde ortaya çıkar. <a href="/wiki/Thales_teoremi_(%C3%A7ember)" title="Thales teoremi (çember)">Thales teoremi</a> ile, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \angle DAB}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∠</mi> <mi>D</mi> <mi>A</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \angle DAB}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65625914b49c57162451d0f96cfa57441f9316be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.109ex; height:2.176ex;" alt="{\displaystyle \angle DAB}"> ve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \angle DCB}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∠</mi> <mi>D</mi> <mi>C</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \angle DCB}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c9237ca0e042297d6aec988710197fc98398957" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.133ex; height:2.176ex;" alt="{\displaystyle \angle DCB}"> her ikisi de dik açıdır. Dik açılı <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle DAB}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mi>A</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle DAB}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aaee66b6b4ee1b13b141c7cfcb8baf81f9497f2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.431ex; height:2.176ex;" alt="{\displaystyle DAB}"> ve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle DCB}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mi>C</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle DCB}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/919b745ff4d7e524f36612b42f1f3cf4859be2f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.455ex; height:2.176ex;" alt="{\displaystyle DCB}"> üçgenlerinin her ikisi de uzunluğu 1 olan <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {BD}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>D</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {BD}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c71fd62ec7ad127356017d4a62c77e118fd00d63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.803ex; height:3.009ex;" alt="{\displaystyle {\overline {BD}}}"> hipotenüsünü paylaşır. Böylece kenar <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {AB}}=\sin \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {AB}}=\sin \alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d47c6f2fcf9f89b3b524b5c7407307af7f9608c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.451ex; height:3.009ex;" alt="{\displaystyle {\overline {AB}}=\sin \alpha }">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {AD}}=\cos \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>D</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {AD}}=\cos \alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9d5510640d62fe08fb6a098640df5a2ea08703f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.867ex; height:3.009ex;" alt="{\displaystyle {\overline {AD}}=\cos \alpha }">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {BC}}=\sin \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {BC}}=\sin \beta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4fb4d2adb704d47062cd5e15ce1b1fc6f83c791" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.386ex; height:3.343ex;" alt="{\displaystyle {\overline {BC}}=\sin \beta }"> ve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {CD}}=\cos \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>C</mi> <mi>D</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {CD}}=\cos \beta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c83d3ef9c5c3c23748e3ad21ba30825f8c42cf07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.734ex; height:3.343ex;" alt="{\displaystyle {\overline {CD}}=\cos \beta }"> olur. <a href="/wiki/%C3%87evre_a%C3%A7%C4%B1" title="Çevre açı">Çevre açı</a> teoremine göre, çemberin merkezindeki <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {AC}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {AC}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea69bf8ccc972402f794c261e00b222f396bcce3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.692ex; height:3.009ex;" alt="{\displaystyle {\overline {AC}}}"> akorunun merkezde oluşturduğu açı <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \angle ADC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∠</mi> <mi>A</mi> <mi>D</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \angle ADC}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/038cc027b12c71c51cfce1a7444f84c70cf6f945" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.112ex; height:2.176ex;" alt="{\displaystyle \angle ADC}"> açısının iki katıdır, yani <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2(\alpha +\beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo stretchy="false">(</mo> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2(\alpha +\beta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac9193cdb619e8234e38712cc1d89988ad995611" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.632ex; height:2.843ex;" alt="{\displaystyle 2(\alpha +\beta )}">. Dolayısıyla, simetrik kırmızı üçgen çiftinin her birinin merkezinde <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha +\beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>+</mo> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha +\beta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99b80a3fdecb9cf75091789bb4335a1bb3561b08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.66ex; height:2.509ex;" alt="{\displaystyle \alpha +\beta }"> açısı vardır. Bu üçgenlerin her birinin <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\frac {1}{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <annotation encoding="application/x-tex">{\frac {1}{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/294aa706548adbc7778fb15bf5616abc4d9ad7be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:1.999ex; height:5.176ex;" alt="{\frac {1}{2}}"> uzunluğunda bir hipotenüsü vardır, dolayısıyla <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {AC}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {AC}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea69bf8ccc972402f794c261e00b222f396bcce3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.692ex; height:3.009ex;" alt="{\displaystyle {\overline {AC}}}"> uzunluğu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\times {\frac {1}{2}}\sin(\alpha +\beta )"> <semantics> <mrow> <mn>2</mn> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> <annotation encoding="application/x-tex">2\times {\frac {1}{2}}\sin(\alpha +\beta )</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6091c7bfe577d13b004a228fe2d2b27662933c5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.713ex; height:5.176ex;" alt="2\times {\frac {1}{2}}\sin(\alpha +\beta )">, yani basitçe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(\alpha +\beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(\alpha +\beta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d14ff9532ccca503cd4271a920864d3bd404e01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.325ex; height:2.843ex;" alt="{\displaystyle \sin(\alpha +\beta )}">. Dörtgenin diğer köşegeni 1 uzunluğundaki çaptır, dolayısıyla köşegenlerin uzunluklarının çarpımı da <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(\alpha +\beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(\alpha +\beta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d14ff9532ccca503cd4271a920864d3bd404e01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.325ex; height:2.843ex;" alt="{\displaystyle \sin(\alpha +\beta )}">'dır. Bu değerler, Batlamyus teoreminin <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |{\overline {AC}}|\cdot |{\overline {BD}}|=|{\overline {AB}}|\cdot |{\overline {CD}}|+|{\overline {AD}}|\cdot |{\overline {BC}}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>D</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>C</mi> <mi>D</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>D</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |{\overline {AC}}|\cdot |{\overline {BD}}|=|{\overline {AB}}|\cdot |{\overline {CD}}|+|{\overline {AD}}|\cdot |{\overline {BC}}|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0d81ff5635f85b487afeb443084db547640272a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.157ex; height:3.509ex;" alt="{\displaystyle |{\overline {AC}}|\cdot |{\overline {BD}}|=|{\overline {AB}}|\cdot |{\overline {CD}}|+|{\overline {AD}}|\cdot |{\overline {BC}}|}"> ifadesinde yerine konulduğunda, sinüs için açı toplamı trigonometrik özdeşliği elde edilir: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(\alpha +\beta )=\sin \alpha \cos \beta +\cos \alpha \sin \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mi>α</mi> <mi>cos</mi> <mo>⁡</mo> <mi>β</mi> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> <mi>sin</mi> <mo>⁡</mo> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(\alpha +\beta )=\sin \alpha \cos \beta +\cos \alpha \sin \beta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/600b028031a41e3d9e37fcd38c2af048374d496d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.159ex; height:2.843ex;" alt="{\displaystyle \sin(\alpha +\beta )=\sin \alpha \cos \beta +\cos \alpha \sin \beta }">. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(\alpha -\beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>−</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(\alpha -\beta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/680bcef1cb476d51f78323083ab28fa9892056ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.325ex; height:2.843ex;" alt="{\displaystyle \sin(\alpha -\beta )}"> için açı farkı formülü, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {CD}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>C</mi> <mi>D</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {CD}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3eb745bd8208e9a5eda418b120c20801bf441814" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.806ex; height:3.009ex;" alt="{\displaystyle {\overline {CD}}}"> kenarının <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {BD}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>D</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {BD}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c71fd62ec7ad127356017d4a62c77e118fd00d63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.803ex; height:3.009ex;" alt="{\displaystyle {\overline {BD}}}"> yerine çap olarak kullanılmasıyla benzer şekilde türetilebilir.<a href="#cite_note-cut-the-knot.org-17">[17]</a> <div class="mw-heading mw-heading2"><h2 id="Açının_katları_ve_yarım_açı_formülleri">Açının katları ve yarım açı formülleri</h2>[<a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&veaction=edit&section=11" title="Değiştirilen bölüm: Açının katları ve yarım açı formülleri" class="mw-editsection-visualeditor">değiştir</a> | <a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&action=edit&section=11" title="Bölümün kaynak kodunu değiştir: Açının katları ve yarım açı formülleri">kaynağı değiştir</a>]</div> <table class="wikitable" style="background-color: background-color:var(--background-color-base);"> <tbody><tr> <th>Tn, ninci <a href="/w/index.php?title=Chebyshev_polinomlar%C4%B1&action=edit&redlink=1" class="new" title="Chebyshev polinomları (sayfa mevcut değil)">Chebyshev polinomudur</a> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(n\theta )=T_{n}(\cos \theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(n\theta )=T_{n}(\cos \theta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0fd5eef19f7f62a21da950d949afe0eeecb2c0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.478ex; height:2.843ex;" alt="{\displaystyle \cos(n\theta )=T_{n}(\cos \theta )}"><a href="#cite_note-mathworld_multiple_angle-18">[18]</a> </td></tr> <tr> <th><a href="/wiki/De_Moivre_form%C3%BCl%C3%BC" title="De Moivre formülü">de Moivre formülü</a>, i <a href="/wiki/%C4%B0_say%C4%B1s%C4%B1" title="İ sayısı">sanal birimdir</a> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(n\theta )+i\sin(n\theta )=(\cos \theta +i\sin \theta )^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(n\theta )+i\sin(n\theta )=(\cos \theta +i\sin \theta )^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0554d3e9311492f7d3a91a464bd29f71c07c379" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.664ex; height:2.843ex;" alt="{\displaystyle \cos(n\theta )+i\sin(n\theta )=(\cos \theta +i\sin \theta )^{n}}"><a href="#cite_note-19">[19]</a> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Açının_katları_formülleri">Açının katları formülleri</h3>[<a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&veaction=edit&section=12" title="Değiştirilen bölüm: Açının katları formülleri" class="mw-editsection-visualeditor">değiştir</a> | <a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&action=edit&section=12" title="Bölümün kaynak kodunu değiştir: Açının katları formülleri">kaynağı değiştir</a>]</div> <div class="mw-heading mw-heading4"><h4 id="Çift_açı_formülleri">Çift açı formülleri</h4>[<a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&veaction=edit&section=13" title="Değiştirilen bölüm: Çift açı formülleri" class="mw-editsection-visualeditor">değiştir</a> | <a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&action=edit&section=13" title="Bölümün kaynak kodunu değiştir: Çift açı formülleri">kaynağı değiştir</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Dosya:Visual_demonstration_of_the_double-angle_trigonometric_identity_for_sine.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0a/Visual_demonstration_of_the_double-angle_trigonometric_identity_for_sine.svg/330px-Visual_demonstration_of_the_double-angle_trigonometric_identity_for_sine.svg.png" decoding="async" width="330" height="163" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0a/Visual_demonstration_of_the_double-angle_trigonometric_identity_for_sine.svg/495px-Visual_demonstration_of_the_double-angle_trigonometric_identity_for_sine.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0a/Visual_demonstration_of_the_double-angle_trigonometric_identity_for_sine.svg/660px-Visual_demonstration_of_the_double-angle_trigonometric_identity_for_sine.svg.png 2x" data-file-width="750" data-file-height="370" /></a><figcaption><div class="center" style="width:auto; margin-left:auto; margin-right:auto;">Sinüs için çift açı formülünün görsel ifadesi. Birim kenarlı ve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/493f1772ffdb7096362a9de2d587b65a79f4b6f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.253ex; height:2.176ex;" alt="{\displaystyle 2\theta }"> açılı yukarıdaki ikizkenar üçgen için alan 1/2 × taban × yükseklik iki yönde hesaplanır. Dik durumdayken alan <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \theta \cos \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \theta \cos \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64e38ab71d25633da1013f8450f6108023b17fab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.309ex; height:2.176ex;" alt="{\displaystyle \sin \theta \cos \theta }"> şeklindedir. Yan yattığında ise aynı alan <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\frac {1}{2}}\sin 2\theta "> <semantics> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mn>2</mn> <mi>θ</mi> </mrow> <annotation encoding="application/x-tex">{\frac {1}{2}}\sin 2\theta</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39a08103ba08fa50225275b56baa8892325b8d96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.881ex; height:5.176ex;" alt="{\frac {1}{2}}\sin 2\theta ">. Bu nedenle, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin 2\theta =2\sin \theta \cos \theta .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mn>2</mn> <mi>θ</mi> <mo>=</mo> <mn>2</mn> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin 2\theta =2\sin \theta \cos \theta .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c4960fcbe1b1efd380362ff3f81d56803a846f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:20.099ex; height:2.176ex;" alt="{\displaystyle \sin 2\theta =2\sin \theta \cos \theta .}"></div></figcaption></figure> Bir açının iki katı için formüller.<a href="#cite_note-STM1-20">[20]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(2\theta )=2\sin \theta \cos \theta =(\sin \theta +\cos \theta )^{2}-1={\frac {2\tan \theta }{1+\tan ^{2}\theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(2\theta )=2\sin \theta \cos \theta =(\sin \theta +\cos \theta )^{2}-1={\frac {2\tan \theta }{1+\tan ^{2}\theta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b664ba05a7cf2449da0ff80930fbd80f4c466da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:56.431ex; height:5.843ex;" alt="{\displaystyle \sin(2\theta )=2\sin \theta \cos \theta =(\sin \theta +\cos \theta )^{2}-1={\frac {2\tan \theta }{1+\tan ^{2}\theta }}}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(2\theta )=\cos ^{2}\theta -\sin ^{2}\theta =2\cos ^{2}\theta -1=1-2\sin ^{2}\theta ={\frac {1-\tan ^{2}\theta }{1+\tan ^{2}\theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mn>2</mn> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(2\theta )=\cos ^{2}\theta -\sin ^{2}\theta =2\cos ^{2}\theta -1=1-2\sin ^{2}\theta ={\frac {1-\tan ^{2}\theta }{1+\tan ^{2}\theta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6f9e9ba9762d2e92822ebdc3edb3640478bdab4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:66.303ex; height:6.176ex;" alt="{\displaystyle \cos(2\theta )=\cos ^{2}\theta -\sin ^{2}\theta =2\cos ^{2}\theta -1=1-2\sin ^{2}\theta ={\frac {1-\tan ^{2}\theta }{1+\tan ^{2}\theta }}}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan(2\theta )={\frac {2\tan \theta }{1-\tan ^{2}\theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan(2\theta )={\frac {2\tan \theta }{1-\tan ^{2}\theta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf0463617c8fb39ff3d7aa1f54294e865be07dbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.251ex; height:5.843ex;" alt="{\displaystyle \tan(2\theta )={\frac {2\tan \theta }{1-\tan ^{2}\theta }}}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cot(2\theta )={\frac {\cot ^{2}\theta -1}{2\cot \theta }}={\frac {1-\tan ^{2}\theta }{2\tan \theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>cot</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mn>2</mn> <mi>cot</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mn>2</mn> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cot(2\theta )={\frac {\cot ^{2}\theta -1}{2\cot \theta }}={\frac {1-\tan ^{2}\theta }{2\tan \theta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec33f11ca4ea52f284435759f3dc6dbb4dcd4ca5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:34.559ex; height:5.843ex;" alt="{\displaystyle \cot(2\theta )={\frac {\cot ^{2}\theta -1}{2\cot \theta }}={\frac {1-\tan ^{2}\theta }{2\tan \theta }}}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sec(2\theta )={\frac {\sec ^{2}\theta }{2-\sec ^{2}\theta }}={\frac {1+\tan ^{2}\theta }{1-\tan ^{2}\theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>sec</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mn>2</mn> <mo>−</mo> <msup> <mi>sec</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sec(2\theta )={\frac {\sec ^{2}\theta }{2-\sec ^{2}\theta }}={\frac {1+\tan ^{2}\theta }{1-\tan ^{2}\theta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2be33f0ae399232258fa67eeab18ccd050e0fc37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:34.322ex; height:6.176ex;" alt="{\displaystyle \sec(2\theta )={\frac {\sec ^{2}\theta }{2-\sec ^{2}\theta }}={\frac {1+\tan ^{2}\theta }{1-\tan ^{2}\theta }}}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \csc(2\theta )={\frac {\sec \theta \csc \theta }{2}}={\frac {1+\tan ^{2}\theta }{2\tan \theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>csc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sec</mi> <mo>⁡</mo> <mi>θ</mi> <mi>csc</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mn>2</mn> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \csc(2\theta )={\frac {\sec \theta \csc \theta }{2}}={\frac {1+\tan ^{2}\theta }{2\tan \theta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a99ddbf2e5f93083d1445c1cab4d62c67423c590" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:34.111ex; height:5.843ex;" alt="{\displaystyle \csc(2\theta )={\frac {\sec \theta \csc \theta }{2}}={\frac {1+\tan ^{2}\theta }{2\tan \theta }}}"></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Üç_kat_açı_formülleri">Üç kat açı formülleri</h4>[<a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&veaction=edit&section=14" title="Değiştirilen bölüm: Üç kat açı formülleri" class="mw-editsection-visualeditor">değiştir</a> | <a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&action=edit&section=14" title="Bölümün kaynak kodunu değiştir: Üç kat açı formülleri">kaynağı değiştir</a>]</div> Üç kat açılar için formüller.<a href="#cite_note-STM1-20">[20]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(3\theta )=3\sin \theta -4\sin ^{3}\theta =4\sin \theta \sin \left({\frac {\pi }{3}}-\theta \right)\sin \left({\frac {\pi }{3}}+\theta \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>3</mn> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <mn>4</mn> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mn>4</mn> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>3</mn> </mfrac> </mrow> <mo>−</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(3\theta )=3\sin \theta -4\sin ^{3}\theta =4\sin \theta \sin \left({\frac {\pi }{3}}-\theta \right)\sin \left({\frac {\pi }{3}}+\theta \right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4aa9c9ab36ae75e820610e27dc910b55858f851" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:58.892ex; height:4.843ex;" alt="{\displaystyle \sin(3\theta )=3\sin \theta -4\sin ^{3}\theta =4\sin \theta \sin \left({\frac {\pi }{3}}-\theta \right)\sin \left({\frac {\pi }{3}}+\theta \right)}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(3\theta )=4\cos ^{3}\theta -3\cos \theta =4\cos \theta \cos \left({\frac {\pi }{3}}-\theta \right)\cos \left({\frac {\pi }{3}}+\theta \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>4</mn> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <mn>3</mn> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mn>4</mn> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>3</mn> </mfrac> </mrow> <mo>−</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(3\theta )=4\cos ^{3}\theta -3\cos \theta =4\cos \theta \cos \left({\frac {\pi }{3}}-\theta \right)\cos \left({\frac {\pi }{3}}+\theta \right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6c40c97578e1f4cd77e8cdb6dc1e54a1e0476e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:60.425ex; height:4.843ex;" alt="{\displaystyle \cos(3\theta )=4\cos ^{3}\theta -3\cos \theta =4\cos \theta \cos \left({\frac {\pi }{3}}-\theta \right)\cos \left({\frac {\pi }{3}}+\theta \right)}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan(3\theta )={\frac {3\tan \theta -\tan ^{3}\theta }{1-3\tan ^{2}\theta }}=\tan \theta \tan \left({\frac {\pi }{3}}-\theta \right)\tan \left({\frac {\pi }{3}}+\theta \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mn>3</mn> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> <mi>tan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>3</mn> </mfrac> </mrow> <mo>−</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> <mi>tan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan(3\theta )={\frac {3\tan \theta -\tan ^{3}\theta }{1-3\tan ^{2}\theta }}=\tan \theta \tan \left({\frac {\pi }{3}}-\theta \right)\tan \left({\frac {\pi }{3}}+\theta \right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de255dcb8512ccdb36c14c807688bd1453435b63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:59.653ex; height:6.176ex;" alt="{\displaystyle \tan(3\theta )={\frac {3\tan \theta -\tan ^{3}\theta }{1-3\tan ^{2}\theta }}=\tan \theta \tan \left({\frac {\pi }{3}}-\theta \right)\tan \left({\frac {\pi }{3}}+\theta \right)}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cot(3\theta )={\frac {3\cot \theta -\cot ^{3}\theta }{1-3\cot ^{2}\theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mi>cot</mi> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <msup> <mi>cot</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mn>3</mn> <msup> <mi>cot</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cot(3\theta )={\frac {3\cot \theta -\cot ^{3}\theta }{1-3\cot ^{2}\theta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c88c94eea3bd262ab04f5717b311e192a82529d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.695ex; height:6.176ex;" alt="{\displaystyle \cot(3\theta )={\frac {3\cot \theta -\cot ^{3}\theta }{1-3\cot ^{2}\theta }}}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sec(3\theta )={\frac {\sec ^{3}\theta }{4-3\sec ^{2}\theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>sec</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mn>4</mn> <mo>−</mo> <mn>3</mn> <msup> <mi>sec</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sec(3\theta )={\frac {\sec ^{3}\theta }{4-3\sec ^{2}\theta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/819c7d1bbf0bd01eb32749d4ec22492f4ffddd87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.043ex; height:6.176ex;" alt="{\displaystyle \sec(3\theta )={\frac {\sec ^{3}\theta }{4-3\sec ^{2}\theta }}}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \csc(3\theta )={\frac {\csc ^{3}\theta }{3\csc ^{2}\theta -4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>csc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>csc</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mn>3</mn> <msup> <mi>csc</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <mn>4</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \csc(3\theta )={\frac {\csc ^{3}\theta }{3\csc ^{2}\theta -4}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3d00235e7b86b059ad8f1b5ba74e608b558e0fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.043ex; height:6.176ex;" alt="{\displaystyle \csc(3\theta )={\frac {\csc ^{3}\theta }{3\csc ^{2}\theta -4}}}"></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Çok_kat_açı_formülleri">Çok kat açı formülleri</h4>[<a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&veaction=edit&section=15" title="Değiştirilen bölüm: Çok kat açı formülleri" class="mw-editsection-visualeditor">değiştir</a> | <a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&action=edit&section=15" title="Bölümün kaynak kodunu değiştir: Çok kat açı formülleri">kaynağı değiştir</a>]</div> Çok katlı açılar için formüller.<a href="#cite_note-21">[21]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\sin(n\theta )&=\sum _{k{\text{ tek}}}(-1)^{\frac {k-1}{2}}{n \choose k}\cos ^{n-k}\theta \sin ^{k}\theta =\sin \theta \sum _{i=0}^{(n+1)/2}\sum _{j=0}^{i}(-1)^{i-j}{n \choose 2i+1}{i \choose j}\cos ^{n-2(i-j)-1}\theta \\{}&=2^{(n-1)}\prod _{k=0}^{n-1}\sin(k\pi /n+\theta )\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> tek</mtext> </mrow> </mrow> </munder> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mi>j</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mrow> <mn>2</mn> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>i</mi> <mi>j</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mi>i</mi> <mo>−</mo> <mi>j</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mo>+</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sin(n\theta )&=\sum _{k{\text{ tek}}}(-1)^{\frac {k-1}{2}}{n \choose k}\cos ^{n-k}\theta \sin ^{k}\theta =\sin \theta \sum _{i=0}^{(n+1)/2}\sum _{j=0}^{i}(-1)^{i-j}{n \choose 2i+1}{i \choose j}\cos ^{n-2(i-j)-1}\theta \\{}&=2^{(n-1)}\prod _{k=0}^{n-1}\sin(k\pi /n+\theta )\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfdc6c3e860b402803250ee01f0484f05778f821" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.913ex; margin-bottom: -0.258ex; width:94.951ex; height:15.509ex;" alt="{\displaystyle {\begin{aligned}\sin(n\theta )&=\sum _{k{\text{ tek}}}(-1)^{\frac {k-1}{2}}{n \choose k}\cos ^{n-k}\theta \sin ^{k}\theta =\sin \theta \sum _{i=0}^{(n+1)/2}\sum _{j=0}^{i}(-1)^{i-j}{n \choose 2i+1}{i \choose j}\cos ^{n-2(i-j)-1}\theta \\{}&=2^{(n-1)}\prod _{k=0}^{n-1}\sin(k\pi /n+\theta )\end{aligned}}}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(n\theta )=\sum _{k{\text{ çift}}}(-1)^{\frac {k}{2}}{n \choose k}\cos ^{n-k}\theta \sin ^{k}\theta =\sum _{i=0}^{n/2}\sum _{j=0}^{i}(-1)^{i-j}{n \choose 2i}{i \choose j}\cos ^{n-2(i-j)}\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> çift</mtext> </mrow> </mrow> </munder> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mn>2</mn> </mfrac> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mi>j</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mrow> <mn>2</mn> <mi>i</mi> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>i</mi> <mi>j</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mi>i</mi> <mo>−</mo> <mi>j</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(n\theta )=\sum _{k{\text{ çift}}}(-1)^{\frac {k}{2}}{n \choose k}\cos ^{n-k}\theta \sin ^{k}\theta =\sum _{i=0}^{n/2}\sum _{j=0}^{i}(-1)^{i-j}{n \choose 2i}{i \choose j}\cos ^{n-2(i-j)}\theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f98ab58a8d03856d751aa672450ec2a83ce23e2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:79.635ex; height:8.676ex;" alt="{\displaystyle \cos(n\theta )=\sum _{k{\text{ çift}}}(-1)^{\frac {k}{2}}{n \choose k}\cos ^{n-k}\theta \sin ^{k}\theta =\sum _{i=0}^{n/2}\sum _{j=0}^{i}(-1)^{i-j}{n \choose 2i}{i \choose j}\cos ^{n-2(i-j)}\theta }"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos((2n+1)\theta )=(-1)^{n}2^{2n}\prod _{k=0}^{2n}\cos(k\pi /(2n+1)-\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </munderover> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos((2n+1)\theta )=(-1)^{n}2^{2n}\prod _{k=0}^{2n}\cos(k\pi /(2n+1)-\theta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8be979c043f2a7e251656e658811fa554e9e51c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:51.35ex; height:7.509ex;" alt="{\displaystyle \cos((2n+1)\theta )=(-1)^{n}2^{2n}\prod _{k=0}^{2n}\cos(k\pi /(2n+1)-\theta )}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(2n\theta )=(-1)^{n}2^{2n-1}\prod _{k=0}^{2n-1}\cos((1+2k)\pi /(4n)-\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mi>k</mi> <mo stretchy="false">)</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>4</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(2n\theta )=(-1)^{n}2^{2n-1}\prod _{k=0}^{2n-1}\cos((1+2k)\pi /(4n)-\theta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f261bc01dc75b640f95b6b82ca0351ddf397ec02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:51.549ex; height:7.509ex;" alt="{\displaystyle \cos(2n\theta )=(-1)^{n}2^{2n-1}\prod _{k=0}^{2n-1}\cos((1+2k)\pi /(4n)-\theta )}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan(n\theta )={\frac {\sum _{k{\text{ tek}}}(-1)^{\frac {k-1}{2}}{n \choose k}\tan ^{k}\theta }{\sum _{k{\text{ çift}}}(-1)^{\frac {k}{2}}{n \choose k}\tan ^{k}\theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> tek</mtext> </mrow> </mrow> </munder> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> çift</mtext> </mrow> </mrow> </munder> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mn>2</mn> </mfrac> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan(n\theta )={\frac {\sum _{k{\text{ tek}}}(-1)^{\frac {k-1}{2}}{n \choose k}\tan ^{k}\theta }{\sum _{k{\text{ çift}}}(-1)^{\frac {k}{2}}{n \choose k}\tan ^{k}\theta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b7b2c3384403ef0dda04f4bd07a6424b5ca0f15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:35.628ex; height:10.009ex;" alt="{\displaystyle \tan(n\theta )={\frac {\sum _{k{\text{ tek}}}(-1)^{\frac {k-1}{2}}{n \choose k}\tan ^{k}\theta }{\sum _{k{\text{ çift}}}(-1)^{\frac {k}{2}}{n \choose k}\tan ^{k}\theta }}}"></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Chebyshev_yöntemi">Chebyshev yöntemi</h4>[<a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&veaction=edit&section=16" title="Değiştirilen bölüm: Chebyshev yöntemi" class="mw-editsection-visualeditor">değiştir</a> | <a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&action=edit&section=16" title="Bölümün kaynak kodunu değiştir: Chebyshev yöntemi">kaynağı değiştir</a>]</div> <a href="/wiki/Pafnuty_Chebyshev" class="mw-redirect" title="Pafnuty Chebyshev">Chebyshev</a> yöntemi, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (n-1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df88c6333caaf6471cf277f24b802ff9931b133e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.207ex; height:2.843ex;" alt="{\displaystyle (n-1)}">inci ve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n-2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (n-2)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a57c71b5e6c79de3b7fb8d7d0a918b434a160f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.207ex; height:2.843ex;" alt="{\displaystyle (n-2)}">inci değerleri bilerek ninci çok katlı açı formülünü bulmak için bir <a href="/wiki/%C3%96zyineleme" title="Özyineleme">özyineleme</a> <a href="/wiki/Algoritma" title="Algoritma">algoritmasıdır</a>.<a href="#cite_note-22">[22]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(nx)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(nx)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a930a1025613b667060f1659b66d739d62576232" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.645ex; height:2.843ex;" alt="{\displaystyle \cos(nx)}"> değeri, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos((n-1)x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos((n-1)x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c37b48abe68455e50e2d8ca88a8fede7dbf0939" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.457ex; height:2.843ex;" alt="{\displaystyle \cos((n-1)x)}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos((n-2)x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos((n-2)x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c4756637677d26dad82f4d6c1a6f191fd6aa92b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.457ex; height:2.843ex;" alt="{\displaystyle \cos((n-2)x)}"> ve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb9af7ed6f44822021b74bb82b431022c7fd66b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.25ex; height:2.843ex;" alt="{\displaystyle \cos(x)}">'den <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(nx)=2\cos x\cos((n-1)x)-\cos((n-2)x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(nx)=2\cos x\cos((n-1)x)-\cos((n-2)x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46eb073a4d4ed4c747f470399d90dbc8cd219148" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.262ex; height:2.843ex;" alt="{\displaystyle \cos(nx)=2\cos x\cos((n-1)x)-\cos((n-2)x)}"> eşitliği yardımıyla hesaplanabilir. Bu durum aşağıdaki formüllerin toplanmasıyla kanıtlanabilir: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\cos((n-1)x+x)&=\cos((n-1)x)\cos x-\sin((n-1)x)\sin x\\\cos((n-1)x-x)&=\cos((n-1)x)\cos x+\sin((n-1)x)\sin x\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>x</mi> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>x</mi> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\cos((n-1)x+x)&=\cos((n-1)x)\cos x-\sin((n-1)x)\sin x\\\cos((n-1)x-x)&=\cos((n-1)x)\cos x+\sin((n-1)x)\sin x\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e323a68098838c00dd5f51bc99ee55280df97df7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:61.15ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}\cos((n-1)x+x)&=\cos((n-1)x)\cos x-\sin((n-1)x)\sin x\\\cos((n-1)x-x)&=\cos((n-1)x)\cos x+\sin((n-1)x)\sin x\end{aligned}}}"> Tümevarım yoluyla <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(nx)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(nx)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a930a1025613b667060f1659b66d739d62576232" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.645ex; height:2.843ex;" alt="{\displaystyle \cos(nx)}">'in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos x,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos x,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/908f0811e6760290d838e5ea7c7769c6d1b25157" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.475ex; height:2.009ex;" alt="{\displaystyle \cos x,}"> 'in bir polinomu olduğu sonucuna varılır, buna birinci türden Chebyshev polinomu denir, bkz. <a href="/w/index.php?title=Chebyshev_polinomlar%C4%B1&action=edit&redlink=1" class="new" title="Chebyshev polinomları (sayfa mevcut değil)">Chebyshev polinomları#Trigonometrik tanım</a>. Benzer şekilde, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(nx)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(nx)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/971f4f57ec36ecf92d5af65eca2d29ffad59c402" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.389ex; height:2.843ex;" alt="{\displaystyle \sin(nx)}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin((n-1)x),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin((n-1)x),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f08bc4486281bfec787b0a358237ad128b444848" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.848ex; height:2.843ex;" alt="{\displaystyle \sin((n-1)x),}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin((n-2)x),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin((n-2)x),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a83f5e725b736a466c91e9633e841317f1db8dff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.848ex; height:2.843ex;" alt="{\displaystyle \sin((n-2)x),}"> ve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/184ba70c3a71df25a25c09f34cd7f8175a9b5280" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.828ex; height:1.676ex;" alt="{\displaystyle \cos x}">'ten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(nx)=2\cos x\sin((n-1)x)-\sin((n-2)x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(nx)=2\cos x\sin((n-1)x)-\sin((n-2)x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa9dc8ff1921a82a6a1c86ea6ddffdd9d7392135" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.496ex; height:2.843ex;" alt="{\displaystyle \sin(nx)=2\cos x\sin((n-1)x)-\sin((n-2)x)}"> yardımıyla hesaplanabilir. Bu, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin((n-1)x+x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>x</mi> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin((n-1)x+x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88e5dbeaa6a1685a2e33c70644e98df8e9256c0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.372ex; height:2.843ex;" alt="{\displaystyle \sin((n-1)x+x)}"> ve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin((n-1)x-x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>x</mi> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin((n-1)x-x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/734986e70f08e557b3e330a710156b2f8dcca3ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.372ex; height:2.843ex;" alt="{\displaystyle \sin((n-1)x-x)}"> formülleri eklenerek kanıtlanabilir. Chebyshev yöntemine benzer bir amaca hizmet ederek, tanjnat için şunu yazabiliriz: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan(nx)={\frac {\tan((n-1)x)+\tan x}{1-\tan((n-1)x)\tan x}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>tan</mi> <mo>⁡</mo> <mi>x</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>tan</mi> <mo>⁡</mo> <mi>x</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan(nx)={\frac {\tan((n-1)x)+\tan x}{1-\tan((n-1)x)\tan x}}\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b87fe648df295e0b3521b1f8bb8dae6a627dcc0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:36.034ex; height:6.509ex;" alt="{\displaystyle \tan(nx)={\frac {\tan((n-1)x)+\tan x}{1-\tan((n-1)x)\tan x}}\,.}"> <div class="mw-heading mw-heading3"><h3 id="Yarım_açı_formülleri">Yarım açı formülleri</h3>[<a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&veaction=edit&section=17" title="Değiştirilen bölüm: Yarım açı formülleri" class="mw-editsection-visualeditor">değiştir</a> | <a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&action=edit&section=17" title="Bölümün kaynak kodunu değiştir: Yarım açı formülleri">kaynağı değiştir</a>]</div> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\sin {\frac {\theta }{2}}&=\operatorname {sgn} \left(\sin {\frac {\theta }{2}}\right){\sqrt {\frac {1-\cos \theta }{2}}}\\[3pt]\cos {\frac {\theta }{2}}&=\operatorname {sgn} \left(\cos {\frac {\theta }{2}}\right){\sqrt {\frac {1+\cos \theta }{2}}}\\[3pt]\tan {\frac {\theta }{2}}&={\frac {1-\cos \theta }{\sin \theta }}={\frac {\sin \theta }{1+\cos \theta }}=\csc \theta -\cot \theta ={\frac {\tan \theta }{1+\sec {\theta }}}\\[6mu]&=\operatorname {sgn}(\sin \theta ){\sqrt {\frac {1-\cos \theta }{1+\cos \theta }}}={\frac {-1+\operatorname {sgn}(\cos \theta ){\sqrt {1+\tan ^{2}\theta }}}{\tan \theta }}\\[3pt]\cot {\frac {\theta }{2}}&={\frac {1+\cos \theta }{\sin \theta }}={\frac {\sin \theta }{1-\cos \theta }}=\csc \theta +\cot \theta =\operatorname {sgn}(\sin \theta ){\sqrt {\frac {1+\cos \theta }{1-\cos \theta }}}\\\sec {\frac {\theta }{2}}&=\operatorname {sgn} \left(\cos {\frac {\theta }{2}}\right){\sqrt {\frac {2}{1+\cos \theta }}}\\\csc {\frac {\theta }{2}}&=\operatorname {sgn} \left(\sin {\frac {\theta }{2}}\right){\sqrt {\frac {2}{1-\cos \theta }}}\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.6em 0.6em 0.633em 0.6em 0.3em 0.3em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>sgn</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mn>2</mn> </mfrac> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>sgn</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mn>2</mn> </mfrac> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>csc</mi> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <mi>cot</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>sec</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </msqrt> </mrow> </mrow> <mrow> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>cot</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>csc</mi> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <mi>cot</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>sec</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>sgn</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn>2</mn> <mrow> <mn>1</mn> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>csc</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>sgn</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn>2</mn> <mrow> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </msqrt> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sin {\frac {\theta }{2}}&=\operatorname {sgn} \left(\sin {\frac {\theta }{2}}\right){\sqrt {\frac {1-\cos \theta }{2}}}\\[3pt]\cos {\frac {\theta }{2}}&=\operatorname {sgn} \left(\cos {\frac {\theta }{2}}\right){\sqrt {\frac {1+\cos \theta }{2}}}\\[3pt]\tan {\frac {\theta }{2}}&={\frac {1-\cos \theta }{\sin \theta }}={\frac {\sin \theta }{1+\cos \theta }}=\csc \theta -\cot \theta ={\frac {\tan \theta }{1+\sec {\theta }}}\\[6mu]&=\operatorname {sgn}(\sin \theta ){\sqrt {\frac {1-\cos \theta }{1+\cos \theta }}}={\frac {-1+\operatorname {sgn}(\cos \theta ){\sqrt {1+\tan ^{2}\theta }}}{\tan \theta }}\\[3pt]\cot {\frac {\theta }{2}}&={\frac {1+\cos \theta }{\sin \theta }}={\frac {\sin \theta }{1-\cos \theta }}=\csc \theta +\cot \theta =\operatorname {sgn}(\sin \theta ){\sqrt {\frac {1+\cos \theta }{1-\cos \theta }}}\\\sec {\frac {\theta }{2}}&=\operatorname {sgn} \left(\cos {\frac {\theta }{2}}\right){\sqrt {\frac {2}{1+\cos \theta }}}\\\csc {\frac {\theta }{2}}&=\operatorname {sgn} \left(\sin {\frac {\theta }{2}}\right){\sqrt {\frac {2}{1-\cos \theta }}}\\\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d53d104c73c4b28f1dd7bfa5d2b0c7207c68cd87" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -24.204ex; margin-bottom: -0.301ex; width:70.887ex; height:50.176ex;" alt="{\displaystyle {\begin{aligned}\sin {\frac {\theta }{2}}&=\operatorname {sgn} \left(\sin {\frac {\theta }{2}}\right){\sqrt {\frac {1-\cos \theta }{2}}}\\[3pt]\cos {\frac {\theta }{2}}&=\operatorname {sgn} \left(\cos {\frac {\theta }{2}}\right){\sqrt {\frac {1+\cos \theta }{2}}}\\[3pt]\tan {\frac {\theta }{2}}&={\frac {1-\cos \theta }{\sin \theta }}={\frac {\sin \theta }{1+\cos \theta }}=\csc \theta -\cot \theta ={\frac {\tan \theta }{1+\sec {\theta }}}\\[6mu]&=\operatorname {sgn}(\sin \theta ){\sqrt {\frac {1-\cos \theta }{1+\cos \theta }}}={\frac {-1+\operatorname {sgn}(\cos \theta ){\sqrt {1+\tan ^{2}\theta }}}{\tan \theta }}\\[3pt]\cot {\frac {\theta }{2}}&={\frac {1+\cos \theta }{\sin \theta }}={\frac {\sin \theta }{1-\cos \theta }}=\csc \theta +\cot \theta =\operatorname {sgn}(\sin \theta ){\sqrt {\frac {1+\cos \theta }{1-\cos \theta }}}\\\sec {\frac {\theta }{2}}&=\operatorname {sgn} \left(\cos {\frac {\theta }{2}}\right){\sqrt {\frac {2}{1+\cos \theta }}}\\\csc {\frac {\theta }{2}}&=\operatorname {sgn} \left(\sin {\frac {\theta }{2}}\right){\sqrt {\frac {2}{1-\cos \theta }}}\\\end{aligned}}}"> <a href="#cite_note-ReferenceA-23">[23]</a><a href="#cite_note-mathworld_half_angle-24">[24]</a> Ayrıca <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\tan {\frac {\eta \pm \theta }{2}}&={\frac {\sin \eta \pm \sin \theta }{\cos \eta +\cos \theta }}\\[3pt]\tan \left({\frac {\theta }{2}}+{\frac {\pi }{4}}\right)&=\sec \theta +\tan \theta \\[3pt]{\sqrt {\frac {1-\sin \theta }{1+\sin \theta }}}&={\frac {\left|1-\tan {\frac {\theta }{2}}\right|}{\left|1+\tan {\frac {\theta }{2}}\right|}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.6em 0.6em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>η</mi> <mo>±</mo> <mi>θ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>η</mi> <mo>±</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>η</mi> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>tan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>sec</mi> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </msqrt> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>|</mo> <mrow> <mn>1</mn> <mo>−</mo> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>|</mo> </mrow> <mrow> <mo>|</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>|</mo> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\tan {\frac {\eta \pm \theta }{2}}&={\frac {\sin \eta \pm \sin \theta }{\cos \eta +\cos \theta }}\\[3pt]\tan \left({\frac {\theta }{2}}+{\frac {\pi }{4}}\right)&=\sec \theta +\tan \theta \\[3pt]{\sqrt {\frac {1-\sin \theta }{1+\sin \theta }}}&={\frac {\left|1-\tan {\frac {\theta }{2}}\right|}{\left|1+\tan {\frac {\theta }{2}}\right|}}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/639270245e163f1afa8682ac8aa12086db43dce1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.505ex; width:30.571ex; height:22.176ex;" alt="{\displaystyle {\begin{aligned}\tan {\frac {\eta \pm \theta }{2}}&={\frac {\sin \eta \pm \sin \theta }{\cos \eta +\cos \theta }}\\[3pt]\tan \left({\frac {\theta }{2}}+{\frac {\pi }{4}}\right)&=\sec \theta +\tan \theta \\[3pt]{\sqrt {\frac {1-\sin \theta }{1+\sin \theta }}}&={\frac {\left|1-\tan {\frac {\theta }{2}}\right|}{\left|1+\tan {\frac {\theta }{2}}\right|}}\end{aligned}}}"> <div class="mw-heading mw-heading3"><h3 id="Tablo">Tablo</h3>[<a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&veaction=edit&section=18" title="Değiştirilen bölüm: Tablo" class="mw-editsection-visualeditor">değiştir</a> | <a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&action=edit&section=18" title="Bölümün kaynak kodunu değiştir: Tablo">kaynağı değiştir</a>]</div> <div role="note" class="hatnote navigation-not-searchable">Ayrıca bakınız: <a href="/w/index.php?title=Tanjant_yar%C4%B1m_a%C3%A7%C4%B1_form%C3%BCl%C3%BC&action=edit&redlink=1" class="new" title="Tanjant yarım açı formülü (sayfa mevcut değil)">Tanjant yarım açı formülü</a></div> Bunlar, toplam ve fark özdeşlikleri ya da çoklu açı formülleri kullanılarak gösterilebilir. <div style="overflow-x:auto;"> <table class="wikitable" style="background-color:var(--background-color-base);"> <tbody><tr> <th></th> <th>Sinüs</th> <th>Kosinüs</th> <th>Tanjant</th> <th>Kotanjant </th></tr> <tr> <th>Çift açı formülü<a href="#cite_note-25">[25]</a><a href="#cite_note-mathworld_double_angle-26">[26]</a> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\sin(2\theta )&=2\sin \theta \cos \theta \ \\&={\frac {2\tan \theta }{1+\tan ^{2}\theta }}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mtext> </mtext> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sin(2\theta )&=2\sin \theta \cos \theta \ \\&={\frac {2\tan \theta }{1+\tan ^{2}\theta }}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a66bfbb4ed79110b8ce3fb784f2143328da44bc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:22.207ex; height:8.843ex;" alt="{\displaystyle {\begin{aligned}\sin(2\theta )&=2\sin \theta \cos \theta \ \\&={\frac {2\tan \theta }{1+\tan ^{2}\theta }}\end{aligned}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\cos(2\theta )&=\cos ^{2}\theta -\sin ^{2}\theta \\&=2\cos ^{2}\theta -1\\&=1-2\sin ^{2}\theta \\&={\frac {1-\tan ^{2}\theta }{1+\tan ^{2}\theta }}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\cos(2\theta )&=\cos ^{2}\theta -\sin ^{2}\theta \\&=2\cos ^{2}\theta -1\\&=1-2\sin ^{2}\theta \\&={\frac {1-\tan ^{2}\theta }{1+\tan ^{2}\theta }}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49a096f6b0f0b866d9286e274ce5404e52aeb902" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.338ex; width:24.894ex; height:15.843ex;" alt="{\displaystyle {\begin{aligned}\cos(2\theta )&=\cos ^{2}\theta -\sin ^{2}\theta \\&=2\cos ^{2}\theta -1\\&=1-2\sin ^{2}\theta \\&={\frac {1-\tan ^{2}\theta }{1+\tan ^{2}\theta }}\end{aligned}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan(2\theta )={\frac {2\tan \theta }{1-\tan ^{2}\theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan(2\theta )={\frac {2\tan \theta }{1-\tan ^{2}\theta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf0463617c8fb39ff3d7aa1f54294e865be07dbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.251ex; height:5.843ex;" alt="{\displaystyle \tan(2\theta )={\frac {2\tan \theta }{1-\tan ^{2}\theta }}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cot(2\theta )={\frac {\cot ^{2}\theta -1}{2\cot \theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>cot</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mn>2</mn> <mi>cot</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cot(2\theta )={\frac {\cot ^{2}\theta -1}{2\cot \theta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ff367e8c14d2b88ca2ce13446f0bfeb4f5bcca3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.73ex; height:5.843ex;" alt="{\displaystyle \cot(2\theta )={\frac {\cot ^{2}\theta -1}{2\cot \theta }}}"> </td></tr> <tr> <th>Üç kat açı formülü<a href="#cite_note-mathworld_multiple_angle-18">[18]</a><a href="#cite_note-Stegun_p._72,_4-27">[27]</a> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\sin(3\theta )&=-\sin ^{3}\theta +3\cos ^{2}\theta \sin \theta \\&=-4\sin ^{3}\theta +3\sin \theta \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <mn>3</mn> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mn>4</mn> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <mn>3</mn> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sin(3\theta )&=-\sin ^{3}\theta +3\cos ^{2}\theta \sin \theta \\&=-4\sin ^{3}\theta +3\sin \theta \end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80c8466c9daeb6ed86e257b9d93221185ce3e064" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:33.104ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}\sin(3\theta )&=-\sin ^{3}\theta +3\cos ^{2}\theta \sin \theta \\&=-4\sin ^{3}\theta +3\sin \theta \end{aligned}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\cos(3\theta )&=\cos ^{3}\theta -3\sin ^{2}\theta \cos \theta \\&=4\cos ^{3}\theta -3\cos \theta \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <mn>3</mn> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>4</mn> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <mn>3</mn> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\cos(3\theta )&=\cos ^{3}\theta -3\sin ^{2}\theta \cos \theta \\&=4\cos ^{3}\theta -3\cos \theta \end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6106f9e626db50781cb8b37ebc89da65f4515e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:31.419ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}\cos(3\theta )&=\cos ^{3}\theta -3\sin ^{2}\theta \cos \theta \\&=4\cos ^{3}\theta -3\cos \theta \end{aligned}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan(3\theta )={\frac {3\tan \theta -\tan ^{3}\theta }{1-3\tan ^{2}\theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mn>3</mn> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan(3\theta )={\frac {3\tan \theta -\tan ^{3}\theta }{1-3\tan ^{2}\theta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d74700e57b121042bdab5df3419d6b4897e65be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.475ex; height:6.176ex;" alt="{\displaystyle \tan(3\theta )={\frac {3\tan \theta -\tan ^{3}\theta }{1-3\tan ^{2}\theta }}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cot(3\theta )={\frac {3\cot \theta -\cot ^{3}\theta }{1-3\cot ^{2}\theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mi>cot</mi> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <msup> <mi>cot</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mn>3</mn> <msup> <mi>cot</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cot(3\theta )={\frac {3\cot \theta -\cot ^{3}\theta }{1-3\cot ^{2}\theta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c88c94eea3bd262ab04f5717b311e192a82529d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.695ex; height:6.176ex;" alt="{\displaystyle \cot(3\theta )={\frac {3\cot \theta -\cot ^{3}\theta }{1-3\cot ^{2}\theta }}}"> </td></tr> <tr> <th>Yarım açı formülü<a href="#cite_note-ReferenceA-23">[23]</a><a href="#cite_note-mathworld_half_angle-24">[24]</a> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&\sin {\frac {\theta }{2}}=\operatorname {sgn} \left(\sin {\frac {\theta }{2}}\right){\sqrt {\frac {1-\cos \theta }{2}}}\\\\&\left({\text{or }}\sin ^{2}{\frac {\theta }{2}}={\frac {1-\cos \theta }{2}}\right)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mi>sgn</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mn>2</mn> </mfrac> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd /> </mtr> <mtr> <mtd /> <mtd> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>or </mtext> </mrow> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&\sin {\frac {\theta }{2}}=\operatorname {sgn} \left(\sin {\frac {\theta }{2}}\right){\sqrt {\frac {1-\cos \theta }{2}}}\\\\&\left({\text{or }}\sin ^{2}{\frac {\theta }{2}}={\frac {1-\cos \theta }{2}}\right)\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/004d5b34a2d03c4390cadc441b59f1d13cacfbd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.01ex; margin-bottom: -0.328ex; width:32.876ex; height:15.843ex;" alt="{\displaystyle {\begin{aligned}&\sin {\frac {\theta }{2}}=\operatorname {sgn} \left(\sin {\frac {\theta }{2}}\right){\sqrt {\frac {1-\cos \theta }{2}}}\\\\&\left({\text{or }}\sin ^{2}{\frac {\theta }{2}}={\frac {1-\cos \theta }{2}}\right)\end{aligned}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&\cos {\frac {\theta }{2}}=\operatorname {sgn} \left(\cos {\frac {\theta }{2}}\right){\sqrt {\frac {1+\cos \theta }{2}}}\\\\&\left({\text{or }}\cos ^{2}{\frac {\theta }{2}}={\frac {1+\cos \theta }{2}}\right)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mi>sgn</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mn>2</mn> </mfrac> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd /> </mtr> <mtr> <mtd /> <mtd> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>or </mtext> </mrow> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&\cos {\frac {\theta }{2}}=\operatorname {sgn} \left(\cos {\frac {\theta }{2}}\right){\sqrt {\frac {1+\cos \theta }{2}}}\\\\&\left({\text{or }}\cos ^{2}{\frac {\theta }{2}}={\frac {1+\cos \theta }{2}}\right)\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/513559b4a16e8a492365ceba9ea3b05bc1d524b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.01ex; margin-bottom: -0.328ex; width:33.387ex; height:15.843ex;" alt="{\displaystyle {\begin{aligned}&\cos {\frac {\theta }{2}}=\operatorname {sgn} \left(\cos {\frac {\theta }{2}}\right){\sqrt {\frac {1+\cos \theta }{2}}}\\\\&\left({\text{or }}\cos ^{2}{\frac {\theta }{2}}={\frac {1+\cos \theta }{2}}\right)\end{aligned}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\tan {\frac {\theta }{2}}&=\csc \theta -\cot \theta \\&=\pm \,{\sqrt {\frac {1-\cos \theta }{1+\cos \theta }}}\\[3pt]&={\frac {\sin \theta }{1+\cos \theta }}\\[3pt]&={\frac {1-\cos \theta }{\sin \theta }}\\[5pt]\tan {\frac {\eta +\theta }{2}}&={\frac {\sin \eta +\sin \theta }{\cos \eta +\cos \theta }}\\[5pt]\tan \left({\frac {\theta }{2}}+{\frac {\pi }{4}}\right)&=\sec \theta +\tan \theta \\[5pt]{\sqrt {\frac {1-\sin \theta }{1+\sin \theta }}}&={\frac {\left|1-\tan {\frac {\theta }{2}}\right|}{\left|1+\tan {\frac {\theta }{2}}\right|}}\\[5pt]\tan {\frac {\theta }{2}}&={\frac {\tan \theta }{1+{\sqrt {1+\tan ^{2}\theta }}}}\\&{\text{for }}\theta \in \left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt 0.6em 0.6em 0.8em 0.8em 0.8em 0.8em 0.3em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>csc</mi> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <mi>cot</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo>±</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>η</mi> <mo>+</mo> <mi>θ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>η</mi> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>η</mi> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>tan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>sec</mi> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </msqrt> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>|</mo> <mrow> <mn>1</mn> <mo>−</mo> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>|</mo> </mrow> <mrow> <mo>|</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>|</mo> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>for </mtext> </mrow> <mi>θ</mi> <mo>∈</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\tan {\frac {\theta }{2}}&=\csc \theta -\cot \theta \\&=\pm \,{\sqrt {\frac {1-\cos \theta }{1+\cos \theta }}}\\[3pt]&={\frac {\sin \theta }{1+\cos \theta }}\\[3pt]&={\frac {1-\cos \theta }{\sin \theta }}\\[5pt]\tan {\frac {\eta +\theta }{2}}&={\frac {\sin \eta +\sin \theta }{\cos \eta +\cos \theta }}\\[5pt]\tan \left({\frac {\theta }{2}}+{\frac {\pi }{4}}\right)&=\sec \theta +\tan \theta \\[5pt]{\sqrt {\frac {1-\sin \theta }{1+\sin \theta }}}&={\frac {\left|1-\tan {\frac {\theta }{2}}\right|}{\left|1+\tan {\frac {\theta }{2}}\right|}}\\[5pt]\tan {\frac {\theta }{2}}&={\frac {\tan \theta }{1+{\sqrt {1+\tan ^{2}\theta }}}}\\&{\text{for }}\theta \in \left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cab50a58d64b8c3341fb4f7646888bbbc67ea0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -30.505ex; width:34.695ex; height:62.176ex;" alt="{\displaystyle {\begin{aligned}\tan {\frac {\theta }{2}}&=\csc \theta -\cot \theta \\&=\pm \,{\sqrt {\frac {1-\cos \theta }{1+\cos \theta }}}\\[3pt]&={\frac {\sin \theta }{1+\cos \theta }}\\[3pt]&={\frac {1-\cos \theta }{\sin \theta }}\\[5pt]\tan {\frac {\eta +\theta }{2}}&={\frac {\sin \eta +\sin \theta }{\cos \eta +\cos \theta }}\\[5pt]\tan \left({\frac {\theta }{2}}+{\frac {\pi }{4}}\right)&=\sec \theta +\tan \theta \\[5pt]{\sqrt {\frac {1-\sin \theta }{1+\sin \theta }}}&={\frac {\left|1-\tan {\frac {\theta }{2}}\right|}{\left|1+\tan {\frac {\theta }{2}}\right|}}\\[5pt]\tan {\frac {\theta }{2}}&={\frac {\tan \theta }{1+{\sqrt {1+\tan ^{2}\theta }}}}\\&{\text{for }}\theta \in \left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)\end{aligned}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\cot {\frac {\theta }{2}}&=\csc \theta +\cot \theta \\&=\pm \,{\sqrt {\frac {1+\cos \theta }{1-\cos \theta }}}\\[3pt]&={\frac {\sin \theta }{1-\cos \theta }}\\[4pt]&={\frac {1+\cos \theta }{\sin \theta }}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt 0.6em 0.7em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>cot</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>csc</mi> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <mi>cot</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo>±</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\cot {\frac {\theta }{2}}&=\csc \theta +\cot \theta \\&=\pm \,{\sqrt {\frac {1+\cos \theta }{1-\cos \theta }}}\\[3pt]&={\frac {\sin \theta }{1-\cos \theta }}\\[4pt]&={\frac {1+\cos \theta }{\sin \theta }}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c9be831a310f112ffc6e5790bc705dc7b070b87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.243ex; margin-bottom: -0.261ex; width:23.282ex; height:26.176ex;" alt="{\displaystyle {\begin{aligned}\cot {\frac {\theta }{2}}&=\csc \theta +\cot \theta \\&=\pm \,{\sqrt {\frac {1+\cos \theta }{1-\cos \theta }}}\\[3pt]&={\frac {\sin \theta }{1-\cos \theta }}\\[4pt]&={\frac {1+\cos \theta }{\sin \theta }}\end{aligned}}}"> </td></tr></tbody></table> </div> Sinüs ve kosinüs için üç kat açı formülünün yalnızca tek bir fonksiyonun kuvvetlerini içermesi, <a href="/w/index.php?title=A%C3%A7%C4%B1y%C4%B1_%C3%BC%C3%A7e_b%C3%B6lme&action=edit&redlink=1" class="new" title="Açıyı üçe bölme (sayfa mevcut değil)">açıyı üçe bölmenin</a> <a href="/wiki/Pergel_ve_%C3%A7izgilik_%C3%A7izimleri" title="Pergel ve çizgilik çizimleri">pergel ve düzeç konstrüksiyonu</a> geometrik problemini <a href="/wiki/%C3%9C%C3%A7%C3%BCnc%C3%BC_dereceden_denklemler" title="Üçüncü dereceden denklemler">kübik denklem</a> çözme cebirsel problemiyle ilişkilendirmeye izin verir, bu da <a href="/wiki/Alan_(matematik)" title="Alan (matematik)">alan teorisi</a> tarafından verilen araçları kullanarak üçlemenin genel olarak imkansız olduğunu kanıtlamaya izin verir.[<a href="/wiki/Vikipedi:Kaynak_g%C3%B6sterme" title="Vikipedi:Kaynak gösterme">kaynak belirtilmeli</a>] Üçte bir açı için trigonometrik özdeşlikleri hesaplamak amacıyla bir formül mevcuttur, ancak bu 4x3 − 3x + d = 0 <a href="/wiki/%C3%9C%C3%A7%C3%BCnc%C3%BC_dereceden_denklemler" title="Üçüncü dereceden denklemler">kübik denklemin</a> sıfırlarını yani köklerini bulmayı gerektirir, burada <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> kosinüs fonksiyonunun üçte birlik açıdaki değeri ve d kosinüs fonksiyonunun tam açıdaki bilinen değeridir. Bununla birlikte, bu denklemin <a href="/wiki/Diskriminant" title="Diskriminant">diskriminantı</a> pozitiftir, bu nedenle bu denklemin üç reel kökü vardır (bunlardan sadece biri üçte birlik açının kosinüsü için çözümdür). Bu çözümlerin hiçbiri <a href="/w/index.php?title=K%C3%BCp_k%C3%B6k&action=edit&redlink=1" class="new" title="Küp kök (sayfa mevcut değil)">küp köklerin</a> altında ara karmaşık sayılar kullandıkları için gerçek bir <a href="/wiki/Cebirsel_ifade" title="Cebirsel ifade">cebirsel ifadeye</a> indirgenemez. <div class="mw-heading mw-heading2"><h2 id="Kuvvet_indirgeme_formülleri">Kuvvet indirgeme formülleri</h2>[<a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&veaction=edit&section=19" title="Değiştirilen bölüm: Kuvvet indirgeme formülleri" class="mw-editsection-visualeditor">değiştir</a> | <a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&action=edit&section=19" title="Bölümün kaynak kodunu değiştir: Kuvvet indirgeme formülleri">kaynağı değiştir</a>]</div> Kosinüs çift açı formülünün ikinci ve üçüncü versiyonlarının çözülmesiyle elde edilir. <table class="wikitable"> <tbody><tr> <th>Sinüs </th> <th>Kosinüs </th> <th>Diğer </th></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin ^{2}\theta ={\frac {1-\cos(2\theta )}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin ^{2}\theta ={\frac {1-\cos(2\theta )}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c92b9e121ea0e8ae53b3e87b5335d27c8494468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.498ex; height:5.676ex;" alt="{\displaystyle \sin ^{2}\theta ={\frac {1-\cos(2\theta )}{2}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos ^{2}\theta ={\frac {1+\cos(2\theta )}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos ^{2}\theta ={\frac {1+\cos(2\theta )}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f1d849df461d73e205b144bc58775502869e132" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.754ex; height:5.676ex;" alt="{\displaystyle \cos ^{2}\theta ={\frac {1+\cos(2\theta )}{2}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin ^{2}\theta \cos ^{2}\theta ={\frac {1-\cos(4\theta )}{8}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>4</mn> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mn>8</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin ^{2}\theta \cos ^{2}\theta ={\frac {1-\cos(4\theta )}{8}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2008e1f7e310a419a03055e69b35650cb97cbe7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:26.528ex; height:5.676ex;" alt="{\displaystyle \sin ^{2}\theta \cos ^{2}\theta ={\frac {1-\cos(4\theta )}{8}}}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin ^{3}\theta ={\frac {3\sin \theta -\sin(3\theta )}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mn>4</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin ^{3}\theta ={\frac {3\sin \theta -\sin(3\theta )}{4}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17cae3a508d2f143f77d0cf1352d4a4116246525" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:24.963ex; height:5.676ex;" alt="{\displaystyle \sin ^{3}\theta ={\frac {3\sin \theta -\sin(3\theta )}{4}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos ^{3}\theta ={\frac {3\cos \theta +\cos(3\theta )}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mn>4</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos ^{3}\theta ={\frac {3\cos \theta +\cos(3\theta )}{4}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af7cf7208a3162c06a234c727c5e85865047d8be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.729ex; height:5.676ex;" alt="{\displaystyle \cos ^{3}\theta ={\frac {3\cos \theta +\cos(3\theta )}{4}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin ^{3}\theta \cos ^{3}\theta ={\frac {3\sin(2\theta )-\sin(6\theta )}{32}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>6</mn> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mn>32</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin ^{3}\theta \cos ^{3}\theta ={\frac {3\sin(2\theta )-\sin(6\theta )}{32}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d6d05a774b0ef8064de4af92f18fbb79c0fa6cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:33.578ex; height:5.676ex;" alt="{\displaystyle \sin ^{3}\theta \cos ^{3}\theta ={\frac {3\sin(2\theta )-\sin(6\theta )}{32}}}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin ^{4}\theta ={\frac {3-4\cos(2\theta )+\cos(4\theta )}{8}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mo>−</mo> <mn>4</mn> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>4</mn> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mn>8</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin ^{4}\theta ={\frac {3-4\cos(2\theta )+\cos(4\theta )}{8}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/426fdf897d222034f749310bcc05cf3cdd7aa738" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:32.061ex; height:5.676ex;" alt="{\displaystyle \sin ^{4}\theta ={\frac {3-4\cos(2\theta )+\cos(4\theta )}{8}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos ^{4}\theta ={\frac {3+4\cos(2\theta )+\cos(4\theta )}{8}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mo>+</mo> <mn>4</mn> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>4</mn> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mn>8</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos ^{4}\theta ={\frac {3+4\cos(2\theta )+\cos(4\theta )}{8}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4af78e15f6d43f36bbcef66ac91270faf5dff5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:32.317ex; height:5.676ex;" alt="{\displaystyle \cos ^{4}\theta ={\frac {3+4\cos(2\theta )+\cos(4\theta )}{8}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin ^{4}\theta \cos ^{4}\theta ={\frac {3-4\cos(4\theta )+\cos(8\theta )}{128}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mo>−</mo> <mn>4</mn> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>4</mn> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>8</mn> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mn>128</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin ^{4}\theta \cos ^{4}\theta ={\frac {3-4\cos(4\theta )+\cos(8\theta )}{128}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4f8d997149837dde03b04d1025fb57905416812" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:38.091ex; height:5.676ex;" alt="{\displaystyle \sin ^{4}\theta \cos ^{4}\theta ={\frac {3-4\cos(4\theta )+\cos(8\theta )}{128}}}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin ^{5}\theta ={\frac {10\sin \theta -5\sin(3\theta )+\sin(5\theta )}{16}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>10</mn> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <mn>5</mn> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>5</mn> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mn>16</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin ^{5}\theta ={\frac {10\sin \theta -5\sin(3\theta )+\sin(5\theta )}{16}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9a2af527323be690168a930dfb5f7e6202955f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:37.433ex; height:5.676ex;" alt="{\displaystyle \sin ^{5}\theta ={\frac {10\sin \theta -5\sin(3\theta )+\sin(5\theta )}{16}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos ^{5}\theta ={\frac {10\cos \theta +5\cos(3\theta )+\cos(5\theta )}{16}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>10</mn> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <mn>5</mn> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>5</mn> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mn>16</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos ^{5}\theta ={\frac {10\cos \theta +5\cos(3\theta )+\cos(5\theta )}{16}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdf873e929be78d09be981037e7d9609e1223c56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:38.455ex; height:5.676ex;" alt="{\displaystyle \cos ^{5}\theta ={\frac {10\cos \theta +5\cos(3\theta )+\cos(5\theta )}{16}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin ^{5}\theta \cos ^{5}\theta ={\frac {10\sin(2\theta )-5\sin(6\theta )+\sin(10\theta )}{512}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>10</mn> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>5</mn> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>6</mn> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>10</mn> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mn>512</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin ^{5}\theta \cos ^{5}\theta ={\frac {10\sin(2\theta )-5\sin(6\theta )+\sin(10\theta )}{512}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7277b387539636dd32d0b7f1ab13953650598d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:47.21ex; height:5.676ex;" alt="{\displaystyle \sin ^{5}\theta \cos ^{5}\theta ={\frac {10\sin(2\theta )-5\sin(6\theta )+\sin(10\theta )}{512}}}"> </td></tr></tbody></table> <div class="center"><div class="thumb tnone"> <div class="thumbinner" style="width:708px;"> <table style="background: transparent; border: 0; padding: 0; margin: 0;" cellspacing="0"> <tbody><tr> <td style="padding: 0; margin: 0;" class="thumbimage"><a href="/wiki/Dosya:Diagram_showing_how_to_derive_the_power_reduction_formula_for_cosine.svg" class="mw-file-description" title="Kosinüs kuvvet indirgeme formülü: açıklayıcı bir şekil. Kırmızı, turuncu ve mavi üçgenlerin hepsi benzerdir ve kırmızı ve turuncu üçgenler eştir. Mavi üçgenin '"`UNIQ--postMath-00000127-QINU`"' hipotenüsü '"`UNIQ--postMath-00000128-QINU`"' uzunluğuna sahiptir. '"`UNIQ--postMath-00000129-QINU`"' açısı '"`UNIQ--postMath-0000012A-QINU`"' olduğundan, bu üçgenin '"`UNIQ--postMath-0000012B-QINU`"' tabanı '"`UNIQ--postMath-0000012C-QINU`"' uzunluğundadır. Bu uzunluk aynı zamanda '"`UNIQ--postMath-0000012D-QINU`"' ve '"`UNIQ--postMath-0000012E-QINU`"' uzunluklarının toplamına eşittir, yani '"`UNIQ--postMath-0000012F-QINU`"'. Bu nedenle, '"`UNIQ--postMath-00000130-QINU`"'. Her iki tarafı '"`UNIQ--postMath-00000131-QINU`"' ile böldüğümüzde kosinüs için kuvvet indirgeme formülü elde edilir: '"`UNIQ--postMath-00000132-QINU`"' '"`UNIQ--postMath-00000133-QINU`"'. Kosinüs için yarım açı formülü '"`UNIQ--postMath-00000134-QINU`"' yerine '"`UNIQ--postMath-00000135-QINU`"' koyarak ve her iki tarafın karekökünü alarak elde edilebilir: '"`UNIQ--postMath-00000136-QINU`"'"><img alt="Kosinüs kuvvet indirgeme formülü: açıklayıcı bir şekil. Kırmızı, turuncu ve mavi üçgenlerin hepsi benzerdir ve kırmızı ve turuncu üçgenler eştir. Mavi üçgenin '"`UNIQ--postMath-00000127-QINU`"' hipotenüsü '"`UNIQ--postMath-00000128-QINU`"' uzunluğuna sahiptir. '"`UNIQ--postMath-00000129-QINU`"' açısı '"`UNIQ--postMath-0000012A-QINU`"' olduğundan, bu üçgenin '"`UNIQ--postMath-0000012B-QINU`"' tabanı '"`UNIQ--postMath-0000012C-QINU`"' uzunluğundadır. Bu uzunluk aynı zamanda '"`UNIQ--postMath-0000012D-QINU`"' ve '"`UNIQ--postMath-0000012E-QINU`"' uzunluklarının toplamına eşittir, yani '"`UNIQ--postMath-0000012F-QINU`"'. Bu nedenle, '"`UNIQ--postMath-00000130-QINU`"'. Her iki tarafı '"`UNIQ--postMath-00000131-QINU`"' ile böldüğümüzde kosinüs için kuvvet indirgeme formülü elde edilir: '"`UNIQ--postMath-00000132-QINU`"' '"`UNIQ--postMath-00000133-QINU`"'. Kosinüs için yarım açı formülü '"`UNIQ--postMath-00000134-QINU`"' yerine '"`UNIQ--postMath-00000135-QINU`"' koyarak ve her iki tarafın karekökünü alarak elde edilebilir: '"`UNIQ--postMath-00000136-QINU`"'" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/88/Diagram_showing_how_to_derive_the_power_reduction_formula_for_cosine.svg/350px-Diagram_showing_how_to_derive_the_power_reduction_formula_for_cosine.svg.png" decoding="async" width="350" height="267" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/88/Diagram_showing_how_to_derive_the_power_reduction_formula_for_cosine.svg/525px-Diagram_showing_how_to_derive_the_power_reduction_formula_for_cosine.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/88/Diagram_showing_how_to_derive_the_power_reduction_formula_for_cosine.svg/700px-Diagram_showing_how_to_derive_the_power_reduction_formula_for_cosine.svg.png 2x" data-file-width="720" data-file-height="550" /></a> </td> <td style="padding: 0; margin: 0; border: 0; width: 2px"> </td> <td style="padding: 0; margin: 0;" class="thumbimage"><a href="/wiki/Dosya:Diagram_showing_how_to_derive_the_power_reducing_formula_for_sine.svg" class="mw-file-description" title="Kosinüs kuvvet indirgeme formülü: açıklayıcı bir şekil. Kırmızı, turuncu ve mavi üçgenlerin hepsi benzerdir ve kırmızı ve turuncu üçgenler eştir. Mavi üçgenin '"`UNIQ--postMath-00000127-QINU`"' hipotenüsü '"`UNIQ--postMath-00000128-QINU`"' uzunluğuna sahiptir. '"`UNIQ--postMath-00000129-QINU`"' açısı '"`UNIQ--postMath-0000012A-QINU`"' olduğundan, bu üçgenin '"`UNIQ--postMath-0000012B-QINU`"' tabanı '"`UNIQ--postMath-0000012C-QINU`"' uzunluğundadır. Bu uzunluk aynı zamanda '"`UNIQ--postMath-0000012D-QINU`"' ve '"`UNIQ--postMath-0000012E-QINU`"' uzunluklarının toplamına eşittir, yani '"`UNIQ--postMath-0000012F-QINU`"'. Bu nedenle, '"`UNIQ--postMath-00000130-QINU`"'. Her iki tarafı '"`UNIQ--postMath-00000131-QINU`"' ile böldüğümüzde kosinüs için kuvvet indirgeme formülü elde edilir: '"`UNIQ--postMath-00000132-QINU`"' '"`UNIQ--postMath-00000133-QINU`"'. Kosinüs için yarım açı formülü '"`UNIQ--postMath-00000134-QINU`"' yerine '"`UNIQ--postMath-00000135-QINU`"' koyarak ve her iki tarafın karekökünü alarak elde edilebilir: '"`UNIQ--postMath-00000136-QINU`"'"><img alt="Kosinüs kuvvet indirgeme formülü: açıklayıcı bir şekil. Kırmızı, turuncu ve mavi üçgenlerin hepsi benzerdir ve kırmızı ve turuncu üçgenler eştir. Mavi üçgenin '"`UNIQ--postMath-00000127-QINU`"' hipotenüsü '"`UNIQ--postMath-00000128-QINU`"' uzunluğuna sahiptir. '"`UNIQ--postMath-00000129-QINU`"' açısı '"`UNIQ--postMath-0000012A-QINU`"' olduğundan, bu üçgenin '"`UNIQ--postMath-0000012B-QINU`"' tabanı '"`UNIQ--postMath-0000012C-QINU`"' uzunluğundadır. Bu uzunluk aynı zamanda '"`UNIQ--postMath-0000012D-QINU`"' ve '"`UNIQ--postMath-0000012E-QINU`"' uzunluklarının toplamına eşittir, yani '"`UNIQ--postMath-0000012F-QINU`"'. Bu nedenle, '"`UNIQ--postMath-00000130-QINU`"'. Her iki tarafı '"`UNIQ--postMath-00000131-QINU`"' ile böldüğümüzde kosinüs için kuvvet indirgeme formülü elde edilir: '"`UNIQ--postMath-00000132-QINU`"' '"`UNIQ--postMath-00000133-QINU`"'. Kosinüs için yarım açı formülü '"`UNIQ--postMath-00000134-QINU`"' yerine '"`UNIQ--postMath-00000135-QINU`"' koyarak ve her iki tarafın karekökünü alarak elde edilebilir: '"`UNIQ--postMath-00000136-QINU`"'" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Diagram_showing_how_to_derive_the_power_reducing_formula_for_sine.svg/350px-Diagram_showing_how_to_derive_the_power_reducing_formula_for_sine.svg.png" decoding="async" width="350" height="350" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Diagram_showing_how_to_derive_the_power_reducing_formula_for_sine.svg/525px-Diagram_showing_how_to_derive_the_power_reducing_formula_for_sine.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Diagram_showing_how_to_derive_the_power_reducing_formula_for_sine.svg/700px-Diagram_showing_how_to_derive_the_power_reducing_formula_for_sine.svg.png 2x" data-file-width="700" data-file-height="700" /></a> </td></tr> <tr> <td style="padding: 0; margin: 0; border: 0;"><div class="thumbcaption"><div class="center" style="width:auto; margin-left:auto; margin-right:auto;">Kosinüs kuvvet indirgeme formülü: açıklayıcı bir şekil. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r33633371">Kırmızı, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r33633371">turuncu ve <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r33633371">mavi üçgenlerin hepsi benzerdir ve <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r33633371">kırmızı ve <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r33633371">turuncu üçgenler eştir. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r33633371">Mavi üçgenin <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {AD}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>D</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {AD}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/baafce21c5650908c09e3ee108d3c07992be82d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.782ex; height:3.009ex;" alt="{\displaystyle {\overline {AD}}}"> hipotenüsü <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\cos \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\cos \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4924df8dd90f4a29430cd8b9e1113e32139741bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.138ex; height:2.176ex;" alt="{\displaystyle 2\cos \theta }"> uzunluğuna sahiptir. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \angle DAE}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∠</mi> <mi>D</mi> <mi>A</mi> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \angle DAE}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d59bc473178744f0b189f30d6a35eb990d89a79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle \angle DAE}"> açısı <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"> olduğundan, bu üçgenin <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {AE}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>E</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {AE}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40c489234007099c91162b222a127b04f2ba081e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.673ex; height:3.009ex;" alt="{\displaystyle {\overline {AE}}}"> tabanı <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\cos ^{2}\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\cos ^{2}\theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/508ecc2fce98d61079e496ea3642f7eb1d6a03cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.192ex; height:2.676ex;" alt="{\displaystyle 2\cos ^{2}\theta }"> uzunluğundadır. Bu uzunluk aynı zamanda <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {BD}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>D</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {BD}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c71fd62ec7ad127356017d4a62c77e118fd00d63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.803ex; height:3.009ex;" alt="{\displaystyle {\overline {BD}}}"> ve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {AF}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>F</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {AF}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4a2a05a6abaf851dbf5ac3a1b4206b9d38dd8d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.759ex; height:3.009ex;" alt="{\displaystyle {\overline {AF}}}"> uzunluklarının toplamına eşittir, yani <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+\cos(2\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+\cos(2\theta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a04f7a5a73831d567bb59e0060f625b01543868" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.176ex; height:2.843ex;" alt="{\displaystyle 1+\cos(2\theta )}">. Bu nedenle, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\cos ^{2}\theta =1+\cos(2\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\cos ^{2}\theta =1+\cos(2\theta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea0da0158b80598aeaea0efeabf8444a622052b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.467ex; height:3.176ex;" alt="{\displaystyle 2\cos ^{2}\theta =1+\cos(2\theta )}">. Her iki tarafı <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}"> ile böldüğümüzde kosinüs için kuvvet indirgeme formülü elde edilir: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos ^{2}\theta =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos ^{2}\theta =}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4218b707d20b7cdbe94a967d71f20f880667baf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.096ex; height:2.676ex;" alt="{\displaystyle \cos ^{2}\theta =}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\frac {1}{2}}(1+\cos(2\theta ))"> <semantics> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>θ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> <annotation encoding="application/x-tex">{\frac {1}{2}}(1+\cos(2\theta ))</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/411bc088de9e52e7631e7ce3170842179381cea7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.984ex; height:5.176ex;" alt="{\frac {1}{2}}(1+\cos(2\theta ))">. Kosinüs için yarım açı formülü <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"> yerine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta /2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta /2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74af4b205463ab5f0642165e85d0b372ff42d442" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.415ex; height:2.843ex;" alt="{\displaystyle \theta /2}"> koyarak ve her iki tarafın karekökünü alarak elde edilebilir: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cos \left(\theta /2\right)=\pm {\sqrt {\left(1+\cos \theta \right)/2}}."> <semantics> <mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </msqrt> </mrow> <mo>.</mo> </mrow> <annotation encoding="application/x-tex">\cos \left(\theta /2\right)=\pm {\sqrt {\left(1+\cos \theta \right)/2}}.</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1641d905abd6a325c5c79a0940282f715aa16d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:29.326ex; height:4.843ex;" alt="\cos \left(\theta /2\right)=\pm {\sqrt {\left(1+\cos \theta \right)/2}}."></div></div> </td> <td style="padding: 0; margin: 0; border: 0;"> </td> <td style="padding: 0; margin: 0; border: 0;"><div class="thumbcaption"><div class="center" style="width:auto; margin-left:auto; margin-right:auto;">Sinüs kuvvet indirgeme formülü: açıklayıcı bir şekil. Gölgeli <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r33633371">mavi ve <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r33633371">yeşil üçgenler ile <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r33633371">kırmızı çizgili <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle EBD}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mi>B</mi> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle EBD}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/374c068ede71e67133333d6f99be1d50a2cea643" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.464ex; height:2.176ex;" alt="{\displaystyle EBD}"> üçgeni dik açılı ve benzerdir ve hepsi <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"> açısını içerir. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r33633371">Kırmızı çizgili üçgenin <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {BD}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>D</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {BD}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c71fd62ec7ad127356017d4a62c77e118fd00d63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.803ex; height:3.009ex;" alt="{\displaystyle {\overline {BD}}}"> hipotenüsünün uzunluğu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\sin \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\sin \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a21b92497e569e3347987b72eaefc7d71d08da6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.883ex; height:2.176ex;" alt="{\displaystyle 2\sin \theta }">, dolayısıyla <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {DE}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>D</mi> <mi>E</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {DE}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a4fcfc4373cde19c969df60998941ce1e27ac57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.854ex; height:3.009ex;" alt="{\displaystyle {\overline {DE}}}"> kenarının uzunluğu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\sin ^{2}\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\sin ^{2}\theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dea5a587d6b2e56728c7dece2bff1084b5b0b07f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.937ex; height:2.676ex;" alt="{\displaystyle 2\sin ^{2}\theta }">'dır. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {AE}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>E</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {AE}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40c489234007099c91162b222a127b04f2ba081e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.673ex; height:3.009ex;" alt="{\displaystyle {\overline {AE}}}"> doğru parçasının uzunluğu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos 2\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mn>2</mn> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos 2\theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ddb1129e0b9873a5ac1b5c7d05be9b9654b8989" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.751ex; height:2.176ex;" alt="{\displaystyle \cos 2\theta }"> ve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {AE}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>E</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {AE}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40c489234007099c91162b222a127b04f2ba081e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.673ex; height:3.009ex;" alt="{\displaystyle {\overline {AE}}}"> ile <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {DE}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>D</mi> <mi>E</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {DE}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a4fcfc4373cde19c969df60998941ce1e27ac57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.854ex; height:3.009ex;" alt="{\displaystyle {\overline {DE}}}"> uzunluklarının toplamı <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {AD}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>D</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {AD}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/baafce21c5650908c09e3ee108d3c07992be82d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.782ex; height:3.009ex;" alt="{\displaystyle {\overline {AD}}}"> uzunluğuna eşittir, yani 1. Dolayısıyla, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos 2\theta +2\sin ^{2}\theta =1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mn>2</mn> <mi>θ</mi> <mo>+</mo> <mn>2</mn> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos 2\theta +2\sin ^{2}\theta =1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ac8a702e463cbc2c59a15e040516619e739dc88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:19.789ex; height:2.843ex;" alt="{\displaystyle \cos 2\theta +2\sin ^{2}\theta =1}">. Her iki taraftan <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos 2\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mn>2</mn> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos 2\theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ddb1129e0b9873a5ac1b5c7d05be9b9654b8989" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.751ex; height:2.176ex;" alt="{\displaystyle \cos 2\theta }"> çıkarıldığında ve 2'ye bölündüğünde sinüs için kuvvet indirgeme formülü elde edilir: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin ^{2}\theta =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin ^{2}\theta =}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb8bc63cbfe9ccee23b9629bd3b61ed49a24af10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.841ex; height:2.676ex;" alt="{\displaystyle \sin ^{2}\theta =}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\frac {1}{2}}(1-\cos(2\theta ))"> <semantics> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>θ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> <annotation encoding="application/x-tex">{\frac {1}{2}}(1-\cos(2\theta ))</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27aff7868d573d2b9d5fd6a6762f7e0ac7cbdf97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.984ex; height:5.176ex;" alt="{\frac {1}{2}}(1-\cos(2\theta ))">. Sinüs için yarım açı formülü, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"> yerine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta /2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta /2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74af4b205463ab5f0642165e85d0b372ff42d442" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.415ex; height:2.843ex;" alt="{\displaystyle \theta /2}"> koyarak ve her iki tarafın karekökünü alarak elde edilebilir: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin \left(\theta /2\right)=\pm {\sqrt {\left(1-\cos \theta \right)/2}}."> <semantics> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </msqrt> </mrow> <mo>.</mo> </mrow> <annotation encoding="application/x-tex">\sin \left(\theta /2\right)=\pm {\sqrt {\left(1-\cos \theta \right)/2}}.</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3a8173019aa1d9e98935bafe89a9d67841202b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:29.07ex; height:4.843ex;" alt="\sin \left(\theta /2\right)=\pm {\sqrt {\left(1-\cos \theta \right)/2}}."> Bu şeklin aynı zamanda <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {EB}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>E</mi> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {EB}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4cba6c29e1bbb53eea3c19877ea913484fb288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.655ex; height:3.009ex;" alt="{\displaystyle {\overline {EB}}}"> dikey doğru parçasında <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin 2\theta =2\sin \theta \cos \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mn>2</mn> <mi>θ</mi> <mo>=</mo> <mn>2</mn> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin 2\theta =2\sin \theta \cos \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6685b59661803da827e8894204d7ac653b0ab66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:19.453ex; height:2.176ex;" alt="{\displaystyle \sin 2\theta =2\sin \theta \cos \theta }"> olduğunu gösterdiğini unutmayın.</div></div> </td></tr></tbody></table> </div> </div></div> <div style="clear:both;"></div> Genel olarak <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efa733f6703578b0c3af870a3170b4ab0dd99c00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.333ex; height:2.176ex;" alt="{\displaystyle \sin \theta }"> veya <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/611e5c70de1d1cf4ebc3b70d2b5467f45d17a483" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.589ex; height:2.176ex;" alt="{\displaystyle \cos \theta }"> kuvvetleri cinsinden aşağıdaki doğrudur ve <a href="/wiki/De_Moivre_form%C3%BCl%C3%BC" title="De Moivre formülü">De Moivre formülü</a>, <a href="/wiki/Euler_form%C3%BCl%C3%BC" title="Euler formülü">Euler formülü</a> ve <a href="/wiki/Binom_teoremi" class="mw-redirect" title="Binom teoremi">binom teoremi</a> kullanılarak çıkarılabilir. <table class="wikitable"> <tbody><tr> <th scope="col">n  ...ise </th> <th scope="col"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos ^{n}\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos ^{n}\theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961fbc8180c54e3b9ddd5c2d7aed828e7334dcf0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.807ex; height:2.343ex;" alt="{\displaystyle \cos ^{n}\theta }"> </th> <th scope="col"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin ^{n}\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin ^{n}\theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cc1b00a3cf7cbd0353d011619f1acfaad44834a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.552ex; height:2.343ex;" alt="{\displaystyle \sin ^{n}\theta }"> </th></tr> <tr> <th scope="row">n tekse </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos ^{n}\theta ={\frac {2}{2^{n}}}\sum _{k=0}^{\frac {n-1}{2}}{\binom {n}{k}}\cos {{\big (}(n-2k)\theta {\big )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mi>k</mi> <mo stretchy="false">)</mo> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos ^{n}\theta ={\frac {2}{2^{n}}}\sum _{k=0}^{\frac {n-1}{2}}{\binom {n}{k}}\cos {{\big (}(n-2k)\theta {\big )}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c748a7f090e555eb84022ee7ca1e50c292b79a71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:36.591ex; height:8.843ex;" alt="{\displaystyle \cos ^{n}\theta ={\frac {2}{2^{n}}}\sum _{k=0}^{\frac {n-1}{2}}{\binom {n}{k}}\cos {{\big (}(n-2k)\theta {\big )}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin ^{n}\theta ={\frac {2}{2^{n}}}\sum _{k=0}^{\frac {n-1}{2}}(-1)^{\left({\frac {n-1}{2}}-k\right)}{\binom {n}{k}}\sin {{\big (}(n-2k)\theta {\big )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>k</mi> </mrow> <mo>)</mo> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mi>k</mi> <mo stretchy="false">)</mo> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin ^{n}\theta ={\frac {2}{2^{n}}}\sum _{k=0}^{\frac {n-1}{2}}(-1)^{\left({\frac {n-1}{2}}-k\right)}{\binom {n}{k}}\sin {{\big (}(n-2k)\theta {\big )}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f03ac7b9b634e09d81aa4feb9be37d0263e24dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:48.145ex; height:8.843ex;" alt="{\displaystyle \sin ^{n}\theta ={\frac {2}{2^{n}}}\sum _{k=0}^{\frac {n-1}{2}}(-1)^{\left({\frac {n-1}{2}}-k\right)}{\binom {n}{k}}\sin {{\big (}(n-2k)\theta {\big )}}}"> </td></tr> <tr> <th scope="row">n çiftse </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos ^{n}\theta ={\frac {1}{2^{n}}}{\binom {n}{\frac {n}{2}}}+{\frac {2}{2^{n}}}\sum _{k=0}^{{\frac {n}{2}}-1}{\binom {n}{k}}\cos {{\big (}(n-2k)\theta {\big )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mi>k</mi> <mo stretchy="false">)</mo> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos ^{n}\theta ={\frac {1}{2^{n}}}{\binom {n}{\frac {n}{2}}}+{\frac {2}{2^{n}}}\sum _{k=0}^{{\frac {n}{2}}-1}{\binom {n}{k}}\cos {{\big (}(n-2k)\theta {\big )}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99b71b87335d2265b64843ff3cb39fb0aa63ec24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:48.275ex; height:8.509ex;" alt="{\displaystyle \cos ^{n}\theta ={\frac {1}{2^{n}}}{\binom {n}{\frac {n}{2}}}+{\frac {2}{2^{n}}}\sum _{k=0}^{{\frac {n}{2}}-1}{\binom {n}{k}}\cos {{\big (}(n-2k)\theta {\big )}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin ^{n}\theta ={\frac {1}{2^{n}}}{\binom {n}{\frac {n}{2}}}+{\frac {2}{2^{n}}}\sum _{k=0}^{{\frac {n}{2}}-1}(-1)^{\left({\frac {n}{2}}-k\right)}{\binom {n}{k}}\cos {{\big (}(n-2k)\theta {\big )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>k</mi> </mrow> <mo>)</mo> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mi>k</mi> <mo stretchy="false">)</mo> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin ^{n}\theta ={\frac {1}{2^{n}}}{\binom {n}{\frac {n}{2}}}+{\frac {2}{2^{n}}}\sum _{k=0}^{{\frac {n}{2}}-1}(-1)^{\left({\frac {n}{2}}-k\right)}{\binom {n}{k}}\cos {{\big (}(n-2k)\theta {\big )}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e1783a72735db28df4f0807cf1627c0f6812c6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:58.379ex; height:8.509ex;" alt="{\displaystyle \sin ^{n}\theta ={\frac {1}{2^{n}}}{\binom {n}{\frac {n}{2}}}+{\frac {2}{2^{n}}}\sum _{k=0}^{{\frac {n}{2}}-1}(-1)^{\left({\frac {n}{2}}-k\right)}{\binom {n}{k}}\cos {{\big (}(n-2k)\theta {\big )}}}"> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Çarpım-toplam_ve_toplam-çarpım_özdeşlikleri">Çarpım-toplam ve toplam-çarpım özdeşlikleri</h2>[<a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&veaction=edit&section=20" title="Değiştirilen bölüm: Çarpım-toplam ve toplam-çarpım özdeşlikleri" class="mw-editsection-visualeditor">değiştir</a> | <a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&action=edit&section=20" title="Bölümün kaynak kodunu değiştir: Çarpım-toplam ve toplam-çarpım özdeşlikleri">kaynağı değiştir</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Dosya:Visual_proof_prosthaphaeresis_cosine_formula.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3c/Visual_proof_prosthaphaeresis_cosine_formula.svg/330px-Visual_proof_prosthaphaeresis_cosine_formula.svg.png" decoding="async" width="330" height="330" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3c/Visual_proof_prosthaphaeresis_cosine_formula.svg/495px-Visual_proof_prosthaphaeresis_cosine_formula.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3c/Visual_proof_prosthaphaeresis_cosine_formula.svg/660px-Visual_proof_prosthaphaeresis_cosine_formula.svg.png 2x" data-file-width="512" data-file-height="512" /></a><figcaption><div class="center" style="width:auto; margin-left:auto; margin-right:auto;">Bir <a href="/wiki/%C4%B0kizkenar_%C3%BC%C3%A7gen" title="İkizkenar üçgen">ikizkenar üçgen</a> kullanarak prostaphaeresis hesaplamaları için toplam ve fark-çarpım kosinüs özdeşliğinin kanıtı</div></figcaption></figure> Çarpım-toplam özdeşlikleri<a href="#cite_note-28">[28]</a> veya <a href="/w/index.php?title=Prosthaphaeresis&action=edit&redlink=1" class="new" title="Prosthaphaeresis (sayfa mevcut değil)">prosthaphaeresis</a> formülleri, <a href="#Açı_toplam_ve_fark_özdeşlikleri">açı toplam teoremleri</a> kullanılarak sağ tarafları genişletilerek kanıtlanabilir. Tarihsel olarak, bunlardan ilk dördü, astronomik hesaplamalar için kullanan <a href="/wiki/Johannes_Werner" title="Johannes Werner">Johannes Werner</a>'den sonra Werner formülleri olarak biliniyordu.<a href="#cite_note-29">[29]</a> Çarpım-toplam formüllerinin bir uygulaması için <a href="/wiki/Genlik_mod%C3%BClasyonu#Standart_GM'nin_basitleştirilmiş_analizi" title="Genlik modülasyonu">genlik modülasyonu</a> ve toplam-çarpım formüllerinin uygulamaları için <a href="/w/index.php?title=Vuru_(akustik)&action=edit&redlink=1" class="new" title="Vuru (akustik) (sayfa mevcut değil)">vuru (akustik)</a> ile <a href="/w/index.php?title=Faz_dedekt%C3%B6r%C3%BC&action=edit&redlink=1" class="new" title="Faz dedektörü (sayfa mevcut değil)">faz dedektörü</a> bölümlerine bakınız. <div class="mw-heading mw-heading3"><h3 id="Çarpım-toplam_özdeşlikleri">Çarpım-toplam özdeşlikleri</h3>[<a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&veaction=edit&section=21" title="Değiştirilen bölüm: Çarpım-toplam özdeşlikleri" class="mw-editsection-visualeditor">değiştir</a> | <a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&action=edit&section=21" title="Bölümün kaynak kodunu değiştir: Çarpım-toplam özdeşlikleri">kaynağı değiştir</a>]</div> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \theta \,\cos \varphi ={\cos(\theta -\varphi )+\cos(\theta +\varphi ) \over 2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace" /> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>−</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>+</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \theta \,\cos \varphi ={\cos(\theta -\varphi )+\cos(\theta +\varphi ) \over 2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/327a89f3259a0c3f78b22012dece9f0f2eb57d04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:37.899ex; height:5.676ex;" alt="{\displaystyle \cos \theta \,\cos \varphi ={\cos(\theta -\varphi )+\cos(\theta +\varphi ) \over 2}}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \theta \,\sin \varphi ={\cos(\theta -\varphi )-\cos(\theta +\varphi ) \over 2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace" /> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>−</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>+</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \theta \,\sin \varphi ={\cos(\theta -\varphi )-\cos(\theta +\varphi ) \over 2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c53f921fff8c794e4024978f37e4a0f9767b33b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:37.388ex; height:5.676ex;" alt="{\displaystyle \sin \theta \,\sin \varphi ={\cos(\theta -\varphi )-\cos(\theta +\varphi ) \over 2}}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \theta \,\cos \varphi ={\sin(\theta +\varphi )+\sin(\theta -\varphi ) \over 2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace" /> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>+</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>−</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \theta \,\cos \varphi ={\sin(\theta +\varphi )+\sin(\theta -\varphi ) \over 2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a9bee52c00a60123d0489c5a80a4818d3a35793" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:37.133ex; height:5.676ex;" alt="{\displaystyle \sin \theta \,\cos \varphi ={\sin(\theta +\varphi )+\sin(\theta -\varphi ) \over 2}}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \theta \,\sin \varphi ={\sin(\theta +\varphi )-\sin(\theta -\varphi ) \over 2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace" /> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>+</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>−</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \theta \,\sin \varphi ={\sin(\theta +\varphi )-\sin(\theta -\varphi ) \over 2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ad20a8a0cac1f030822e4721018d3707394ce15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:37.133ex; height:5.676ex;" alt="{\displaystyle \cos \theta \,\sin \varphi ={\sin(\theta +\varphi )-\sin(\theta -\varphi ) \over 2}}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan \theta \,\tan \varphi ={\frac {\cos(\theta -\varphi )-\cos(\theta +\varphi )}{\cos(\theta -\varphi )+\cos(\theta +\varphi )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace" /> <mi>tan</mi> <mo>⁡</mo> <mi>φ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>−</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>+</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>−</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>+</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan \theta \,\tan \varphi ={\frac {\cos(\theta -\varphi )-\cos(\theta +\varphi )}{\cos(\theta -\varphi )+\cos(\theta +\varphi )}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/979c1b61da456f6fff77bb97d24d28356acdce1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:38.396ex; height:6.509ex;" alt="{\displaystyle \tan \theta \,\tan \varphi ={\frac {\cos(\theta -\varphi )-\cos(\theta +\varphi )}{\cos(\theta -\varphi )+\cos(\theta +\varphi )}}}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan \theta \,\cot \varphi ={\frac {\sin(\theta +\varphi )+\sin(\theta -\varphi )}{\sin(\theta +\varphi )-\sin(\theta -\varphi )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace" /> <mi>cot</mi> <mo>⁡</mo> <mi>φ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>+</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>−</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>+</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>−</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan \theta \,\cot \varphi ={\frac {\sin(\theta +\varphi )+\sin(\theta -\varphi )}{\sin(\theta +\varphi )-\sin(\theta -\varphi )}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9aca2b2508b283e441e6aacd046d0a41ea69741" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:37.625ex; height:6.509ex;" alt="{\displaystyle \tan \theta \,\cot \varphi ={\frac {\sin(\theta +\varphi )+\sin(\theta -\varphi )}{\sin(\theta +\varphi )-\sin(\theta -\varphi )}}}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\prod _{k=1}^{n}\cos \theta _{k}&={\frac {1}{2^{n}}}\sum _{e\in S}\cos(e_{1}\theta _{1}+\cdots +e_{n}\theta _{n})\\[6pt]&{\text{burada }}e=(e_{1},\ldots ,e_{n})\in S=\{1,-1\}^{n}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.9em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> <mo>∈</mo> <mi>S</mi> </mrow> </munder> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>burada </mtext> </mrow> <mi>e</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>S</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <msup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\prod _{k=1}^{n}\cos \theta _{k}&={\frac {1}{2^{n}}}\sum _{e\in S}\cos(e_{1}\theta _{1}+\cdots +e_{n}\theta _{n})\\[6pt]&{\text{burada }}e=(e_{1},\ldots ,e_{n})\in S=\{1,-1\}^{n}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43ab464a5d09ae80c48997340721704036836b77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:49.238ex; height:11.509ex;" alt="{\displaystyle {\begin{aligned}\prod _{k=1}^{n}\cos \theta _{k}&={\frac {1}{2^{n}}}\sum _{e\in S}\cos(e_{1}\theta _{1}+\cdots +e_{n}\theta _{n})\\[6pt]&{\text{burada }}e=(e_{1},\ldots ,e_{n})\in S=\{1,-1\}^{n}\end{aligned}}}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \prod _{k=1}^{n}\sin \theta _{k}={\frac {(-1)^{\left\lfloor {\frac {n}{2}}\right\rfloor }}{2^{n}}}{\begin{cases}\displaystyle \sum _{e\in S}\cos(e_{1}\theta _{1}+\cdots +e_{n}\theta _{n})\prod _{j=1}^{n}e_{j}\;n\;{\text{çift ise}},\\\displaystyle \sum _{e\in S}\sin(e_{1}\theta _{1}+\cdots +e_{n}\theta _{n})\prod _{j=1}^{n}e_{j}\;n\;{\text{tek ise}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>⌊</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> <mo>⌋</mo> </mrow> </mrow> </msup> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> <mo>∈</mo> <mi>S</mi> </mrow> </munder> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mspace width="thickmathspace" /> <mi>n</mi> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>çift ise</mtext> </mrow> <mo>,</mo> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> <mo>∈</mo> <mi>S</mi> </mrow> </munder> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mspace width="thickmathspace" /> <mi>n</mi> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>tek ise</mtext> </mrow> </mstyle> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \prod _{k=1}^{n}\sin \theta _{k}={\frac {(-1)^{\left\lfloor {\frac {n}{2}}\right\rfloor }}{2^{n}}}{\begin{cases}\displaystyle \sum _{e\in S}\cos(e_{1}\theta _{1}+\cdots +e_{n}\theta _{n})\prod _{j=1}^{n}e_{j}\;n\;{\text{çift ise}},\\\displaystyle \sum _{e\in S}\sin(e_{1}\theta _{1}+\cdots +e_{n}\theta _{n})\prod _{j=1}^{n}e_{j}\;n\;{\text{tek ise}}\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47fa593a34d79492a7d9ab63e8fb6620d20a5d32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:65.8ex; height:14.509ex;" alt="{\displaystyle \prod _{k=1}^{n}\sin \theta _{k}={\frac {(-1)^{\left\lfloor {\frac {n}{2}}\right\rfloor }}{2^{n}}}{\begin{cases}\displaystyle \sum _{e\in S}\cos(e_{1}\theta _{1}+\cdots +e_{n}\theta _{n})\prod _{j=1}^{n}e_{j}\;n\;{\text{çift ise}},\\\displaystyle \sum _{e\in S}\sin(e_{1}\theta _{1}+\cdots +e_{n}\theta _{n})\prod _{j=1}^{n}e_{j}\;n\;{\text{tek ise}}\end{cases}}}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Toplam-çarpım_özdeşlikleri">Toplam-çarpım özdeşlikleri</h3>[<a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&veaction=edit&section=22" title="Değiştirilen bölüm: Toplam-çarpım özdeşlikleri" class="mw-editsection-visualeditor">değiştir</a> | <a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&action=edit&section=22" title="Bölümün kaynak kodunu değiştir: Toplam-çarpım özdeşlikleri">kaynağı değiştir</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Dosya:Diagram_illustrating_sum_to_product_identities_for_sine_and_cosine.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/Diagram_illustrating_sum_to_product_identities_for_sine_and_cosine.svg/350px-Diagram_illustrating_sum_to_product_identities_for_sine_and_cosine.svg.png" decoding="async" width="350" height="328" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/Diagram_illustrating_sum_to_product_identities_for_sine_and_cosine.svg/525px-Diagram_illustrating_sum_to_product_identities_for_sine_and_cosine.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/37/Diagram_illustrating_sum_to_product_identities_for_sine_and_cosine.svg/700px-Diagram_illustrating_sum_to_product_identities_for_sine_and_cosine.svg.png 2x" data-file-width="800" data-file-height="750" /></a><figcaption><div class="center" style="width:auto; margin-left:auto; margin-right:auto;">Sinüs ve kosinüs için toplam-çarpım özdeşliklerini gösteren şekil. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r33633371">Mavi dik açılı üçgen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"> açısına ve <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r33633371">kırmızı dik açılı üçgen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"> açısına sahiptir. Her ikisinin de hipotenüs uzunluğu 1'dir. Burada <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> ve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"> olarak adlandırılan yardımcı açılar, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=(\theta +\varphi )/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>+</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=(\theta +\varphi )/2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aea4ddc3bf78577357a0ec9ec3f6a93d99ae69a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:13.943ex; height:2.843ex;" alt="{\displaystyle p=(\theta +\varphi )/2}"> ve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=(\theta -\varphi )/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>−</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q=(\theta -\varphi )/2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/851045a6d2c31271e8f451bdd621d695e60d64ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.753ex; height:2.843ex;" alt="{\displaystyle q=(\theta -\varphi )/2}"> olacak şekilde oluşturulur. Bu nedenle, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =p+q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mi>p</mi> <mo>+</mo> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =p+q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a4547ab06f7720b7c9824de47fdc8f5c129dfa9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.268ex; height:2.509ex;" alt="{\displaystyle \theta =p+q}"> ve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi =p-q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>=</mo> <mi>p</mi> <mo>−</mo> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi =p-q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6642e89fe4128353f6f1087245887d2e0963108" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.698ex; height:2.509ex;" alt="{\displaystyle \varphi =p-q}">. Bu, her biri hipotenüs <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adf15d6740573f6e78cd246504c008e6996823a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.568ex; height:2.009ex;" alt="{\displaystyle \cos q}"> ve tabanlarında <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> açısı olan iki eş mor dış çizgi üçgenin <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle AFG}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>F</mi> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AFG}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c125a0a5581866bec98a6d1c8136e60f4dbb2169" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.311ex; height:2.176ex;" alt="{\displaystyle AFG}"> ve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle FCE}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mi>C</mi> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle FCE}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2da3f08ac3add9b601bb21bb451bb062ca433dd1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.283ex; height:2.176ex;" alt="{\displaystyle FCE}"> inşa edilmesini sağlar. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r33633371">Kırmızı ve <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r33633371">mavi üçgenlerin yüksekliklerinin toplamı <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \theta +\sin \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \theta +\sin \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92114d2d874dd1fbe28f856a7700500fc5736df4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.936ex; height:2.676ex;" alt="{\displaystyle \sin \theta +\sin \varphi }">'dir ve bu bir <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r33633371">mor üçgenin yüksekliğinin iki katına eşittir, yani <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\sin p\cos q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>sin</mi> <mo>⁡</mo> <mi>p</mi> <mi>cos</mi> <mo>⁡</mo> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\sin p\cos q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/226c2aa53c59c290788efce066b6a47e066fd336" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.917ex; height:2.509ex;" alt="{\displaystyle 2\sin p\cos q}">. Bu denklemdeki <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> ve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"> değerlerini <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"> ve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"> cinsinden yazmak sinüs için bir toplam-çarpım özdeşliği verir: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \theta +\sin \varphi =2\sin \left({\frac {\theta +\varphi }{2}}\right)\cos \left({\frac {\theta -\varphi }{2}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo>=</mo> <mn>2</mn> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>θ</mi> <mo>+</mo> <mi>φ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>θ</mi> <mo>−</mo> <mi>φ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \theta +\sin \varphi =2\sin \left({\frac {\theta +\varphi }{2}}\right)\cos \left({\frac {\theta -\varphi }{2}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4996910d4d4d71e2ac769cc70b775051c4b1b167" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:42.355ex; height:6.176ex;" alt="{\displaystyle \sin \theta +\sin \varphi =2\sin \left({\frac {\theta +\varphi }{2}}\right)\cos \left({\frac {\theta -\varphi }{2}}\right)}">. Benzer şekilde, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r33633371">kırmızı ve <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r33633371">mavi üçgenlerin genişliklerinin toplamı kosinüs için karşılık gelen özdeşliği verir.</div></figcaption></figure> Toplam-çarpım özdeşlikleri aşağıdaki gibidir:<a href="#cite_note-30">[30]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \theta \pm \sin \varphi =2\sin \left({\frac {\theta \pm \varphi }{2}}\right)\cos \left({\frac {\theta \mp \varphi }{2}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>±</mo> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo>=</mo> <mn>2</mn> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>θ</mi> <mo>±</mo> <mi>φ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>θ</mi> <mo>∓</mo> <mi>φ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \theta \pm \sin \varphi =2\sin \left({\frac {\theta \pm \varphi }{2}}\right)\cos \left({\frac {\theta \mp \varphi }{2}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/838468c4f91a3bd2a833279c980893762e20931a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:42.355ex; height:6.176ex;" alt="{\displaystyle \sin \theta \pm \sin \varphi =2\sin \left({\frac {\theta \pm \varphi }{2}}\right)\cos \left({\frac {\theta \mp \varphi }{2}}\right)}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \theta +\cos \varphi =2\cos \left({\frac {\theta +\varphi }{2}}\right)\cos \left({\frac {\theta -\varphi }{2}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>=</mo> <mn>2</mn> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>θ</mi> <mo>+</mo> <mi>φ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>θ</mi> <mo>−</mo> <mi>φ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \theta +\cos \varphi =2\cos \left({\frac {\theta +\varphi }{2}}\right)\cos \left({\frac {\theta -\varphi }{2}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/677cfdcfc861384c210778089fcc67c5a6b67676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:43.121ex; height:6.176ex;" alt="{\displaystyle \cos \theta +\cos \varphi =2\cos \left({\frac {\theta +\varphi }{2}}\right)\cos \left({\frac {\theta -\varphi }{2}}\right)}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \theta -\cos \varphi =-2\sin \left({\frac {\theta +\varphi }{2}}\right)\sin \left({\frac {\theta -\varphi }{2}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>θ</mi> <mo>+</mo> <mi>φ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>θ</mi> <mo>−</mo> <mi>φ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \theta -\cos \varphi =-2\sin \left({\frac {\theta +\varphi }{2}}\right)\sin \left({\frac {\theta -\varphi }{2}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3dcc9046722cad9ba4ad9b5a97d7ee980dc2fac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:44.418ex; height:6.176ex;" alt="{\displaystyle \cos \theta -\cos \varphi =-2\sin \left({\frac {\theta +\varphi }{2}}\right)\sin \left({\frac {\theta -\varphi }{2}}\right)}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan \theta \pm \tan \varphi ={\frac {\sin(\theta \pm \varphi )}{\cos \theta \,\cos \varphi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> <mo>±</mo> <mi>tan</mi> <mo>⁡</mo> <mi>φ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>±</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace" /> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan \theta \pm \tan \varphi ={\frac {\sin(\theta \pm \varphi )}{\cos \theta \,\cos \varphi }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5a91cc17d681682fda0cfafa2c9f9ba733e784e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.26ex; height:6.343ex;" alt="{\displaystyle \tan \theta \pm \tan \varphi ={\frac {\sin(\theta \pm \varphi )}{\cos \theta \,\cos \varphi }}}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Hermite_kotanjant_özdeşliği">Hermite kotanjant özdeşliği</h3>[<a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&veaction=edit&section=23" title="Değiştirilen bölüm: Hermite kotanjant özdeşliği" class="mw-editsection-visualeditor">değiştir</a> | <a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&action=edit&section=23" title="Bölümün kaynak kodunu değiştir: Hermite kotanjant özdeşliği">kaynağı değiştir</a>]</div> <div role="note" class="hatnote navigation-not-searchable">Ana madde: <a href="/w/index.php?title=Hermite_kotanjant_%C3%B6zde%C5%9Fli%C4%9Fi&action=edit&redlink=1" class="new" title="Hermite kotanjant özdeşliği (sayfa mevcut değil)">Hermite kotanjant özdeşliği</a></div> <a href="/wiki/Charles_Hermite" title="Charles Hermite">Charles Hermite</a> aşağıdaki özdeşliği göstermiştir.<a href="#cite_note-31">[31]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1},\ldots ,a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1},\ldots ,a_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/451345cc97e2ed923dd4656fcc400c3f37119cca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.911ex; height:2.009ex;" alt="{\displaystyle a_{1},\ldots ,a_{n}}"> sayılarının, hiçbiri π'nin bir tam sayı katı kadar farklı olmayan <a href="/wiki/Karma%C5%9F%C4%B1k_say%C4%B1" title="Karmaşık sayı">karmaşık sayılar</a> olduğunu varsayalım. Varsayalım ki <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{n,k}=\prod _{\begin{smallmatrix}1\leq j\leq n\\j\neq k\end{smallmatrix}}\cot(a_{k}-a_{j})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mn>1</mn> <mo>≤</mo> <mi>j</mi> <mo>≤</mo> <mi>n</mi> </mtd> </mtr> <mtr> <mtd> <mi>j</mi> <mo>≠</mo> <mi>k</mi> </mtd> </mtr> </mtable> </mstyle> </mrow> </munder> <mi>cot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{n,k}=\prod _{\begin{smallmatrix}1\leq j\leq n\\j\neq k\end{smallmatrix}}\cot(a_{k}-a_{j})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add4d3e70d16463f3a6941af196afc1e6c337ad1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.338ex; width:25.762ex; height:7.843ex;" alt="{\displaystyle A_{n,k}=\prod _{\begin{smallmatrix}1\leq j\leq n\\j\neq k\end{smallmatrix}}\cot(a_{k}-a_{j})}"> (özellikle, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1,1},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{1,1},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7af060aaa5ce8adabd73ae0d1ddf2863b64122f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.724ex; height:2.843ex;" alt="{\displaystyle A_{1,1},}"> bir <a href="/w/index.php?title=Bo%C5%9F_%C3%A7arp%C4%B1m&action=edit&redlink=1" class="new" title="Boş çarpım (sayfa mevcut değil)">boş çarpım</a> olmak üzere, 1'dir).O halde <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cot(z-a_{1})\cdots \cot(z-a_{n})=\cos {\frac {n\pi }{2}}+\sum _{k=1}^{n}A_{n,k}\cot(z-a_{k}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋯</mo> <mi>cot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mi>cot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cot(z-a_{1})\cdots \cot(z-a_{n})=\cos {\frac {n\pi }{2}}+\sum _{k=1}^{n}A_{n,k}\cot(z-a_{k}).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9db60873e93178b3b00e9cdea04ca45727748be1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:59.112ex; height:6.843ex;" alt="{\displaystyle \cot(z-a_{1})\cdots \cot(z-a_{n})=\cos {\frac {n\pi }{2}}+\sum _{k=1}^{n}A_{n,k}\cot(z-a_{k}).}"> Aşikar olmayan en basit örnek n = 2 durumudur: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cot(z-a_{1})\cot(z-a_{2})=-1+\cot(a_{1}-a_{2})\cot(z-a_{1})+\cot(a_{2}-a_{1})\cot(z-a_{2}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>cot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mi>cot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>cot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mi>cot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>cot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cot(z-a_{1})\cot(z-a_{2})=-1+\cot(a_{1}-a_{2})\cot(z-a_{1})+\cot(a_{2}-a_{1})\cot(z-a_{2}).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b24df9b1cd3d225a90be137a1e195b9314bd873c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:82.678ex; height:2.843ex;" alt="{\displaystyle \cot(z-a_{1})\cot(z-a_{2})=-1+\cot(a_{1}-a_{2})\cot(z-a_{1})+\cot(a_{2}-a_{1})\cot(z-a_{2}).}"> <div class="mw-heading mw-heading3"><h3 id="Trigonometrik_fonksiyonların_sonlu_çarpımları">Trigonometrik fonksiyonların sonlu çarpımları</h3>[<a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&veaction=edit&section=24" title="Değiştirilen bölüm: Trigonometrik fonksiyonların sonlu çarpımları" class="mw-editsection-visualeditor">değiştir</a> | <a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&action=edit&section=24" title="Bölümün kaynak kodunu değiştir: Trigonometrik fonksiyonların sonlu çarpımları">kaynağı değiştir</a>]</div> n, m <a href="/wiki/Aralar%C4%B1nda_asal" title="Aralarında asal">aralarında asal</a> tam sayıları için <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \prod _{k=1}^{n}\left(2a+2\cos \left({\frac {2\pi km}{n}}+x\right)\right)=2\left(T_{n}(a)+{(-1)}^{n+m}\cos(nx)\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>a</mi> <mo>+</mo> <mn>2</mn> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>k</mi> <mi>m</mi> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>+</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mi>m</mi> </mrow> </msup> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \prod _{k=1}^{n}\left(2a+2\cos \left({\frac {2\pi km}{n}}+x\right)\right)=2\left(T_{n}(a)+{(-1)}^{n+m}\cos(nx)\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35a18f9f21ae92883699255c30f14c0e044a4b91" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:62.829ex; height:6.843ex;" alt="{\displaystyle \prod _{k=1}^{n}\left(2a+2\cos \left({\frac {2\pi km}{n}}+x\right)\right)=2\left(T_{n}(a)+{(-1)}^{n+m}\cos(nx)\right)}"> burada Tn <a href="/w/index.php?title=Chebyshev_polinomlar%C4%B1&action=edit&redlink=1" class="new" title="Chebyshev polinomları (sayfa mevcut değil)">Chebyshev polinomudur</a>.[<a href="/wiki/Vikipedi:Kaynak_g%C3%B6sterme" title="Vikipedi:Kaynak gösterme">kaynak belirtilmeli</a>] Sinüs fonksiyonu için aşağıdaki ilişki geçerlidir; <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \prod _{k=1}^{n-1}\sin \left({\frac {k\pi }{n}}\right)={\frac {n}{2^{n-1}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mi>π</mi> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \prod _{k=1}^{n-1}\sin \left({\frac {k\pi }{n}}\right)={\frac {n}{2^{n-1}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f62d62eb63c131d80662c86ce573c766dff38e02" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:22.193ex; height:7.343ex;" alt="{\displaystyle \prod _{k=1}^{n-1}\sin \left({\frac {k\pi }{n}}\right)={\frac {n}{2^{n-1}}}.}"> Daha genel olarak bir n > 0 tam sayı için<a href="#cite_note-32">[32]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(nx)=2^{n-1}\prod _{k=0}^{n-1}\sin \left({\frac {k}{n}}\pi +x\right)=2^{n-1}\prod _{k=1}^{n}\sin \left({\frac {k}{n}}\pi -x\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mi>n</mi> </mfrac> </mrow> <mi>π</mi> <mo>+</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mi>n</mi> </mfrac> </mrow> <mi>π</mi> <mo>−</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(nx)=2^{n-1}\prod _{k=0}^{n-1}\sin \left({\frac {k}{n}}\pi +x\right)=2^{n-1}\prod _{k=1}^{n}\sin \left({\frac {k}{n}}\pi -x\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39bc0df571510f63829f6bfd13bdef039393b693" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:58.82ex; height:7.509ex;" alt="{\displaystyle \sin(nx)=2^{n-1}\prod _{k=0}^{n-1}\sin \left({\frac {k}{n}}\pi +x\right)=2^{n-1}\prod _{k=1}^{n}\sin \left({\frac {k}{n}}\pi -x\right).}"> veya <a href="/wiki/Kiri%C5%9F_(geometri)" title="Kiriş (geometri)">kiriş</a> fonksiyonu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \operatorname {crd} x\equiv 2\sin {\tfrac {1}{2}}x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>crd</mi> <mo>⁡</mo> <mi>x</mi> <mo>≡</mo> <mn>2</mn> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \operatorname {crd} x\equiv 2\sin {\tfrac {1}{2}}x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fb65dde5e2ce70be95524a9bd166bb4cce2e68f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:15.832ex; height:3.509ex;" alt="{\textstyle \operatorname {crd} x\equiv 2\sin {\tfrac {1}{2}}x}"> cinsinden yazılabilir, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {crd} (nx)=\prod _{k=1}^{n}\operatorname {crd} \left({\frac {k}{n}}2\pi -x\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>crd</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>crd</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mi>n</mi> </mfrac> </mrow> <mn>2</mn> <mi>π</mi> <mo>−</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {crd} (nx)=\prod _{k=1}^{n}\operatorname {crd} \left({\frac {k}{n}}2\pi -x\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c1514dfa1463e38c7b295984ae5abc5f078d0fe" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:30.425ex; height:6.843ex;" alt="{\displaystyle \operatorname {crd} (nx)=\prod _{k=1}^{n}\operatorname {crd} \left({\frac {k}{n}}2\pi -x\right).}"> Bu, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle z^{n}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle z^{n}-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e027adce13551f723570e5a8f5a9009a64dc7479" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.312ex; height:2.343ex;" alt="{\textstyle z^{n}-1}"> polinomunun doğrusal <a href="/w/index.php?title=Polinomlar%C4%B1n_%C3%A7arpanlara_ayr%C4%B1lmas%C4%B1&action=edit&redlink=1" class="new" title="Polinomların çarpanlara ayrılması (sayfa mevcut değil)">çarpanlara ayrılmasından</a> gelir. (bkz. <a href="/w/index.php?title=Birimin_k%C3%B6k%C3%BC&action=edit&redlink=1" class="new" title="Birimin kökü (sayfa mevcut değil)">birimin kökü</a>): Herhangi bir karmaşık z ve bir tam sayı n > 0 için, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{n}-1=\prod _{k=1}^{n}\left(z-\exp {\Bigl (}{\frac {k}{n}}2\pi i{\Bigr )}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo>=</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <mi>z</mi> <mo>−</mo> <mi>exp</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mi>n</mi> </mfrac> </mrow> <mn>2</mn> <mi>π</mi> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z^{n}-1=\prod _{k=1}^{n}\left(z-\exp {\Bigl (}{\frac {k}{n}}2\pi i{\Bigr )}\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ab3339ad78d71762afc41f0713bbc004552dfd9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:33.393ex; height:6.843ex;" alt="{\displaystyle z^{n}-1=\prod _{k=1}^{n}\left(z-\exp {\Bigl (}{\frac {k}{n}}2\pi i{\Bigr )}\right).}"> <div class="mw-heading mw-heading2"><h2 id="Doğrusal_kombinasyonlar">Doğrusal kombinasyonlar</h2>[<a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&veaction=edit&section=25" title="Değiştirilen bölüm: Doğrusal kombinasyonlar" class="mw-editsection-visualeditor">değiştir</a> | <a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&action=edit&section=25" title="Bölümün kaynak kodunu değiştir: Doğrusal kombinasyonlar">kaynağı değiştir</a>]</div> Bazı amaçlar için, aynı periyot veya frekansta ancak farklı <a href="/wiki/Faz_(dalga)" title="Faz (dalga)">faz kaymaları</a> olan sinüs dalgalarının herhangi bir doğrusal kombinasyonunun da aynı periyot veya frekansa ancak farklı bir faz kaymasına sahip bir sinüs dalgası olduğunu bilmek önemlidir. Bu <a href="/wiki/Sin%C3%BCs_dalgas%C4%B1" title="Sinüs dalgası">sinüzoidal</a> <a href="/wiki/E%C4%9Fri_uydurma" title="Eğri uydurma">veri uydurma</a> için kullanışlıdır. Ölçülen veya gözlemlenen veriler, aşağıdaki <a href="/w/index.php?title=Faz_i%C3%A7i_ve_kareleme_bile%C5%9Fenleri&action=edit&redlink=1" class="new" title="Faz içi ve kareleme bileşenleri (sayfa mevcut değil)">faz içi ve kareleme bileşenleri</a> temelinin a ve b bilinmeyenleri ile doğrusal olarak ilişkili olduğundan, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"> ve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"> ile karşılaştırıldığında daha basit bir <a href="/wiki/Jacobi_matrisi" title="Jacobi matrisi">Jacobyen</a> ile sonuçlanır. <div class="mw-heading mw-heading3"><h3 id="Sinüs_ve_kosinüs">Sinüs ve kosinüs</h3>[<a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&veaction=edit&section=26" title="Değiştirilen bölüm: Sinüs ve kosinüs" class="mw-editsection-visualeditor">değiştir</a> | <a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&action=edit&section=26" title="Bölümün kaynak kodunu değiştir: Sinüs ve kosinüs">kaynağı değiştir</a>]</div> Sinüs ve kosinüs dalgalarının doğrusal kombinasyonu veya harmonik toplamı, faz kayması ve ölçeklendirilmiş genliğe sahip tek bir sinüs dalgasına eşdeğerdir,<a href="#cite_note-33">[33]</a><a href="#cite_note-ReferenceB-34">[34]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cos x+b\sin x=c\cos(x+\varphi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mi>b</mi> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mi>c</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cos x+b\sin x=c\cos(x+\varphi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f79ce517fe8e938d761046d7313d2f430361ad83" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.345ex; height:2.843ex;" alt="{\displaystyle a\cos x+b\sin x=c\cos(x+\varphi )}"> burada <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\neq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f455a7f96d74aa94573d8e32da3b240ab0aa294f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.491ex; height:2.676ex;" alt="{\displaystyle a\neq 0}"> olduğu göz önüne alındığında <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"> ve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"> şu şekilde tanımlanır: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}c&=\operatorname {sgn}(a){\sqrt {a^{2}+b^{2}}},\\\varphi &={\arctan }{\bigl (}{-b/a}{\bigr )},\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>c</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>φ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>arctan</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}c&=\operatorname {sgn}(a){\sqrt {a^{2}+b^{2}}},\\\varphi &={\arctan }{\bigl (}{-b/a}{\bigr )},\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5ce47a28fee1135805243b831d9c5e34baa8be0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:21.927ex; height:6.843ex;" alt="{\displaystyle {\begin{aligned}c&=\operatorname {sgn}(a){\sqrt {a^{2}+b^{2}}},\\\varphi &={\arctan }{\bigl (}{-b/a}{\bigr )},\end{aligned}}}"> <div class="mw-heading mw-heading3"><h3 id="Keyfi_faz_kayması">Keyfi faz kayması</h3>[<a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&veaction=edit&section=27" title="Değiştirilen bölüm: Keyfi faz kayması" class="mw-editsection-visualeditor">değiştir</a> | <a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&action=edit&section=27" title="Bölümün kaynak kodunu değiştir: Keyfi faz kayması">kaynağı değiştir</a>]</div> Daha genel olarak, keyfi faz kaymaları için <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\sin(x+\theta _{a})+b\sin(x+\theta _{b})=c\sin(x+\varphi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mi>b</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>c</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\sin(x+\theta _{a})+b\sin(x+\theta _{b})=c\sin(x+\varphi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aeaf09e4b009ca81c036134c9e1ada87d2b38fe" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.58ex; height:2.843ex;" alt="{\displaystyle a\sin(x+\theta _{a})+b\sin(x+\theta _{b})=c\sin(x+\varphi )}"> Burada <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"> ve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"> aşağıdaki ifadeleri sağlar: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}c^{2}&=a^{2}+b^{2}+2ab\cos \left(\theta _{a}-\theta _{b}\right),\\\tan \varphi &={\frac {a\sin \theta _{a}+b\sin \theta _{b}}{a\cos \theta _{a}+b\cos \theta _{b}}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>a</mi> <mi>b</mi> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>tan</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>+</mo> <mi>b</mi> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mrow> <mrow> <mi>a</mi> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>+</mo> <mi>b</mi> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}c^{2}&=a^{2}+b^{2}+2ab\cos \left(\theta _{a}-\theta _{b}\right),\\\tan \varphi &={\frac {a\sin \theta _{a}+b\sin \theta _{b}}{a\cos \theta _{a}+b\cos \theta _{b}}}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2dde06d5e8234657dfeb7d591d5a672688256e4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:35.538ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}c^{2}&=a^{2}+b^{2}+2ab\cos \left(\theta _{a}-\theta _{b}\right),\\\tan \varphi &={\frac {a\sin \theta _{a}+b\sin \theta _{b}}{a\cos \theta _{a}+b\cos \theta _{b}}}.\end{aligned}}}"> <div class="mw-heading mw-heading3"><h3 id="İkiden_fazla_sinüzoid">İkiden fazla sinüzoid</h3>[<a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&veaction=edit&section=28" title="Değiştirilen bölüm: İkiden fazla sinüzoid" class="mw-editsection-visualeditor">değiştir</a> | <a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&action=edit&section=28" title="Bölümün kaynak kodunu değiştir: İkiden fazla sinüzoid">kaynağı değiştir</a>]</div> <div role="note" class="hatnote navigation-not-searchable">Ayrıca bakınız: <a href="/wiki/Faz%C3%B6r#Toplama" title="Fazör">Fazör toplama</a></div> Genel durum şu şekildedir<a href="#cite_note-ReferenceB-34">[34]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i}a_{i}\sin(x+\theta _{i})=a\sin(x+\theta ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i}a_{i}\sin(x+\theta _{i})=a\sin(x+\theta ),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e0f58cc33fbd8f27ae1221a7cab1d60a49f6986" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:32.171ex; height:5.509ex;" alt="{\displaystyle \sum _{i}a_{i}\sin(x+\theta _{i})=a\sin(x+\theta ),}"> burada <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{2}=\sum _{i,j}a_{i}a_{j}\cos(\theta _{i}-\theta _{j})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{2}=\sum _{i,j}a_{i}a_{j}\cos(\theta _{i}-\theta _{j})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54e2a23b18ed4268ef834d3dad3dbf03a19b176b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:25.332ex; height:5.843ex;" alt="{\displaystyle a^{2}=\sum _{i,j}a_{i}a_{j}\cos(\theta _{i}-\theta _{j})}"> ve <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan \theta ={\frac {\sum _{i}a_{i}\sin \theta _{i}}{\sum _{i}a_{i}\cos \theta _{i}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan \theta ={\frac {\sum _{i}a_{i}\sin \theta _{i}}{\sum _{i}a_{i}\cos \theta _{i}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/108579447fd00d2a5d3f4b97b76f484ccd13a9a0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.864ex; height:6.509ex;" alt="{\displaystyle \tan \theta ={\frac {\sum _{i}a_{i}\sin \theta _{i}}{\sum _{i}a_{i}\cos \theta _{i}}}.}"> <div class="mw-heading mw-heading2"><h2 id="Lagrange_trigonometrik_özdeşlikleri">Lagrange trigonometrik özdeşlikleri</h2>[<a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&veaction=edit&section=29" title="Değiştirilen bölüm: Lagrange trigonometrik özdeşlikleri" class="mw-editsection-visualeditor">değiştir</a> | <a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&action=edit&section=29" title="Bölümün kaynak kodunu değiştir: Lagrange trigonometrik özdeşlikleri">kaynağı değiştir</a>]</div> Adını <a href="/wiki/Joseph_Louis_Lagrange" class="mw-redirect" title="Joseph Louis Lagrange">Joseph Louis Lagrange</a>'dan alan bu özdeşlikler şunlardır:<a href="#cite_note-Muniz-35">[35]</a><a href="#cite_note-36">[36]</a><a href="#cite_note-37">[37]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\sum _{k=0}^{n}\sin k\theta &={\frac {\cos {\tfrac {1}{2}}\theta -\cos \left(\left(n+{\tfrac {1}{2}}\right)\theta \right)}{2\sin {\tfrac {1}{2}}\theta }}\\[5pt]\sum _{k=0}^{n}\cos k\theta &={\frac {\sin {\tfrac {1}{2}}\theta +\sin \left(\left(n+{\tfrac {1}{2}}\right)\theta \right)}{2\sin {\tfrac {1}{2}}\theta }}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.8em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>sin</mi> <mo>⁡</mo> <mi>k</mi> <mi>θ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>θ</mi> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mrow> <mo>)</mo> </mrow> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>θ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>cos</mi> <mo>⁡</mo> <mi>k</mi> <mi>θ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>θ</mi> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mrow> <mo>)</mo> </mrow> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>θ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sum _{k=0}^{n}\sin k\theta &={\frac {\cos {\tfrac {1}{2}}\theta -\cos \left(\left(n+{\tfrac {1}{2}}\right)\theta \right)}{2\sin {\tfrac {1}{2}}\theta }}\\[5pt]\sum _{k=0}^{n}\cos k\theta &={\frac {\sin {\tfrac {1}{2}}\theta +\sin \left(\left(n+{\tfrac {1}{2}}\right)\theta \right)}{2\sin {\tfrac {1}{2}}\theta }}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c850aa2dd7cc306d87d5d68d5802cab64f8572a0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.005ex; width:38.057ex; height:17.176ex;" alt="{\displaystyle {\begin{aligned}\sum _{k=0}^{n}\sin k\theta &={\frac {\cos {\tfrac {1}{2}}\theta -\cos \left(\left(n+{\tfrac {1}{2}}\right)\theta \right)}{2\sin {\tfrac {1}{2}}\theta }}\\[5pt]\sum _{k=0}^{n}\cos k\theta &={\frac {\sin {\tfrac {1}{2}}\theta +\sin \left(\left(n+{\tfrac {1}{2}}\right)\theta \right)}{2\sin {\tfrac {1}{2}}\theta }}\end{aligned}}}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta \not \equiv 0{\pmod {2\pi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>≢</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>2</mn> <mi>π</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta \not \equiv 0{\pmod {2\pi }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/791cd472e03864c1184dbc66ea38954151fc739e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.531ex; height:2.843ex;" alt="{\displaystyle \theta \not \equiv 0{\pmod {2\pi }}}"> için. İlgili bir fonksiyon <a href="/w/index.php?title=Dirichlet_%C3%A7ekirde%C4%9Fi&action=edit&redlink=1" class="new" title="Dirichlet çekirdeği (sayfa mevcut değil)">Dirichlet çekirdeğidir</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{n}(\theta )=1+2\sum _{k=1}^{n}\cos k\theta ={\frac {\sin \left(\left(n+{\tfrac {1}{2}}\right)\theta \right)}{\sin {\tfrac {1}{2}}\theta }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mn>2</mn> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>cos</mi> <mo>⁡</mo> <mi>k</mi> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mrow> <mo>)</mo> </mrow> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>θ</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{n}(\theta )=1+2\sum _{k=1}^{n}\cos k\theta ={\frac {\sin \left(\left(n+{\tfrac {1}{2}}\right)\theta \right)}{\sin {\tfrac {1}{2}}\theta }}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b01dc407ca91aa189a3988781a3924bf9e556c4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:43.303ex; height:7.843ex;" alt="{\displaystyle D_{n}(\theta )=1+2\sum _{k=1}^{n}\cos k\theta ={\frac {\sin \left(\left(n+{\tfrac {1}{2}}\right)\theta \right)}{\sin {\tfrac {1}{2}}\theta }}.}"> Benzer bir özdeşlik<a href="#cite_note-38">[38]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=1}^{n}\cos(2k-1)\alpha ={\frac {\sin(2n\alpha )}{2\sin \alpha }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>α</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <mi>sin</mi> <mo>⁡</mo> <mi>α</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{n}\cos(2k-1)\alpha ={\frac {\sin(2n\alpha )}{2\sin \alpha }}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8b255b3a83c20795d48290b1ed04f38600947e1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:29.818ex; height:6.843ex;" alt="{\displaystyle \sum _{k=1}^{n}\cos(2k-1)\alpha ={\frac {\sin(2n\alpha )}{2\sin \alpha }}.}"> Kanıt aşağıdaki gibidir. <a href="#Açı_toplam_ve_fark_özdeşlikleri">açı toplam ve fark özdeşlikleri</a> kullanılarak, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(A+B)-\sin(A-B)=2\cos A\sin B.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>+</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>−</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mi>cos</mi> <mo>⁡</mo> <mi>A</mi> <mi>sin</mi> <mo>⁡</mo> <mi>B</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(A+B)-\sin(A-B)=2\cos A\sin B.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3312923622de75ef5c0d257446c597b13c91536" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.795ex; height:2.843ex;" alt="{\displaystyle \sin(A+B)-\sin(A-B)=2\cos A\sin B.}"> O zaman aşağıdaki formülü inceleyelim, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\sin \alpha \sum _{k=1}^{n}\cos(2k-1)\alpha =2\sin \alpha \cos \alpha +2\sin \alpha \cos 3\alpha +2\sin \alpha \cos 5\alpha +\ldots +2\sin \alpha \cos(2n-1)\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>sin</mi> <mo>⁡</mo> <mi>α</mi> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>α</mi> <mo>=</mo> <mn>2</mn> <mi>sin</mi> <mo>⁡</mo> <mi>α</mi> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> <mo>+</mo> <mn>2</mn> <mi>sin</mi> <mo>⁡</mo> <mi>α</mi> <mi>cos</mi> <mo>⁡</mo> <mn>3</mn> <mi>α</mi> <mo>+</mo> <mn>2</mn> <mi>sin</mi> <mo>⁡</mo> <mi>α</mi> <mi>cos</mi> <mo>⁡</mo> <mn>5</mn> <mi>α</mi> <mo>+</mo> <mo>…</mo> <mo>+</mo> <mn>2</mn> <mi>sin</mi> <mo>⁡</mo> <mi>α</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\sin \alpha \sum _{k=1}^{n}\cos(2k-1)\alpha =2\sin \alpha \cos \alpha +2\sin \alpha \cos 3\alpha +2\sin \alpha \cos 5\alpha +\ldots +2\sin \alpha \cos(2n-1)\alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10bb5c3842f00af0f5e9a1a1d97fe3ff573062b4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:97.295ex; height:6.843ex;" alt="{\displaystyle 2\sin \alpha \sum _{k=1}^{n}\cos(2k-1)\alpha =2\sin \alpha \cos \alpha +2\sin \alpha \cos 3\alpha +2\sin \alpha \cos 5\alpha +\ldots +2\sin \alpha \cos(2n-1)\alpha }"> ve bu formül yukarıdaki özdeşlik kullanılarak yazılabilir, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&2\sin \alpha \sum _{k=1}^{n}\cos(2k-1)\alpha \\&\quad =\sum _{k=1}^{n}(\sin(2k\alpha )-\sin(2(k-1)\alpha ))\\&\quad =(\sin 2\alpha -\sin 0)+(\sin 4\alpha -\sin 2\alpha )+(\sin 6\alpha -\sin 4\alpha )+\ldots +(\sin(2n\alpha )-\sin(2(n-1)\alpha ))\\&\quad =\sin(2n\alpha ).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mn>2</mn> <mi>sin</mi> <mo>⁡</mo> <mi>α</mi> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>α</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mspace width="1em" /> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mi>α</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mspace width="1em" /> <mo>=</mo> <mo stretchy="false">(</mo> <mi>sin</mi> <mo>⁡</mo> <mn>2</mn> <mi>α</mi> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>sin</mi> <mo>⁡</mo> <mn>4</mn> <mi>α</mi> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mn>2</mn> <mi>α</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>sin</mi> <mo>⁡</mo> <mn>6</mn> <mi>α</mi> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mn>4</mn> <mi>α</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo>…</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mi>α</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mspace width="1em" /> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mi>α</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&2\sin \alpha \sum _{k=1}^{n}\cos(2k-1)\alpha \\&\quad =\sum _{k=1}^{n}(\sin(2k\alpha )-\sin(2(k-1)\alpha ))\\&\quad =(\sin 2\alpha -\sin 0)+(\sin 4\alpha -\sin 2\alpha )+(\sin 6\alpha -\sin 4\alpha )+\ldots +(\sin(2n\alpha )-\sin(2(n-1)\alpha ))\\&\quad =\sin(2n\alpha ).\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d00ee200d61869170023f7e6a62611bbd3b1bb5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.671ex; width:95.957ex; height:20.509ex;" alt="{\displaystyle {\begin{aligned}&2\sin \alpha \sum _{k=1}^{n}\cos(2k-1)\alpha \\&\quad =\sum _{k=1}^{n}(\sin(2k\alpha )-\sin(2(k-1)\alpha ))\\&\quad =(\sin 2\alpha -\sin 0)+(\sin 4\alpha -\sin 2\alpha )+(\sin 6\alpha -\sin 4\alpha )+\ldots +(\sin(2n\alpha )-\sin(2(n-1)\alpha ))\\&\quad =\sin(2n\alpha ).\end{aligned}}}"> Dolayısıyla, bu formülü <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\sin \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>sin</mi> <mo>⁡</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\sin \alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ca076a7f20158e3519c78ed0eccf32889766e64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.28ex; height:2.176ex;" alt="{\displaystyle 2\sin \alpha }"> ile bölmek kanıtı tamamlar. <div class="mw-heading mw-heading2"><h2 id="Belirli_doğrusal_kesirli_dönüşümler">Belirli doğrusal kesirli dönüşümler</h2>[<a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&veaction=edit&section=30" title="Değiştirilen bölüm: Belirli doğrusal kesirli dönüşümler" class="mw-editsection-visualeditor">değiştir</a> | <a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&action=edit&section=30" title="Bölümün kaynak kodunu değiştir: Belirli doğrusal kesirli dönüşümler">kaynağı değiştir</a>]</div> Eğer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"> <a href="/w/index.php?title=M%C3%B6bius_d%C3%B6n%C3%BC%C5%9F%C3%BCm%C3%BC&action=edit&redlink=1" class="new" title="Möbius dönüşümü (sayfa mevcut değil)">doğrusal kesirli dönüşüm</a> tarafından veriliyorsa <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)={\frac {(\cos \alpha )x-\sin \alpha }{(\sin \alpha )x+\cos \alpha }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mi>x</mi> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>α</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>sin</mi> <mo>⁡</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mi>x</mi> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)={\frac {(\cos \alpha )x-\sin \alpha }{(\sin \alpha )x+\cos \alpha }},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbc62585cae3c00794e045d021d6bb010afee79b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:24.695ex; height:6.509ex;" alt="{\displaystyle f(x)={\frac {(\cos \alpha )x-\sin \alpha }{(\sin \alpha )x+\cos \alpha }},}"> ve benzer şekilde <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)={\frac {(\cos \beta )x-\sin \beta }{(\sin \beta )x+\cos \beta }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mi>x</mi> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>β</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>sin</mi> <mo>⁡</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mi>x</mi> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mi>β</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)={\frac {(\cos \beta )x-\sin \beta }{(\sin \beta )x+\cos \beta }},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd4ca80eca844d6808928b5c9c6bbfeec9d09f53" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:24.221ex; height:6.509ex;" alt="{\displaystyle g(x)={\frac {(\cos \beta )x-\sin \beta }{(\sin \beta )x+\cos \beta }},}"> öyleyse <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f{\big (}g(x){\big )}=g{\big (}f(x){\big )}={\frac {{\big (}\cos(\alpha +\beta ){\big )}x-\sin(\alpha +\beta )}{{\big (}\sin(\alpha +\beta ){\big )}x+\cos(\alpha +\beta )}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mi>x</mi> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mi>x</mi> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f{\big (}g(x){\big )}=g{\big (}f(x){\big )}={\frac {{\big (}\cos(\alpha +\beta ){\big )}x-\sin(\alpha +\beta )}{{\big (}\sin(\alpha +\beta ){\big )}x+\cos(\alpha +\beta )}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/733b8bbdcbd48e17c233d2afc1a1aebe527d3e84" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:50.599ex; height:7.343ex;" alt="{\displaystyle f{\big (}g(x){\big )}=g{\big (}f(x){\big )}={\frac {{\big (}\cos(\alpha +\beta ){\big )}x-\sin(\alpha +\beta )}{{\big (}\sin(\alpha +\beta ){\big )}x+\cos(\alpha +\beta )}}.}"> Daha açık bir ifadeyle, eğer tüm <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"> için <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\alpha }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a741210b1c3e47122356f562a1661f264b01df7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.423ex; height:2.509ex;" alt="{\displaystyle f_{\alpha }}"> yukarıda <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> olarak adlandırdığımız şey olsun. <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\alpha }\circ f_{\beta }=f_{\alpha +\beta }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>∘</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>+</mo> <mi>β</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\alpha }\circ f_{\beta }=f_{\alpha +\beta }.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b30dbcc591485c992d716801500d94924f1283ce" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.321ex; height:2.843ex;" alt="{\displaystyle f_{\alpha }\circ f_{\beta }=f_{\alpha +\beta }.}"> Eğer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> bir doğrunun eğimi ise, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"> doğrunun <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6807df17ce845a127ca43612e2ffd5cf62b7adc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.296ex; height:2.176ex;" alt="{\displaystyle -\alpha }"> açısı boyunca dönüşünün eğimidir. <div class="mw-heading mw-heading2"><h2 id="Karmaşık_üstel_fonksiyon_ile_ilişkisi">Karmaşık üstel fonksiyon ile ilişkisi</h2>[<a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&veaction=edit&section=31" title="Değiştirilen bölüm: Karmaşık üstel fonksiyon ile ilişkisi" class="mw-editsection-visualeditor">değiştir</a> | <a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&action=edit&section=31" title="Bölümün kaynak kodunu değiştir: Karmaşık üstel fonksiyon ile ilişkisi">kaynağı değiştir</a>]</div> <div role="note" class="hatnote navigation-not-searchable">Ana madde: <a href="/wiki/Euler_form%C3%BCl%C3%BC" title="Euler formülü">Euler formülü</a></div> Euler'in formülü, herhangi bir gerçek sayı x için:<a href="#cite_note-39">[39]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{ix}=\cos x+i\sin x,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>x</mi> </mrow> </msup> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{ix}=\cos x+i\sin x,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aab1fcd1a6db5cc6678bb9cbd871580eeeb86eda" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.999ex; height:3.009ex;" alt="{\displaystyle e^{ix}=\cos x+i\sin x,}"> burada i <a href="/wiki/%C4%B0_say%C4%B1s%C4%B1" title="İ sayısı">sanal birimdir</a>. x yerine -x koyduğumuzda aşağıdaki ifadeyi elde ederiz: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-ix}=\cos(-x)+i\sin(-x)=\cos x-i\sin x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>x</mi> </mrow> </msup> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>−</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-ix}=\cos(-x)+i\sin(-x)=\cos x-i\sin x.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04cef40d5b6d8b3132de5092a2098e84fa9cabec" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.267ex; height:3.176ex;" alt="{\displaystyle e^{-ix}=\cos(-x)+i\sin(-x)=\cos x-i\sin x.}"> Bu iki denklem, kosinüs ve sinüsü <a href="/wiki/%C3%9Cstel_fonksiyon" title="Üstel fonksiyon">üstel fonksiyon</a> cinsinden çözmek için kullanılabilir. Spesifik olarak,<a href="#cite_note-40">[40]</a><a href="#cite_note-41">[41]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos x={\frac {e^{ix}+e^{-ix}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>x</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>x</mi> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos x={\frac {e^{ix}+e^{-ix}}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e16aed66a59735cc85653fb255bd6ba5b3e0db4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.528ex; height:5.676ex;" alt="{\displaystyle \cos x={\frac {e^{ix}+e^{-ix}}{2}}}"> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin x={\frac {e^{ix}-e^{-ix}}{2i}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>x</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>x</mi> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>i</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin x={\frac {e^{ix}-e^{-ix}}{2i}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a19a293592d5d5dda850bf2de5b92aba3c9764f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.273ex; height:5.676ex;" alt="{\displaystyle \sin x={\frac {e^{ix}-e^{-ix}}{2i}}}"> Bu formüller, diğer birçok trigonometrik özdeşliği kanıtlamak için kullanışlıdır. Örneğin, ei(θ+φ) = eiθ eiφ demek oluyor ki <div style="padding-left: 1.5em; padding-right:0em; overflow: hidden;">cos(θ + φ) + i sin(θ + φ) = (cos θ + i sin θ) (cos φ + i sin φ) = (cos θ cos φ − sin θ sin φ) + i (cos θ sin φ + sin θ cos φ).</div> Sol tarafın reel kısmının, sağ tarafın reel kısmına eşit olması kosinüs için bir açı toplama formülüdür. Sanal kısımların eşitliği sinüs için bir açı toplama formülü verir. Aşağıdaki tablo trigonometrik fonksiyonları ve bunların terslerini üstel fonksiyon ve <a href="/w/index.php?title=Karma%C5%9F%C4%B1k_logaritma&action=edit&redlink=1" class="new" title="Karmaşık logaritma (sayfa mevcut değil)">karmaşık logaritma</a> cinsinden ifade etmektedir. <table class="wikitable" style="background-color:var(--background-color-base)"> <tbody><tr> <th>Fonksiyon </th> <th>Ters fonksiyon<a href="#cite_note-42">[42]</a> </th></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \theta ={\frac {e^{i\theta }-e^{-i\theta }}{2i}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>θ</mi> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>i</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \theta ={\frac {e^{i\theta }-e^{-i\theta }}{2i}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06b4aaf0bb6dff52bfabe5efe8bf06b39458e73a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.695ex; height:5.676ex;" alt="{\displaystyle \sin \theta ={\frac {e^{i\theta }-e^{-i\theta }}{2i}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arcsin x=-i\,\ln \left(ix+{\sqrt {1-x^{2}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arcsin</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mi>i</mi> <mspace width="thinmathspace" /> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arcsin x=-i\,\ln \left(ix+{\sqrt {1-x^{2}}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd2627b1d52cec6b80ad8b88325c26b3bdd508eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:32.56ex; height:4.843ex;" alt="{\displaystyle \arcsin x=-i\,\ln \left(ix+{\sqrt {1-x^{2}}}\right)}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \theta ={\frac {e^{i\theta }+e^{-i\theta }}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>θ</mi> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \theta ={\frac {e^{i\theta }+e^{-i\theta }}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e128422fe183eff61ef1383fe0d8230f4f41b85c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.951ex; height:5.676ex;" alt="{\displaystyle \cos \theta ={\frac {e^{i\theta }+e^{-i\theta }}{2}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arccos x=-i\ln \left(x+{\sqrt {x^{2}-1}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccos</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mi>i</mi> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arccos x=-i\ln \left(x+{\sqrt {x^{2}-1}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/525915e698d3c65f8b7ce6481b580e27e6b4b278" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.626ex; height:4.843ex;" alt="{\displaystyle \arccos x=-i\ln \left(x+{\sqrt {x^{2}-1}}\right)}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan \theta =-i\,{\frac {e^{i\theta }-e^{-i\theta }}{e^{i\theta }+e^{-i\theta }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mo>−</mo> <mi>i</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>θ</mi> </mrow> </msup> </mrow> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>θ</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan \theta =-i\,{\frac {e^{i\theta }-e^{-i\theta }}{e^{i\theta }+e^{-i\theta }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8b347d8fc30d80b8f39bb0117fb7cdf722f3e84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.197ex; height:6.176ex;" alt="{\displaystyle \tan \theta =-i\,{\frac {e^{i\theta }-e^{-i\theta }}{e^{i\theta }+e^{-i\theta }}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arctan x={\frac {i}{2}}\ln \left({\frac {i+x}{i-x}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arctan</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mn>2</mn> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <mo>+</mo> <mi>x</mi> </mrow> <mrow> <mi>i</mi> <mo>−</mo> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arctan x={\frac {i}{2}}\ln \left({\frac {i+x}{i-x}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ef3d37c25e24cd8e8f10d64b645d483b5af4b9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.836ex; height:6.176ex;" alt="{\displaystyle \arctan x={\frac {i}{2}}\ln \left({\frac {i+x}{i-x}}\right)}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \csc \theta ={\frac {2i}{e^{i\theta }-e^{-i\theta }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>csc</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>i</mi> </mrow> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>θ</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \csc \theta ={\frac {2i}{e^{i\theta }-e^{-i\theta }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8df9e91acb45bca4c7eacfb43f001e79135ec1ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.821ex; height:5.676ex;" alt="{\displaystyle \csc \theta ={\frac {2i}{e^{i\theta }-e^{-i\theta }}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {arccsc} x=-i\,\ln \left({\frac {i}{x}}+{\sqrt {1-{\frac {1}{x^{2}}}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccsc</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mi>i</mi> <mspace width="thinmathspace" /> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mi>x</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {arccsc} x=-i\,\ln \left({\frac {i}{x}}+{\sqrt {1-{\frac {1}{x^{2}}}}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40d92cc1a94ae99881cb8f6444c679a900b2f827" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.201ex; height:6.343ex;" alt="{\displaystyle \operatorname {arccsc} x=-i\,\ln \left({\frac {i}{x}}+{\sqrt {1-{\frac {1}{x^{2}}}}}\right)}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sec \theta ={\frac {2}{e^{i\theta }+e^{-i\theta }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sec</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>θ</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sec \theta ={\frac {2}{e^{i\theta }+e^{-i\theta }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5ab3b8feffd73949e6c96ee07e6ad889002bb8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.821ex; height:5.676ex;" alt="{\displaystyle \sec \theta ={\frac {2}{e^{i\theta }+e^{-i\theta }}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {arcsec} x=-i\,\ln \left({\frac {1}{x}}+i{\sqrt {1-{\frac {1}{x^{2}}}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arcsec</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mi>i</mi> <mspace width="thinmathspace" /> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mo>+</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {arcsec} x=-i\,\ln \left({\frac {1}{x}}+i{\sqrt {1-{\frac {1}{x^{2}}}}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84410aec2775b35a5da69b1b5ec4a67e7782432d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:35.003ex; height:6.343ex;" alt="{\displaystyle \operatorname {arcsec} x=-i\,\ln \left({\frac {1}{x}}+i{\sqrt {1-{\frac {1}{x^{2}}}}}\right)}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cot \theta =i\,{\frac {e^{i\theta }+e^{-i\theta }}{e^{i\theta }-e^{-i\theta }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cot</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>i</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>θ</mi> </mrow> </msup> </mrow> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>θ</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cot \theta =i\,{\frac {e^{i\theta }+e^{-i\theta }}{e^{i\theta }-e^{-i\theta }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8ad07ba68a0e82ef4963c9a8e8f26423bc14f8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.129ex; height:6.176ex;" alt="{\displaystyle \cot \theta =i\,{\frac {e^{i\theta }+e^{-i\theta }}{e^{i\theta }-e^{-i\theta }}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {arccot} x={\frac {i}{2}}\ln \left({\frac {x-i}{x+i}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccot</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mn>2</mn> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>−</mo> <mi>i</mi> </mrow> <mrow> <mi>x</mi> <mo>+</mo> <mi>i</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {arccot} x={\frac {i}{2}}\ln \left({\frac {x-i}{x+i}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97cbf62906aa131d1ca159ac0251e1839f2f0e89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.576ex; height:6.176ex;" alt="{\displaystyle \operatorname {arccot} x={\frac {i}{2}}\ln \left({\frac {x-i}{x+i}}\right)}"> </td></tr> <tr> <td><a href="/w/index.php?title=Cis_(matematik)&action=edit&redlink=1" class="new" title="Cis (matematik) (sayfa mevcut değil)"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cis} \theta =e^{i\theta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cis</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {cis} \theta =e^{i\theta }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3935f8cb3c9ce5c8d4ad26498e08c7a4bcbc3fc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.826ex; height:2.676ex;" alt="{\displaystyle \operatorname {cis} \theta =e^{i\theta }}"></a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {arccis} x=-i\ln x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccis</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mi>i</mi> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {arccis} x=-i\ln x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71eb03debd43622da2486c3dfb4e53b4227de5a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:17.171ex; height:2.343ex;" alt="{\displaystyle \operatorname {arccis} x=-i\ln x}"> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Seri_açılımları">Seri açılımları</h2>[<a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&veaction=edit&section=32" title="Değiştirilen bölüm: Seri açılımları" class="mw-editsection-visualeditor">değiştir</a> | <a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&action=edit&section=32" title="Bölümün kaynak kodunu değiştir: Seri açılımları">kaynağı değiştir</a>]</div> Trigonometrik fonksiyonları tanımlamak için bir <a href="/wiki/Kuvvet_serisi" title="Kuvvet serisi">kuvvet serisi</a> açılımı kullanıldığında, aşağıdaki özdeşlikler elde edilir:<a href="#cite_note-43">[43]</a> <dl><dd><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin x=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots =\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n+1}}{(2n+1)!}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mi>x</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow> <mn>3</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mrow> <mn>5</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mrow> <mn>7</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin x=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots =\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n+1}}{(2n+1)!}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f410e287da836452159f9769be8dacaaf3be038" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:52.538ex; height:7.009ex;" alt="{\displaystyle \sin x=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots =\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n+1}}{(2n+1)!}},}"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos x=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots =\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n}}{(2n)!}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mrow> <mn>4</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mrow> <mn>6</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos x=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots =\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n}}{(2n)!}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e3d56fe29cc48a9b8329a4747da71ac62a18ff3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:50.526ex; height:7.009ex;" alt="{\displaystyle \cos x=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots =\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n}}{(2n)!}}.}"></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Sonsuz_çarpım_formülleri">Sonsuz çarpım formülleri</h2>[<a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&veaction=edit&section=33" title="Değiştirilen bölüm: Sonsuz çarpım formülleri" class="mw-editsection-visualeditor">değiştir</a> | <a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&action=edit&section=33" title="Bölümün kaynak kodunu değiştir: Sonsuz çarpım formülleri">kaynağı değiştir</a>]</div> <a href="/wiki/%C3%96zel_fonksiyonlar" title="Özel fonksiyonlar">Özel fonksiyon</a> uygulamaları için, trigonometrik fonksiyonlar için aşağıdaki <a href="/w/index.php?title=Sonsuz_%C3%A7arp%C4%B1m&action=edit&redlink=1" class="new" title="Sonsuz çarpım (sayfa mevcut değil)">sonsuz çarpım</a> formülleri kullanışlıdır:<a href="#cite_note-44">[44]</a><a href="#cite_note-45">[45]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\sin x&=x\prod _{n=1}^{\infty }\left(1-{\frac {x^{2}}{\pi ^{2}n^{2}}}\right),&\cos x&=\prod _{n=1}^{\infty }\left(1-{\frac {x^{2}}{\pi ^{2}\left(n-{\frac {1}{2}}\right)\!{\vphantom {)}}^{2}}}\right),\\[10mu]\sinh x&=x\prod _{n=1}^{\infty }\left(1+{\frac {x^{2}}{\pi ^{2}n^{2}}}\right),&\cosh x&=\prod _{n=1}^{\infty }\left(1+{\frac {x^{2}}{\pi ^{2}\left(n-{\frac {1}{2}}\right)\!{\vphantom {)}}^{2}}}\right).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.856em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>x</mi> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mspace width="negativethinmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mpadded width="0"> <mphantom> <mo stretchy="false">)</mo> </mphantom> </mpadded> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>sinh</mi> <mo>⁡</mo> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>x</mi> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mtd> <mtd> <mi>cosh</mi> <mo>⁡</mo> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mspace width="negativethinmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mpadded width="0"> <mphantom> <mo stretchy="false">)</mo> </mphantom> </mpadded> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sin x&=x\prod _{n=1}^{\infty }\left(1-{\frac {x^{2}}{\pi ^{2}n^{2}}}\right),&\cos x&=\prod _{n=1}^{\infty }\left(1-{\frac {x^{2}}{\pi ^{2}\left(n-{\frac {1}{2}}\right)\!{\vphantom {)}}^{2}}}\right),\\[10mu]\sinh x&=x\prod _{n=1}^{\infty }\left(1+{\frac {x^{2}}{\pi ^{2}n^{2}}}\right),&\cosh x&=\prod _{n=1}^{\infty }\left(1+{\frac {x^{2}}{\pi ^{2}\left(n-{\frac {1}{2}}\right)\!{\vphantom {)}}^{2}}}\right).\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d43e1176a38b1e6b653837eb782ae79735783ac" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.005ex; width:67.782ex; height:17.176ex;" alt="{\displaystyle {\begin{aligned}\sin x&=x\prod _{n=1}^{\infty }\left(1-{\frac {x^{2}}{\pi ^{2}n^{2}}}\right),&\cos x&=\prod _{n=1}^{\infty }\left(1-{\frac {x^{2}}{\pi ^{2}\left(n-{\frac {1}{2}}\right)\!{\vphantom {)}}^{2}}}\right),\\[10mu]\sinh x&=x\prod _{n=1}^{\infty }\left(1+{\frac {x^{2}}{\pi ^{2}n^{2}}}\right),&\cosh x&=\prod _{n=1}^{\infty }\left(1+{\frac {x^{2}}{\pi ^{2}\left(n-{\frac {1}{2}}\right)\!{\vphantom {)}}^{2}}}\right).\end{aligned}}}"> <div class="mw-heading mw-heading2"><h2 id="Ters_trigonometrik_fonksiyonlar">Ters trigonometrik fonksiyonlar</h2>[<a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&veaction=edit&section=34" title="Değiştirilen bölüm: Ters trigonometrik fonksiyonlar" class="mw-editsection-visualeditor">değiştir</a> | <a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&action=edit&section=34" title="Bölümün kaynak kodunu değiştir: Ters trigonometrik fonksiyonlar">kaynağı değiştir</a>]</div> <div role="note" class="hatnote navigation-not-searchable">Ana madde: <a href="/wiki/Ters_trigonometrik_fonksiyonlar" title="Ters trigonometrik fonksiyonlar">Ters trigonometrik fonksiyonlar</a></div> Aşağıdaki özdeşlikler, bir trigonometrik fonksiyonun bir ters trigonometrik fonksiyonla bileşiminin sonucunu verir.<a href="#cite_note-46">[46]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\sin(\arcsin x)&=x&\cos(\arcsin x)&={\sqrt {1-x^{2}}}&\tan(\arcsin x)&={\frac {x}{\sqrt {1-x^{2}}}}\\\sin(\arccos x)&={\sqrt {1-x^{2}}}&\cos(\arccos x)&=x&\tan(\arccos x)&={\frac {\sqrt {1-x^{2}}}{x}}\\\sin(\arctan x)&={\frac {x}{\sqrt {1+x^{2}}}}&\cos(\arctan x)&={\frac {1}{\sqrt {1+x^{2}}}}&\tan(\arctan x)&=x\\\sin(\operatorname {arccsc} x)&={\frac {1}{x}}&\cos(\operatorname {arccsc} x)&={\frac {\sqrt {x^{2}-1}}{x}}&\tan(\operatorname {arccsc} x)&={\frac {1}{\sqrt {x^{2}-1}}}\\\sin(\operatorname {arcsec} x)&={\frac {\sqrt {x^{2}-1}}{x}}&\cos(\operatorname {arcsec} x)&={\frac {1}{x}}&\tan(\operatorname {arcsec} x)&={\sqrt {x^{2}-1}}\\\sin(\operatorname {arccot} x)&={\frac {1}{\sqrt {1+x^{2}}}}&\cos(\operatorname {arccot} x)&={\frac {x}{\sqrt {1+x^{2}}}}&\tan(\operatorname {arccot} x)&={\frac {1}{x}}\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arcsin</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>x</mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arcsin</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mtd> <mtd> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arcsin</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arccos</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arccos</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>x</mi> </mtd> <mtd> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arccos</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> <mi>x</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arctan</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arctan</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mtd> <mtd> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arctan</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arccsc</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arccsc</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </msqrt> <mi>x</mi> </mfrac> </mrow> </mtd> <mtd> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arccsc</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </msqrt> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arcsec</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </msqrt> <mi>x</mi> </mfrac> </mrow> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arcsec</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> </mtd> <mtd> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arcsec</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arccot</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arccot</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mtd> <mtd> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arccot</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sin(\arcsin x)&=x&\cos(\arcsin x)&={\sqrt {1-x^{2}}}&\tan(\arcsin x)&={\frac {x}{\sqrt {1-x^{2}}}}\\\sin(\arccos x)&={\sqrt {1-x^{2}}}&\cos(\arccos x)&=x&\tan(\arccos x)&={\frac {\sqrt {1-x^{2}}}{x}}\\\sin(\arctan x)&={\frac {x}{\sqrt {1+x^{2}}}}&\cos(\arctan x)&={\frac {1}{\sqrt {1+x^{2}}}}&\tan(\arctan x)&=x\\\sin(\operatorname {arccsc} x)&={\frac {1}{x}}&\cos(\operatorname {arccsc} x)&={\frac {\sqrt {x^{2}-1}}{x}}&\tan(\operatorname {arccsc} x)&={\frac {1}{\sqrt {x^{2}-1}}}\\\sin(\operatorname {arcsec} x)&={\frac {\sqrt {x^{2}-1}}{x}}&\cos(\operatorname {arcsec} x)&={\frac {1}{x}}&\tan(\operatorname {arcsec} x)&={\sqrt {x^{2}-1}}\\\sin(\operatorname {arccot} x)&={\frac {1}{\sqrt {1+x^{2}}}}&\cos(\operatorname {arccot} x)&={\frac {x}{\sqrt {1+x^{2}}}}&\tan(\operatorname {arccot} x)&={\frac {1}{x}}\\\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73e11f774f4e8cc2a43d8046f65215030625bde7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -19.671ex; width:87.28ex; height:40.509ex;" alt="{\displaystyle {\begin{aligned}\sin(\arcsin x)&=x&\cos(\arcsin x)&={\sqrt {1-x^{2}}}&\tan(\arcsin x)&={\frac {x}{\sqrt {1-x^{2}}}}\\\sin(\arccos x)&={\sqrt {1-x^{2}}}&\cos(\arccos x)&=x&\tan(\arccos x)&={\frac {\sqrt {1-x^{2}}}{x}}\\\sin(\arctan x)&={\frac {x}{\sqrt {1+x^{2}}}}&\cos(\arctan x)&={\frac {1}{\sqrt {1+x^{2}}}}&\tan(\arctan x)&=x\\\sin(\operatorname {arccsc} x)&={\frac {1}{x}}&\cos(\operatorname {arccsc} x)&={\frac {\sqrt {x^{2}-1}}{x}}&\tan(\operatorname {arccsc} x)&={\frac {1}{\sqrt {x^{2}-1}}}\\\sin(\operatorname {arcsec} x)&={\frac {\sqrt {x^{2}-1}}{x}}&\cos(\operatorname {arcsec} x)&={\frac {1}{x}}&\tan(\operatorname {arcsec} x)&={\sqrt {x^{2}-1}}\\\sin(\operatorname {arccot} x)&={\frac {1}{\sqrt {1+x^{2}}}}&\cos(\operatorname {arccot} x)&={\frac {x}{\sqrt {1+x^{2}}}}&\tan(\operatorname {arccot} x)&={\frac {1}{x}}\\\end{aligned}}}"> Yukarıdaki her bir denklemin her iki tarafının <a href="/w/index.php?title=%C3%87arp%C4%B1msal_ters&action=edit&redlink=1" class="new" title="Çarpımsal ters (sayfa mevcut değil)">çarpımsal tersi</a> alındığında <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \csc ={\frac {1}{\sin }},\;\sec ={\frac {1}{\cos }},{\text{ ve }}\cot ={\frac {1}{\tan }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>csc</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>sin</mi> </mfrac> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mi>sec</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>cos</mi> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> ve </mtext> </mrow> <mi>cot</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>tan</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \csc ={\frac {1}{\sin }},\;\sec ={\frac {1}{\cos }},{\text{ ve }}\cot ={\frac {1}{\tan }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2948e061062f0070b300888f84e98ffe54252832" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:36.713ex; height:5.176ex;" alt="{\displaystyle \csc ={\frac {1}{\sin }},\;\sec ={\frac {1}{\cos }},{\text{ ve }}\cot ={\frac {1}{\tan }}}"> denklemleri elde edilir. Yukarıdaki formülün sağ tarafı her zaman ters çevrilecektir. Örneğin, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cot(\arcsin x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arcsin</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cot(\arcsin x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46ff5b8a1d9fad99884720e39fc7456145bd7ac3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.588ex; height:2.843ex;" alt="{\displaystyle \cot(\arcsin x)}"> için denklem şöyledir: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cot(\arcsin x)={\frac {1}{\tan(\arcsin x)}}={\frac {1}{\frac {x}{\sqrt {1-x^{2}}}}}={\frac {\sqrt {1-x^{2}}}{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arcsin</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arcsin</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mfrac> <mi>x</mi> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> <mi>x</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cot(\arcsin x)={\frac {1}{\tan(\arcsin x)}}={\frac {1}{\frac {x}{\sqrt {1-x^{2}}}}}={\frac {\sqrt {1-x^{2}}}{x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/918258736bef3575995edac47e9ee9221899089d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:52.301ex; height:8.343ex;" alt="{\displaystyle \cot(\arcsin x)={\frac {1}{\tan(\arcsin x)}}={\frac {1}{\frac {x}{\sqrt {1-x^{2}}}}}={\frac {\sqrt {1-x^{2}}}{x}}}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \csc(\arccos x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>csc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arccos</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \csc(\arccos x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57f7876efcc11629f64a31bf30632bd80a793144" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.725ex; height:2.843ex;" alt="{\displaystyle \csc(\arccos x)}"> ve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sec(\arccos x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arccos</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sec(\arccos x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc02667a452583137f5a4abd3d733097015b03c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.725ex; height:2.843ex;" alt="{\displaystyle \sec(\arccos x)}"> için denklemler ise şöyledir: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \csc(\arccos x)={\frac {1}{\sin(\arccos x)}}={\frac {1}{\sqrt {1-x^{2}}}}\qquad {\text{ ve }}\quad \sec(\arccos x)={\frac {1}{\cos(\arccos x)}}={\frac {1}{x}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>csc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arccos</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arccos</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> ve </mtext> </mrow> <mspace width="1em" /> <mi>sec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arccos</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arccos</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \csc(\arccos x)={\frac {1}{\sin(\arccos x)}}={\frac {1}{\sqrt {1-x^{2}}}}\qquad {\text{ ve }}\quad \sec(\arccos x)={\frac {1}{\cos(\arccos x)}}={\frac {1}{x}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cec04293de8e0fe36ccf4d26ea2771eeb16b5a3f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:88.105ex; height:6.509ex;" alt="{\displaystyle \csc(\arccos x)={\frac {1}{\sin(\arccos x)}}={\frac {1}{\sqrt {1-x^{2}}}}\qquad {\text{ ve }}\quad \sec(\arccos x)={\frac {1}{\cos(\arccos x)}}={\frac {1}{x}}.}"> Aşağıdaki özdeşlikler, <a href="#Yansımalar">yansıma özdeşlikleri</a> tarafından ortaya konmuştur. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,r,s,-x,-r,{\text{ ve }}-s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>r</mi> <mo>,</mo> <mi>s</mi> <mo>,</mo> <mo>−</mo> <mi>x</mi> <mo>,</mo> <mo>−</mo> <mi>r</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> ve </mtext> </mrow> <mo>−</mo> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,r,s,-x,-r,{\text{ ve }}-s}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/410b8adf5e3df47ee596b9d43a87e00bf7c24f69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.985ex; height:2.343ex;" alt="{\displaystyle x,r,s,-x,-r,{\text{ ve }}-s}"> ilgili fonksiyonların etki alanlarında olduğunda geçerlidirler. <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{alignedat}{9}{\frac {\pi }{2}}~&=~\arcsin(x)&&+\arccos(x)~&&=~\arctan(r)&&+\operatorname {arccot}(r)~&&=~\operatorname {arcsec}(s)&&+\operatorname {arccsc}(s)\\[0.4ex]\pi ~&=~\arccos(x)&&+\arccos(-x)~&&=~\operatorname {arccot}(r)&&+\operatorname {arccot}(-r)~&&=~\operatorname {arcsec}(s)&&+\operatorname {arcsec}(-s)\\[0.4ex]0~&=~\arcsin(x)&&+\arcsin(-x)~&&=~\arctan(r)&&+\arctan(-r)~&&=~\operatorname {arccsc}(s)&&+\operatorname {arccsc}(-s)\\[1.0ex]\end{alignedat}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left right left right left right left" rowspacing="0.472em 0.472em 0.73em" columnspacing="0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mtext> </mtext> </mtd> <mtd> <mi></mi> <mo>=</mo> <mtext> </mtext> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd /> <mtd> <mi></mi> <mo>+</mo> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mtd> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mtext> </mtext> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mtd> <mtd /> <mtd> <mi></mi> <mo>+</mo> <mi>arccot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mtd> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mtext> </mtext> <mi>arcsec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mtd> <mtd /> <mtd> <mi></mi> <mo>+</mo> <mi>arccsc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>π</mi> <mtext> </mtext> </mtd> <mtd> <mi></mi> <mo>=</mo> <mtext> </mtext> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd /> <mtd> <mi></mi> <mo>+</mo> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mtd> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mtext> </mtext> <mi>arccot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mtd> <mtd /> <mtd> <mi></mi> <mo>+</mo> <mi>arccot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mtd> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mtext> </mtext> <mi>arcsec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mtd> <mtd /> <mtd> <mi></mi> <mo>+</mo> <mi>arcsec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> <mtext> </mtext> </mtd> <mtd> <mi></mi> <mo>=</mo> <mtext> </mtext> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd /> <mtd> <mi></mi> <mo>+</mo> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mtd> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mtext> </mtext> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mtd> <mtd /> <mtd> <mi></mi> <mo>+</mo> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mtd> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mtext> </mtext> <mi>arccsc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mtd> <mtd /> <mtd> <mi></mi> <mo>+</mo> <mi>arccsc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{9}{\frac {\pi }{2}}~&=~\arcsin(x)&&+\arccos(x)~&&=~\arctan(r)&&+\operatorname {arccot}(r)~&&=~\operatorname {arcsec}(s)&&+\operatorname {arccsc}(s)\\[0.4ex]\pi ~&=~\arccos(x)&&+\arccos(-x)~&&=~\operatorname {arccot}(r)&&+\operatorname {arccot}(-r)~&&=~\operatorname {arcsec}(s)&&+\operatorname {arcsec}(-s)\\[0.4ex]0~&=~\arcsin(x)&&+\arcsin(-x)~&&=~\arctan(r)&&+\arctan(-r)~&&=~\operatorname {arccsc}(s)&&+\operatorname {arccsc}(-s)\\[1.0ex]\end{alignedat}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/839db041f421b4af5890106450e2b4d3e9e183d4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.042ex; margin-bottom: -0.296ex; width:86.141ex; height:11.843ex;" alt="{\displaystyle {\begin{alignedat}{9}{\frac {\pi }{2}}~&=~\arcsin(x)&&+\arccos(x)~&&=~\arctan(r)&&+\operatorname {arccot}(r)~&&=~\operatorname {arcsec}(s)&&+\operatorname {arccsc}(s)\\[0.4ex]\pi ~&=~\arccos(x)&&+\arccos(-x)~&&=~\operatorname {arccot}(r)&&+\operatorname {arccot}(-r)~&&=~\operatorname {arcsec}(s)&&+\operatorname {arcsec}(-s)\\[0.4ex]0~&=~\arcsin(x)&&+\arcsin(-x)~&&=~\arctan(r)&&+\arctan(-r)~&&=~\operatorname {arccsc}(s)&&+\operatorname {arccsc}(-s)\\[1.0ex]\end{alignedat}}}"> Aynı zamanda,<a href="#cite_note-Wu-47">[47]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\arctan x+\arctan {\dfrac {1}{x}}&={\begin{cases}{\frac {\pi }{2}},&x>0\ \ {\text{ise}}\\-{\frac {\pi }{2}},&x<0\ \ {\text{ise}}\end{cases}}\\\operatorname {arccot} x+\operatorname {arccot} {\dfrac {1}{x}}&={\begin{cases}{\frac {\pi }{2}},&x>0\ \ {\text{ise}}\\{\frac {3\pi }{2}},&x<0\ \ {\text{ise}}\end{cases}}\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>arctan</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mi>arctan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> </mtd> <mtd> <mi>x</mi> <mo>></mo> <mn>0</mn> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>ise</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> </mtd> <mtd> <mi>x</mi> <mo><</mo> <mn>0</mn> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>ise</mtext> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>arccot</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mi>arccot</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> </mtd> <mtd> <mi>x</mi> <mo>></mo> <mn>0</mn> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>ise</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> </mtd> <mtd> <mi>x</mi> <mo><</mo> <mn>0</mn> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>ise</mtext> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\arctan x+\arctan {\dfrac {1}{x}}&={\begin{cases}{\frac {\pi }{2}},&x>0\ \ {\text{ise}}\\-{\frac {\pi }{2}},&x<0\ \ {\text{ise}}\end{cases}}\\\operatorname {arccot} x+\operatorname {arccot} {\dfrac {1}{x}}&={\begin{cases}{\frac {\pi }{2}},&x>0\ \ {\text{ise}}\\{\frac {3\pi }{2}},&x<0\ \ {\text{ise}}\end{cases}}\\\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1550eb261168535eb2d732bae21217ff991b01e7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:42.29ex; height:14.176ex;" alt="{\displaystyle {\begin{aligned}\arctan x+\arctan {\dfrac {1}{x}}&={\begin{cases}{\frac {\pi }{2}},&x>0\ \ {\text{ise}}\\-{\frac {\pi }{2}},&x<0\ \ {\text{ise}}\end{cases}}\\\operatorname {arccot} x+\operatorname {arccot} {\dfrac {1}{x}}&={\begin{cases}{\frac {\pi }{2}},&x>0\ \ {\text{ise}}\\{\frac {3\pi }{2}},&x<0\ \ {\text{ise}}\end{cases}}\\\end{aligned}}}"> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arccos {\frac {1}{x}}=\operatorname {arcsec} x\qquad {\text{ ve }}\qquad \operatorname {arcsec} {\frac {1}{x}}=\arccos x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mo>=</mo> <mi>arcsec</mi> <mo>⁡</mo> <mi>x</mi> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> ve </mtext> </mrow> <mspace width="2em" /> <mi>arcsec</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mo>=</mo> <mi>arccos</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arccos {\frac {1}{x}}=\operatorname {arcsec} x\qquad {\text{ ve }}\qquad \operatorname {arcsec} {\frac {1}{x}}=\arccos x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2f242507937fb37a9e34074d11bcf3d53e744a4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:52.445ex; height:5.176ex;" alt="{\displaystyle \arccos {\frac {1}{x}}=\operatorname {arcsec} x\qquad {\text{ ve }}\qquad \operatorname {arcsec} {\frac {1}{x}}=\arccos x}"> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arcsin {\frac {1}{x}}=\operatorname {arccsc} x\qquad {\text{ ve }}\qquad \operatorname {arccsc} {\frac {1}{x}}=\arcsin x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arcsin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mo>=</mo> <mi>arccsc</mi> <mo>⁡</mo> <mi>x</mi> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> ve </mtext> </mrow> <mspace width="2em" /> <mi>arccsc</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mo>=</mo> <mi>arcsin</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arcsin {\frac {1}{x}}=\operatorname {arccsc} x\qquad {\text{ ve }}\qquad \operatorname {arccsc} {\frac {1}{x}}=\arcsin x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f90d88a7d59fbd1a45e04f5f7d29cf875ea621bf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:51.934ex; height:5.176ex;" alt="{\displaystyle \arcsin {\frac {1}{x}}=\operatorname {arccsc} x\qquad {\text{ ve }}\qquad \operatorname {arccsc} {\frac {1}{x}}=\arcsin x}"> <a href="/wiki/Ters_trigonometrik_fonksiyonlar" title="Ters trigonometrik fonksiyonlar">Arktanjant</a> fonksiyonu bir seri olarak genişletilebilir:<a href="#cite_note-48">[48]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arctan(nx)=\sum _{m=1}^{n}\arctan {\frac {x}{1+(m-1)mx^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>arctan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mrow> <mn>1</mn> <mo>+</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>m</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arctan(nx)=\sum _{m=1}^{n}\arctan {\frac {x}{1+(m-1)mx^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2f2701b97013583d1969ced44ba8bc0ab5191ea" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:41.998ex; height:6.843ex;" alt="{\displaystyle \arctan(nx)=\sum _{m=1}^{n}\arctan {\frac {x}{1+(m-1)mx^{2}}}}"> <div class="mw-heading mw-heading2"><h2 id="Değişken_içermeyen_özdeşlikler">Değişken içermeyen özdeşlikler</h2>[<a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&veaction=edit&section=35" title="Değiştirilen bölüm: Değişken içermeyen özdeşlikler" class="mw-editsection-visualeditor">değiştir</a> | <a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&action=edit&section=35" title="Bölümün kaynak kodunu değiştir: Değişken içermeyen özdeşlikler">kaynağı değiştir</a>]</div> <a href="/wiki/Ters_trigonometrik_fonksiyonlar" title="Ters trigonometrik fonksiyonlar">Arktanjant</a> fonksiyonu cinsinden aşağıdaki ifadelere sahibiz;<a href="#cite_note-Wu-47">[47]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arctan {\frac {1}{2}}=\arctan {\frac {1}{3}}+\arctan {\frac {1}{7}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arctan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mi>arctan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>7</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arctan {\frac {1}{2}}=\arctan {\frac {1}{3}}+\arctan {\frac {1}{7}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f4bf5f97aed47c8fb5c1510fc5114ff1f5c8cfb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:33.141ex; height:5.343ex;" alt="{\displaystyle \arctan {\frac {1}{2}}=\arctan {\frac {1}{3}}+\arctan {\frac {1}{7}}.}"> <a href="/w/index.php?title=Morrie_yasas%C4%B1&action=edit&redlink=1" class="new" title="Morrie yasası (sayfa mevcut değil)">Morrie yasası</a> olarak bilinen ilginç özdeşlik, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos 20^{\circ }\cdot \cos 40^{\circ }\cdot \cos 80^{\circ }={\frac {1}{8}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <msup> <mn>20</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <msup> <mn>40</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <msup> <mn>80</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos 20^{\circ }\cdot \cos 40^{\circ }\cdot \cos 80^{\circ }={\frac {1}{8}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c2dd4aef667106a64004bd68d9f2f0d601aa314" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:29.734ex; height:5.176ex;" alt="{\displaystyle \cos 20^{\circ }\cdot \cos 40^{\circ }\cdot \cos 80^{\circ }={\frac {1}{8}},}"> tek değişken içeren bir özdeşliğin özel bir durumudur: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \prod _{j=0}^{k-1}\cos \left(2^{j}x\right)={\frac {\sin \left(2^{k}x\right)}{2^{k}\sin x}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \prod _{j=0}^{k-1}\cos \left(2^{j}x\right)={\frac {\sin \left(2^{k}x\right)}{2^{k}\sin x}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbae68746c87fad8b959c7f3bb156c9ce6eb521d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:25.147ex; height:7.676ex;" alt="{\displaystyle \prod _{j=0}^{k-1}\cos \left(2^{j}x\right)={\frac {\sin \left(2^{k}x\right)}{2^{k}\sin x}}.}"> Benzer şekilde, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin 20^{\circ }\cdot \sin 40^{\circ }\cdot \sin 80^{\circ }={\frac {\sqrt {3}}{8}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <msup> <mn>20</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <msup> <mn>40</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <msup> <mn>80</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>3</mn> </msqrt> <mn>8</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin 20^{\circ }\cdot \sin 40^{\circ }\cdot \sin 80^{\circ }={\frac {\sqrt {3}}{8}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d48d9efac6609e756a9b6224248b123be594a1a6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:30.257ex; height:5.843ex;" alt="{\displaystyle \sin 20^{\circ }\cdot \sin 40^{\circ }\cdot \sin 80^{\circ }={\frac {\sqrt {3}}{8}}}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=20^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msup> <mn>20</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=20^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/662aa3374f5cee87a18e4d1b49dffe6594a06791" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.807ex; height:2.343ex;" alt="{\displaystyle x=20^{\circ }}"> olan bir özdeşliğin özel bir durumudur: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin x\cdot \sin \left(60^{\circ }-x\right)\cdot \sin \left(60^{\circ }+x\right)={\frac {\sin 3x}{4}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msup> <mn>60</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>−</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msup> <mn>60</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>+</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mn>3</mn> <mi>x</mi> </mrow> <mn>4</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin x\cdot \sin \left(60^{\circ }-x\right)\cdot \sin \left(60^{\circ }+x\right)={\frac {\sin 3x}{4}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0f378baa62e963c75f55b29522a59bdfc061101" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:42.675ex; height:5.176ex;" alt="{\displaystyle \sin x\cdot \sin \left(60^{\circ }-x\right)\cdot \sin \left(60^{\circ }+x\right)={\frac {\sin 3x}{4}}.}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=15^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msup> <mn>15</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=15^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/396dc4950f4912c3c41367f83469d66e7bdccab3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.807ex; height:2.343ex;" alt="{\displaystyle x=15^{\circ }}"> durumu için, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\sin 15^{\circ }\cdot \sin 45^{\circ }\cdot \sin 75^{\circ }&={\frac {\sqrt {2}}{8}},\\\sin 15^{\circ }\cdot \sin 75^{\circ }&={\frac {1}{4}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <msup> <mn>15</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <msup> <mn>45</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <msup> <mn>75</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>2</mn> </msqrt> <mn>8</mn> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <msup> <mn>15</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <msup> <mn>75</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sin 15^{\circ }\cdot \sin 45^{\circ }\cdot \sin 75^{\circ }&={\frac {\sqrt {2}}{8}},\\\sin 15^{\circ }\cdot \sin 75^{\circ }&={\frac {1}{4}}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1677fd23685cc495eaadfda1e01916efd1fca152" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:31.655ex; height:11.176ex;" alt="{\displaystyle {\begin{aligned}\sin 15^{\circ }\cdot \sin 45^{\circ }\cdot \sin 75^{\circ }&={\frac {\sqrt {2}}{8}},\\\sin 15^{\circ }\cdot \sin 75^{\circ }&={\frac {1}{4}}.\end{aligned}}}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=10^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=10^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3df93affdf270c071e4a1fd5fca36b4ecf363f14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.807ex; height:2.343ex;" alt="{\displaystyle x=10^{\circ }}"> durumu için, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin 10^{\circ }\cdot \sin 50^{\circ }\cdot \sin 70^{\circ }={\frac {1}{8}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <msup> <mn>50</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <msup> <mn>70</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin 10^{\circ }\cdot \sin 50^{\circ }\cdot \sin 70^{\circ }={\frac {1}{8}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/481c028cc1989226779b0fd8083f4c7ecfaaa289" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.968ex; height:5.176ex;" alt="{\displaystyle \sin 10^{\circ }\cdot \sin 50^{\circ }\cdot \sin 70^{\circ }={\frac {1}{8}}.}"> Aynı kosinüs özdeşliği <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos x\cdot \cos \left(60^{\circ }-x\right)\cdot \cos \left(60^{\circ }+x\right)={\frac {\cos 3x}{4}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msup> <mn>60</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>−</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msup> <mn>60</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>+</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>cos</mi> <mo>⁡</mo> <mn>3</mn> <mi>x</mi> </mrow> <mn>4</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos x\cdot \cos \left(60^{\circ }-x\right)\cdot \cos \left(60^{\circ }+x\right)={\frac {\cos 3x}{4}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9da763698e9dcb5902e0bb0efaa29f2028eca45f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:43.697ex; height:5.176ex;" alt="{\displaystyle \cos x\cdot \cos \left(60^{\circ }-x\right)\cdot \cos \left(60^{\circ }+x\right)={\frac {\cos 3x}{4}}.}"> Benzer şekilde, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\cos 10^{\circ }\cdot \cos 50^{\circ }\cdot \cos 70^{\circ }&={\frac {\sqrt {3}}{8}},\\\cos 15^{\circ }\cdot \cos 45^{\circ }\cdot \cos 75^{\circ }&={\frac {\sqrt {2}}{8}},\\\cos 15^{\circ }\cdot \cos 75^{\circ }&={\frac {1}{4}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <msup> <mn>50</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <msup> <mn>70</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>3</mn> </msqrt> <mn>8</mn> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <msup> <mn>15</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <msup> <mn>45</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <msup> <mn>75</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>2</mn> </msqrt> <mn>8</mn> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <msup> <mn>15</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <msup> <mn>75</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\cos 10^{\circ }\cdot \cos 50^{\circ }\cdot \cos 70^{\circ }&={\frac {\sqrt {3}}{8}},\\\cos 15^{\circ }\cdot \cos 45^{\circ }\cdot \cos 75^{\circ }&={\frac {\sqrt {2}}{8}},\\\cos 15^{\circ }\cdot \cos 75^{\circ }&={\frac {1}{4}}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b188f6a11f571a505691ab9aa55cfd0b37b5e700" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.005ex; width:32.422ex; height:17.176ex;" alt="{\displaystyle {\begin{aligned}\cos 10^{\circ }\cdot \cos 50^{\circ }\cdot \cos 70^{\circ }&={\frac {\sqrt {3}}{8}},\\\cos 15^{\circ }\cdot \cos 45^{\circ }\cdot \cos 75^{\circ }&={\frac {\sqrt {2}}{8}},\\\cos 15^{\circ }\cdot \cos 75^{\circ }&={\frac {1}{4}}.\end{aligned}}}"> Benzer şekilde, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\tan 50^{\circ }\cdot \tan 60^{\circ }\cdot \tan 70^{\circ }&=\tan 80^{\circ },\\\tan 40^{\circ }\cdot \tan 30^{\circ }\cdot \tan 20^{\circ }&=\tan 10^{\circ }.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>tan</mi> <mo>⁡</mo> <msup> <mn>50</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>⋅</mo> <mi>tan</mi> <mo>⁡</mo> <msup> <mn>60</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>⋅</mo> <mi>tan</mi> <mo>⁡</mo> <msup> <mn>70</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>tan</mi> <mo>⁡</mo> <msup> <mn>80</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>tan</mi> <mo>⁡</mo> <msup> <mn>40</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>⋅</mo> <mi>tan</mi> <mo>⁡</mo> <msup> <mn>30</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>⋅</mo> <mi>tan</mi> <mo>⁡</mo> <msup> <mn>20</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>tan</mi> <mo>⁡</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\tan 50^{\circ }\cdot \tan 60^{\circ }\cdot \tan 70^{\circ }&=\tan 80^{\circ },\\\tan 40^{\circ }\cdot \tan 30^{\circ }\cdot \tan 20^{\circ }&=\tan 10^{\circ }.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5dc032dfbc6766ea7e13f892ce8b97d933b1a5bf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:36.358ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}\tan 50^{\circ }\cdot \tan 60^{\circ }\cdot \tan 70^{\circ }&=\tan 80^{\circ },\\\tan 40^{\circ }\cdot \tan 30^{\circ }\cdot \tan 20^{\circ }&=\tan 10^{\circ }.\end{aligned}}}"> Aşağıdakiler, değişkenleri içeren bir özdeşliğe kolayca genelleştirilemeyebilir (ancak aşağıdaki açıklamaya bakınız): <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos 24^{\circ }+\cos 48^{\circ }+\cos 96^{\circ }+\cos 168^{\circ }={\frac {1}{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <msup> <mn>24</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <msup> <mn>48</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <msup> <mn>96</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <msup> <mn>168</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos 24^{\circ }+\cos 48^{\circ }+\cos 96^{\circ }+\cos 168^{\circ }={\frac {1}{2}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4006eb68677ca45758bbbb6eb2a348dfe881d4c8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:42.937ex; height:5.176ex;" alt="{\displaystyle \cos 24^{\circ }+\cos 48^{\circ }+\cos 96^{\circ }+\cos 168^{\circ }={\frac {1}{2}}.}"> Bu özdeşliği, paydalarda 21 ile düşündüğümüzde derece ölçüsü radyan ölçüsünden daha isabetli olmaktan çıkar: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos {\frac {2\pi }{21}}+\cos \left(2\cdot {\frac {2\pi }{21}}\right)+\cos \left(4\cdot {\frac {2\pi }{21}}\right)+\cos \left(5\cdot {\frac {2\pi }{21}}\right)+\cos \left(8\cdot {\frac {2\pi }{21}}\right)+\cos \left(10\cdot {\frac {2\pi }{21}}\right)={\frac {1}{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mn>21</mn> </mfrac> </mrow> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mn>21</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>4</mn> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mn>21</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>5</mn> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mn>21</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>8</mn> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mn>21</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>10</mn> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mn>21</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos {\frac {2\pi }{21}}+\cos \left(2\cdot {\frac {2\pi }{21}}\right)+\cos \left(4\cdot {\frac {2\pi }{21}}\right)+\cos \left(5\cdot {\frac {2\pi }{21}}\right)+\cos \left(8\cdot {\frac {2\pi }{21}}\right)+\cos \left(10\cdot {\frac {2\pi }{21}}\right)={\frac {1}{2}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/970eaaf23b9ffe1e690d9f563b1b532beb1d1506" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:91.459ex; height:6.176ex;" alt="{\displaystyle \cos {\frac {2\pi }{21}}+\cos \left(2\cdot {\frac {2\pi }{21}}\right)+\cos \left(4\cdot {\frac {2\pi }{21}}\right)+\cos \left(5\cdot {\frac {2\pi }{21}}\right)+\cos \left(8\cdot {\frac {2\pi }{21}}\right)+\cos \left(10\cdot {\frac {2\pi }{21}}\right)={\frac {1}{2}}.}"> 1, 2, 4, 5, 8, 10 çarpanları, modeli netleştirmeye başlayabilir: bunlar 21/2'den küçük olan ve 21 ile <a href="/wiki/Aralar%C4%B1nda_asal" title="Aralarında asal">göreceli asal</a> olan (veya ortak <a href="/wiki/Asal_%C3%A7arpan" title="Asal çarpan">asal çarpanları</a> olmayan) tam sayılardır. Son birkaç örnek indirgenemez <a href="/w/index.php?title=Siklotomik_polinom&action=edit&redlink=1" class="new" title="Siklotomik polinom (sayfa mevcut değil)">siklotomik polinomlarla</a> ilgili temel bir gerçeğin sonucudur: kosinüsler bu polinomların sıfırlarının gerçel kısımlarıdır; sıfırların toplamı (yukarıdaki son durumda) 21'de değerlendirilen <a href="/wiki/M%C3%B6bius_fonksiyonu" title="Möbius fonksiyonu">Möbius fonksiyonudur</a>; sıfırların sadece yarısı yukarıda mevcuttur. Bu sonuncusundan önceki iki özdeşlik, 21 yerine sırasıyla 10 ve 15 konduğunda aynı şekilde ortaya çıkar. Diğer kosinüs özdeşlikleri şunlardır:<a href="#cite_note-49">[49]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}2\cos {\frac {\pi }{3}}&=1,\\2\cos {\frac {\pi }{5}}\times 2\cos {\frac {2\pi }{5}}&=1,\\2\cos {\frac {\pi }{7}}\times 2\cos {\frac {2\pi }{7}}\times 2\cos {\frac {3\pi }{7}}&=1,\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mn>2</mn> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>3</mn> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>5</mn> </mfrac> </mrow> <mo>×</mo> <mn>2</mn> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mn>5</mn> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>7</mn> </mfrac> </mrow> <mo>×</mo> <mn>2</mn> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mn>7</mn> </mfrac> </mrow> <mo>×</mo> <mn>2</mn> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>7</mn> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}2\cos {\frac {\pi }{3}}&=1,\\2\cos {\frac {\pi }{5}}\times 2\cos {\frac {2\pi }{5}}&=1,\\2\cos {\frac {\pi }{7}}\times 2\cos {\frac {2\pi }{7}}\times 2\cos {\frac {3\pi }{7}}&=1,\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26b9aba9f4182d9ae190b677d829917647c40fe6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.006ex; margin-bottom: -0.332ex; width:35.313ex; height:15.843ex;" alt="{\displaystyle {\begin{aligned}2\cos {\frac {\pi }{3}}&=1,\\2\cos {\frac {\pi }{5}}\times 2\cos {\frac {2\pi }{5}}&=1,\\2\cos {\frac {\pi }{7}}\times 2\cos {\frac {2\pi }{7}}\times 2\cos {\frac {3\pi }{7}}&=1,\end{aligned}}}"> ve tüm tek sayılar için böyle devam eder, dolayısıyla <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos {\frac {\pi }{3}}+\cos {\frac {\pi }{5}}\times \cos {\frac {2\pi }{5}}+\cos {\frac {\pi }{7}}\times \cos {\frac {2\pi }{7}}\times \cos {\frac {3\pi }{7}}+\dots =1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>5</mn> </mfrac> </mrow> <mo>×</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mn>5</mn> </mfrac> </mrow> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>7</mn> </mfrac> </mrow> <mo>×</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mn>7</mn> </mfrac> </mrow> <mo>×</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>7</mn> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos {\frac {\pi }{3}}+\cos {\frac {\pi }{5}}\times \cos {\frac {2\pi }{5}}+\cos {\frac {\pi }{7}}\times \cos {\frac {2\pi }{7}}\times \cos {\frac {3\pi }{7}}+\dots =1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d363b4652ea359de079e5e62c91809c3f5d142b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:62.159ex; height:5.343ex;" alt="{\displaystyle \cos {\frac {\pi }{3}}+\cos {\frac {\pi }{5}}\times \cos {\frac {2\pi }{5}}+\cos {\frac {\pi }{7}}\times \cos {\frac {2\pi }{7}}\times \cos {\frac {3\pi }{7}}+\dots =1.}"> Bu ilginç özdeşliklerin birçoğu aşağıdaki gibi daha genel gerçeklerden kaynaklanmaktadır:<a href="#cite_note-50">[50]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \prod _{k=1}^{n-1}\sin {\frac {k\pi }{n}}={\frac {n}{2^{n-1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mi>π</mi> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \prod _{k=1}^{n-1}\sin {\frac {k\pi }{n}}={\frac {n}{2^{n-1}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3353129e74dab4d1c88156b3feaed39a87edecb2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.512ex; height:7.343ex;" alt="{\displaystyle \prod _{k=1}^{n-1}\sin {\frac {k\pi }{n}}={\frac {n}{2^{n-1}}}}"> ve <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \prod _{k=1}^{n-1}\cos {\frac {k\pi }{n}}={\frac {\sin {\frac {\pi n}{2}}}{2^{n-1}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mi>π</mi> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>π</mi> <mi>n</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \prod _{k=1}^{n-1}\cos {\frac {k\pi }{n}}={\frac {\sin {\frac {\pi n}{2}}}{2^{n-1}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23771366582f5eb40a1b922f2622a7d49e6196c8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.94ex; height:7.343ex;" alt="{\displaystyle \prod _{k=1}^{n-1}\cos {\frac {k\pi }{n}}={\frac {\sin {\frac {\pi n}{2}}}{2^{n-1}}}.}"> Bunları birleştirmek bize şunları verir; <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \prod _{k=1}^{n-1}\tan {\frac {k\pi }{n}}={\frac {n}{\sin {\frac {\pi n}{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mi>π</mi> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>π</mi> <mi>n</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \prod _{k=1}^{n-1}\tan {\frac {k\pi }{n}}={\frac {n}{\sin {\frac {\pi n}{2}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c859389844bfc0b7e89a545f1707d94f348a073e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.541ex; height:7.343ex;" alt="{\displaystyle \prod _{k=1}^{n-1}\tan {\frac {k\pi }{n}}={\frac {n}{\sin {\frac {\pi n}{2}}}}}"> Eğer n bir tek sayı ise (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=2m+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>2</mn> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=2m+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d89f9bd84d50b23c0f8f2bb0d67c524a13d440f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.699ex; height:2.343ex;" alt="{\displaystyle n=2m+1}">) simetrilerden yararlanarak şu sonucu elde edebiliriz <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \prod _{k=1}^{m}\tan {\frac {k\pi }{2m+1}}={\sqrt {2m+1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mi>π</mi> </mrow> <mrow> <mn>2</mn> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>m</mi> <mo>+</mo> <mn>1</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \prod _{k=1}^{m}\tan {\frac {k\pi }{2m+1}}={\sqrt {2m+1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdd27a671ce4c2ace2f03869e2f93b6687ea362c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:27.385ex; height:6.843ex;" alt="{\displaystyle \prod _{k=1}^{m}\tan {\frac {k\pi }{2m+1}}={\sqrt {2m+1}}}"> <a href="/wiki/Butterworth_filtre" title="Butterworth filtre">Butterworth alçak geçiren filtresinin</a> transfer fonksiyonu polinom ve kutuplar cinsinden ifade edilebilir. Frekansı kesim frekansı olarak belirleyerek, aşağıdaki özdeşlik kanıtlanabilir: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \prod _{k=1}^{n}\sin {\frac {\left(2k-1\right)\pi }{4n}}=\prod _{k=1}^{n}\cos {\frac {\left(2k-1\right)\pi }{4n}}={\frac {\sqrt {2}}{2^{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mi>π</mi> </mrow> <mrow> <mn>4</mn> <mi>n</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mi>π</mi> </mrow> <mrow> <mn>4</mn> <mi>n</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>2</mn> </msqrt> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \prod _{k=1}^{n}\sin {\frac {\left(2k-1\right)\pi }{4n}}=\prod _{k=1}^{n}\cos {\frac {\left(2k-1\right)\pi }{4n}}={\frac {\sqrt {2}}{2^{n}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1b66a3fb7f313ab994957a0a4356f061ac6758e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:45.068ex; height:7.009ex;" alt="{\displaystyle \prod _{k=1}^{n}\sin {\frac {\left(2k-1\right)\pi }{4n}}=\prod _{k=1}^{n}\cos {\frac {\left(2k-1\right)\pi }{4n}}={\frac {\sqrt {2}}{2^{n}}}}"> <div class="mw-heading mw-heading3"><h3 id="π'nin_hesaplanması">π'nin hesaplanması</h3>[<a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&veaction=edit&section=36" title="Değiştirilen bölüm: π'nin hesaplanması" class="mw-editsection-visualeditor">değiştir</a> | <a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&action=edit&section=36" title="Bölümün kaynak kodunu değiştir: π'nin hesaplanması">kaynağı değiştir</a>]</div> <a href="/wiki/Pi_say%C4%B1s%C4%B1" title="Pi sayısı">π</a>'yi <a href="/w/index.php?title=Pi_say%C4%B1s%C4%B1n%C4%B1n_yakla%C5%9F%C4%B1k_de%C4%9Ferleri&action=edit&redlink=1" class="new" title="Pi sayısının yaklaşık değerleri (sayfa mevcut değil)">çok sayıda basamağa kadar</a> hesaplamanın etkili bir yolu, <a href="/wiki/John_Machin" title="John Machin">Machin</a>'den kaynaklanan aşağıdaki değişkensiz özdeşliğe dayanır. Bu <a href="/w/index.php?title=Machin_benzeri_form%C3%BCl&action=edit&redlink=1" class="new" title="Machin benzeri formül (sayfa mevcut değil)">Machin benzeri formül</a> olarak bilinir: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\pi }{4}}=4\arctan {\frac {1}{5}}-\arctan {\frac {1}{239}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mo>=</mo> <mn>4</mn> <mi>arctan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mrow> <mo>−</mo> <mi>arctan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>239</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\pi }{4}}=4\arctan {\frac {1}{5}}-\arctan {\frac {1}{239}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba362ff207097dc35ca873f9a16bcda21a96b278" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:29.685ex; height:5.176ex;" alt="{\displaystyle {\frac {\pi }{4}}=4\arctan {\frac {1}{5}}-\arctan {\frac {1}{239}}}"> veya alternatif olarak <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a>'in bir özdeşliğini kullanarak: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\pi }{4}}=5\arctan {\frac {1}{7}}+2\arctan {\frac {3}{79}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mo>=</mo> <mn>5</mn> <mi>arctan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>7</mn> </mfrac> </mrow> <mo>+</mo> <mn>2</mn> <mi>arctan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>79</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\pi }{4}}=5\arctan {\frac {1}{7}}+2\arctan {\frac {3}{79}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a92944742871304556f99c70f1d0dcddbfdd6d19" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:30.072ex; height:5.343ex;" alt="{\displaystyle {\frac {\pi }{4}}=5\arctan {\frac {1}{7}}+2\arctan {\frac {3}{79}}}"> veya <a href="/wiki/Pisagor_%C3%BC%C3%A7l%C3%BCs%C3%BC" title="Pisagor üçlüsü">Pisagor üçlülerini</a> kullanarak: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi =\arccos {\frac {4}{5}}+\arccos {\frac {5}{13}}+\arccos {\frac {16}{65}}=\arcsin {\frac {3}{5}}+\arcsin {\frac {12}{13}}+\arcsin {\frac {63}{65}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>=</mo> <mi>arccos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>5</mn> </mfrac> </mrow> <mo>+</mo> <mi>arccos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>13</mn> </mfrac> </mrow> <mo>+</mo> <mi>arccos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>16</mn> <mn>65</mn> </mfrac> </mrow> <mo>=</mo> <mi>arcsin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>5</mn> </mfrac> </mrow> <mo>+</mo> <mi>arcsin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>12</mn> <mn>13</mn> </mfrac> </mrow> <mo>+</mo> <mi>arcsin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>63</mn> <mn>65</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi =\arccos {\frac {4}{5}}+\arccos {\frac {5}{13}}+\arccos {\frac {16}{65}}=\arcsin {\frac {3}{5}}+\arcsin {\frac {12}{13}}+\arcsin {\frac {63}{65}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/868e08992d38cdf1db57d2fe871465994374dece" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:75.04ex; height:5.176ex;" alt="{\displaystyle \pi =\arccos {\frac {4}{5}}+\arccos {\frac {5}{13}}+\arccos {\frac {16}{65}}=\arcsin {\frac {3}{5}}+\arcsin {\frac {12}{13}}+\arcsin {\frac {63}{65}}.}"> Diğerleri ise şunlardır:<a href="#cite_note-Wu-47">[47]</a><a href="#cite_note-Harris-51">[51]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\pi }{4}}=\arctan {\frac {1}{2}}+\arctan {\frac {1}{3}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>arctan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\pi }{4}}=\arctan {\frac {1}{2}}+\arctan {\frac {1}{3}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70a2d0d101a812a677f9de8c4f26ad60d3d2ae5c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:26.457ex; height:5.176ex;" alt="{\displaystyle {\frac {\pi }{4}}=\arctan {\frac {1}{2}}+\arctan {\frac {1}{3}},}"> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi =\arctan 1+\arctan 2+\arctan 3,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mn>1</mn> <mo>+</mo> <mi>arctan</mi> <mo>⁡</mo> <mn>2</mn> <mo>+</mo> <mi>arctan</mi> <mo>⁡</mo> <mn>3</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi =\arctan 1+\arctan 2+\arctan 3,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eaaf9d009d99a22e3f36dc3e2de8bcc9521fff86" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:34.805ex; height:2.509ex;" alt="{\displaystyle \pi =\arctan 1+\arctan 2+\arctan 3,}"> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\pi }{4}}=2\arctan {\frac {1}{3}}+\arctan {\frac {1}{7}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> <mi>arctan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mi>arctan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>7</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\pi }{4}}=2\arctan {\frac {1}{3}}+\arctan {\frac {1}{7}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de45b7f0d9700152d497c26fe7114051d28859cd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:28.007ex; height:5.343ex;" alt="{\displaystyle {\frac {\pi }{4}}=2\arctan {\frac {1}{3}}+\arctan {\frac {1}{7}}.}"> Genel olarak, θn = Σn−1k=1 arctan tk ∈ (π/4, 3π/4) olan t1, ..., tn−1 ∈ (−1, 1) sayıları için tn = tan(π/2 − θn) = cot θn olsun. Bu son ifade, tanjantları t1, ..., tn−1 olan bir açılar toplamının kotanjantı için formül kullanılarak doğrudan hesaplanabilir ve değeri (-1, 1) içinde olacaktır. Özellikle, tüm t1, ..., tn−1 değerleri rasyonel olduğunda hesaplanan tn de rasyonel olacaktır. Bu değerlerle, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {\pi }{2}}&=\sum _{k=1}^{n}\arctan(t_{k})\\\pi &=\sum _{k=1}^{n}\operatorname {sgn}(t_{k})\arccos \left({\frac {1-t_{k}^{2}}{1+t_{k}^{2}}}\right)\\\pi &=\sum _{k=1}^{n}\arcsin \left({\frac {2t_{k}}{1+t_{k}^{2}}}\right)\\\pi &=\sum _{k=1}^{n}\arctan \left({\frac {2t_{k}}{1-t_{k}^{2}}}\right)\,,\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>π</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mi>arccos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <msubsup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msubsup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>π</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>arcsin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msubsup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>π</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msubsup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {\pi }{2}}&=\sum _{k=1}^{n}\arctan(t_{k})\\\pi &=\sum _{k=1}^{n}\operatorname {sgn}(t_{k})\arccos \left({\frac {1-t_{k}^{2}}{1+t_{k}^{2}}}\right)\\\pi &=\sum _{k=1}^{n}\arcsin \left({\frac {2t_{k}}{1+t_{k}^{2}}}\right)\\\pi &=\sum _{k=1}^{n}\arctan \left({\frac {2t_{k}}{1-t_{k}^{2}}}\right)\,,\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0b63307dd37f5a18a7d8aa871813a45851f50c2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -14.338ex; width:33.922ex; height:29.843ex;" alt="{\displaystyle {\begin{aligned}{\frac {\pi }{2}}&=\sum _{k=1}^{n}\arctan(t_{k})\\\pi &=\sum _{k=1}^{n}\operatorname {sgn}(t_{k})\arccos \left({\frac {1-t_{k}^{2}}{1+t_{k}^{2}}}\right)\\\pi &=\sum _{k=1}^{n}\arcsin \left({\frac {2t_{k}}{1+t_{k}^{2}}}\right)\\\pi &=\sum _{k=1}^{n}\arctan \left({\frac {2t_{k}}{1-t_{k}^{2}}}\right)\,,\end{aligned}}}"> burada ilk ifade hariç hepsinde tanjnat yarım açı formüllerini kullandık. İlk iki formül, tk değerlerinden biri veya daha fazlası (-1, 1) içinde olmasa bile çalışır. t = p/q rasyonel ise, yukarıdaki formüllerdeki (2t, 1 − t2, 1 + t2) değerlerinin Pisagor üçlüsü (2pq, q2 − p2, q2 + p2) ile orantılı olduğunu unutmayın. Örneğin, n = 3 terimleri için, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\pi }{2}}=\arctan \left({\frac {a}{b}}\right)+\arctan \left({\frac {c}{d}}\right)+\arctan \left({\frac {bd-ac}{ad+bc}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mi>d</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mi>d</mi> <mo>−</mo> <mi>a</mi> <mi>c</mi> </mrow> <mrow> <mi>a</mi> <mi>d</mi> <mo>+</mo> <mi>b</mi> <mi>c</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\pi }{2}}=\arctan \left({\frac {a}{b}}\right)+\arctan \left({\frac {c}{d}}\right)+\arctan \left({\frac {bd-ac}{ad+bc}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cc40e73e89c8e9b173fc9b1b5f5a821fcf7344e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:51.562ex; height:6.176ex;" alt="{\displaystyle {\frac {\pi }{2}}=\arctan \left({\frac {a}{b}}\right)+\arctan \left({\frac {c}{d}}\right)+\arctan \left({\frac {bd-ac}{ad+bc}}\right)}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall \ \ a,b,c,d>0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mtext> </mtext> <mtext> </mtext> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>,</mo> <mi>d</mi> <mo>></mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \ \ a,b,c,d>0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6eac621972779d67d452b0904904f8b37571ddb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.913ex; height:2.509ex;" alt="{\displaystyle \forall \ \ a,b,c,d>0.}"> <div class="mw-heading mw-heading3"><h3 id="Öklid'in_bir_özdeşliği">Öklid'in bir özdeşliği</h3>[<a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&veaction=edit&section=37" title="Değiştirilen bölüm: Öklid'in bir özdeşliği" class="mw-editsection-visualeditor">değiştir</a> | <a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&action=edit&section=37" title="Bölümün kaynak kodunu değiştir: Öklid'in bir özdeşliği">kaynağı değiştir</a>]</div> <a href="/wiki/%C3%96klid" title="Öklid">Öklid</a>, <a href="/wiki/%C3%96klid%27in_Elementleri" title="Öklid'in Elementleri">Elementler</a> adlı eserinin XIII. Kitabı, 10. Önermesinde bir çemberin içine yerleştirilmiş düzgün beşgenin kenarındaki karenin alanının, aynı çemberin içine yerleştirilmiş düzgün altıgen ve düzgün ongenin kenarlarındaki karelerin alanlarının toplamına eşit olduğunu göstermiştir. Modern trigonometri dilinde bu şöyle ifade edilir: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin ^{2}18^{\circ }+\sin ^{2}30^{\circ }=\sin ^{2}36^{\circ }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <msup> <mn>18</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>+</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <msup> <mn>30</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>=</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <msup> <mn>36</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin ^{2}18^{\circ }+\sin ^{2}30^{\circ }=\sin ^{2}36^{\circ }.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f5bcb6987a9a34dfab0acd02f02ae1d7565176c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:29.614ex; height:2.843ex;" alt="{\displaystyle \sin ^{2}18^{\circ }+\sin ^{2}30^{\circ }=\sin ^{2}36^{\circ }.}"> <a href="/wiki/Batlamyus" title="Batlamyus">Batlamyus</a> bu önermeyi <a href="/wiki/Almagest" title="Almagest">Almagest</a>'in I. Kitap, 11. Bölümünde <a href="/wiki/Batlamyus_kiri%C5%9Fler_tablosu" title="Batlamyus kirişler tablosu">Batlamyus kirişler tablosundaki</a> bazı açıları hesaplamak için kullanmıştır. <div class="mw-heading mw-heading2"><h2 id="Trigonometrik_fonksiyonların_bileşimi">Trigonometrik fonksiyonların bileşimi</h2>[<a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&veaction=edit&section=38" title="Değiştirilen bölüm: Trigonometrik fonksiyonların bileşimi" class="mw-editsection-visualeditor">değiştir</a> | <a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&action=edit&section=38" title="Bölümün kaynak kodunu değiştir: Trigonometrik fonksiyonların bileşimi">kaynağı değiştir</a>]</div> Bu özdeşlikler, trigonometrik bir fonksiyonun trigonometrik bir fonksiyonunu içerir:<a href="#cite_note-52">[52]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(t\sin x)=J_{0}(t)+2\sum _{k=1}^{\infty }J_{2k}(t)\cos(2kx)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(t\sin x)=J_{0}(t)+2\sum _{k=1}^{\infty }J_{2k}(t)\cos(2kx)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4405aef6354af781dd00f4e5d8ccd01e4077fa2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:41.804ex; height:6.843ex;" alt="{\displaystyle \cos(t\sin x)=J_{0}(t)+2\sum _{k=1}^{\infty }J_{2k}(t)\cos(2kx)}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(t\sin x)=2\sum _{k=0}^{\infty }J_{2k+1}(t)\sin {\big (}(2k+1)x{\big )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(t\sin x)=2\sum _{k=0}^{\infty }J_{2k+1}(t)\sin {\big (}(2k+1)x{\big )}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7962c294ff1ea6846bbd8688106876c0fa37446" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:42.08ex; height:7.009ex;" alt="{\displaystyle \sin(t\sin x)=2\sum _{k=0}^{\infty }J_{2k+1}(t)\sin {\big (}(2k+1)x{\big )}}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(t\cos x)=J_{0}(t)+2\sum _{k=1}^{\infty }(-1)^{k}J_{2k}(t)\cos(2kx)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(t\cos x)=J_{0}(t)+2\sum _{k=1}^{\infty }(-1)^{k}J_{2k}(t)\cos(2kx)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94587da38f7f3c09db62e82c46485f69bb6fd4d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:47.541ex; height:6.843ex;" alt="{\displaystyle \cos(t\cos x)=J_{0}(t)+2\sum _{k=1}^{\infty }(-1)^{k}J_{2k}(t)\cos(2kx)}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(t\cos x)=2\sum _{k=0}^{\infty }(-1)^{k}J_{2k+1}(t)\cos {\big (}(2k+1)x{\big )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(t\cos x)=2\sum _{k=0}^{\infty }(-1)^{k}J_{2k+1}(t)\cos {\big (}(2k+1)x{\big )}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65a85f8ef4f0e10a54c2395cc598941841501a2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:48.072ex; height:7.009ex;" alt="{\displaystyle \sin(t\cos x)=2\sum _{k=0}^{\infty }(-1)^{k}J_{2k+1}(t)\cos {\big (}(2k+1)x{\big )}}"></dd></dl> burada Ji <a href="/wiki/Bessel_fonksiyonu" title="Bessel fonksiyonu">Bessel fonksiyonlarıdır</a>. <div class="mw-heading mw-heading2"><h2 id="α_+_β_+_γ_=_180°_durumu_için_diğer_"koşullu"_özdeşlikler">α + β + γ = 180° durumu için diğer "koşullu" özdeşlikler</h2>[<a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&veaction=edit&section=39" title="Değiştirilen bölüm: α + β + γ = 180° durumu için diğer "koşullu" özdeşlikler" class="mw-editsection-visualeditor">değiştir</a> | <a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&action=edit&section=39" title="Bölümün kaynak kodunu değiştir: α + β + γ = 180° durumu için diğer "koşullu" özdeşlikler">kaynağı değiştir</a>]</div> Bir koşullu trigonometrik özdeşlik, trigonometrik fonksiyonların argümanları üzerinde belirtilen koşullar sağlandığında geçerli olan bir trigonometrik özdeşliktir.<a href="#cite_note-53">[53]</a> Aşağıdaki formüller rastgele düzlem üçgenler için geçerlidir ve formüllerde yer alan fonksiyonlar iyi tanımlandığı sürece <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha +\beta +\gamma =180^{\circ },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo>+</mo> <mi>γ</mi> <mo>=</mo> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha +\beta +\gamma =180^{\circ },}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/928612a6876aa87cd721d27af92267baadba3238" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.05ex; height:2.843ex;" alt="{\displaystyle \alpha +\beta +\gamma =180^{\circ },}"> formülünden takip edilir (ikincisi sadece tanjant ve kotanjantların yer aldığı formüller için geçerlidir). <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\tan \alpha +\tan \beta +\tan \gamma &=\tan \alpha \tan \beta \tan \gamma \\1&=\cot \beta \cot \gamma +\cot \gamma \cot \alpha +\cot \alpha \cot \beta \\\cot \left({\frac {\alpha }{2}}\right)+\cot \left({\frac {\beta }{2}}\right)+\cot \left({\frac {\gamma }{2}}\right)&=\cot \left({\frac {\alpha }{2}}\right)\cot \left({\frac {\beta }{2}}\right)\cot \left({\frac {\gamma }{2}}\right)\\1&=\tan \left({\frac {\beta }{2}}\right)\tan \left({\frac {\gamma }{2}}\right)+\tan \left({\frac {\gamma }{2}}\right)\tan \left({\frac {\alpha }{2}}\right)+\tan \left({\frac {\alpha }{2}}\right)\tan \left({\frac {\beta }{2}}\right)\\\sin \alpha +\sin \beta +\sin \gamma &=4\cos \left({\frac {\alpha }{2}}\right)\cos \left({\frac {\beta }{2}}\right)\cos \left({\frac {\gamma }{2}}\right)\\-\sin \alpha +\sin \beta +\sin \gamma &=4\cos \left({\frac {\alpha }{2}}\right)\sin \left({\frac {\beta }{2}}\right)\sin \left({\frac {\gamma }{2}}\right)\\\cos \alpha +\cos \beta +\cos \gamma &=4\sin \left({\frac {\alpha }{2}}\right)\sin \left({\frac {\beta }{2}}\right)\sin \left({\frac {\gamma }{2}}\right)+1\\-\cos \alpha +\cos \beta +\cos \gamma &=4\sin \left({\frac {\alpha }{2}}\right)\cos \left({\frac {\beta }{2}}\right)\cos \left({\frac {\gamma }{2}}\right)-1\\\sin(2\alpha )+\sin(2\beta )+\sin(2\gamma )&=4\sin \alpha \sin \beta \sin \gamma \\-\sin(2\alpha )+\sin(2\beta )+\sin(2\gamma )&=4\sin \alpha \cos \beta \cos \gamma \\\cos(2\alpha )+\cos(2\beta )+\cos(2\gamma )&=-4\cos \alpha \cos \beta \cos \gamma -1\\-\cos(2\alpha )+\cos(2\beta )+\cos(2\gamma )&=-4\cos \alpha \sin \beta \sin \gamma +1\\\sin ^{2}\alpha +\sin ^{2}\beta +\sin ^{2}\gamma &=2\cos \alpha \cos \beta \cos \gamma +2\\-\sin ^{2}\alpha +\sin ^{2}\beta +\sin ^{2}\gamma &=2\cos \alpha \sin \beta \sin \gamma \\\cos ^{2}\alpha +\cos ^{2}\beta +\cos ^{2}\gamma &=-2\cos \alpha \cos \beta \cos \gamma +1\\-\cos ^{2}\alpha +\cos ^{2}\beta +\cos ^{2}\gamma &=-2\cos \alpha \sin \beta \sin \gamma +1\\\sin ^{2}(2\alpha )+\sin ^{2}(2\beta )+\sin ^{2}(2\gamma )&=-2\cos(2\alpha )\cos(2\beta )\cos(2\gamma )+2\\\cos ^{2}(2\alpha )+\cos ^{2}(2\beta )+\cos ^{2}(2\gamma )&=2\cos(2\alpha )\,\cos(2\beta )\,\cos(2\gamma )+1\\1&=\sin ^{2}\left({\frac {\alpha }{2}}\right)+\sin ^{2}\left({\frac {\beta }{2}}\right)+\sin ^{2}\left({\frac {\gamma }{2}}\right)+2\sin \left({\frac {\alpha }{2}}\right)\,\sin \left({\frac {\beta }{2}}\right)\,\sin \left({\frac {\gamma }{2}}\right)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>tan</mi> <mo>⁡</mo> <mi>α</mi> <mo>+</mo> <mi>tan</mi> <mo>⁡</mo> <mi>β</mi> <mo>+</mo> <mi>tan</mi> <mo>⁡</mo> <mi>γ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>tan</mi> <mo>⁡</mo> <mi>α</mi> <mi>tan</mi> <mo>⁡</mo> <mi>β</mi> <mi>tan</mi> <mo>⁡</mo> <mi>γ</mi> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>cot</mi> <mo>⁡</mo> <mi>β</mi> <mi>cot</mi> <mo>⁡</mo> <mi>γ</mi> <mo>+</mo> <mi>cot</mi> <mo>⁡</mo> <mi>γ</mi> <mi>cot</mi> <mo>⁡</mo> <mi>α</mi> <mo>+</mo> <mi>cot</mi> <mo>⁡</mo> <mi>α</mi> <mi>cot</mi> <mo>⁡</mo> <mi>β</mi> </mtd> </mtr> <mtr> <mtd> <mi>cot</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>cot</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>β</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>cot</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>γ</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>cot</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>cot</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>β</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>cot</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>γ</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>tan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>β</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>tan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>γ</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>tan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>γ</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>tan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>tan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>tan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>β</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>α</mi> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mi>β</mi> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mi>γ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>4</mn> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>β</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>γ</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>α</mi> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mi>β</mi> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mi>γ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>4</mn> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>β</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>γ</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mi>β</mi> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mi>γ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>4</mn> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>β</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>γ</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mi>β</mi> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mi>γ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>4</mn> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>β</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>γ</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>α</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>β</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>γ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>4</mn> <mi>sin</mi> <mo>⁡</mo> <mi>α</mi> <mi>sin</mi> <mo>⁡</mo> <mi>β</mi> <mi>sin</mi> <mo>⁡</mo> <mi>γ</mi> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>α</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>β</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>γ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>4</mn> <mi>sin</mi> <mo>⁡</mo> <mi>α</mi> <mi>cos</mi> <mo>⁡</mo> <mi>β</mi> <mi>cos</mi> <mo>⁡</mo> <mi>γ</mi> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>α</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>β</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>γ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mn>4</mn> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> <mi>cos</mi> <mo>⁡</mo> <mi>β</mi> <mi>cos</mi> <mo>⁡</mo> <mi>γ</mi> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>α</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>β</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>γ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mn>4</mn> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> <mi>sin</mi> <mo>⁡</mo> <mi>β</mi> <mi>sin</mi> <mo>⁡</mo> <mi>γ</mi> <mo>+</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>α</mi> <mo>+</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>β</mi> <mo>+</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>γ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> <mi>cos</mi> <mo>⁡</mo> <mi>β</mi> <mi>cos</mi> <mo>⁡</mo> <mi>γ</mi> <mo>+</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>α</mi> <mo>+</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>β</mi> <mo>+</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>γ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> <mi>sin</mi> <mo>⁡</mo> <mi>β</mi> <mi>sin</mi> <mo>⁡</mo> <mi>γ</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>α</mi> <mo>+</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>β</mi> <mo>+</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>γ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> <mi>cos</mi> <mo>⁡</mo> <mi>β</mi> <mi>cos</mi> <mo>⁡</mo> <mi>γ</mi> <mo>+</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>α</mi> <mo>+</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>β</mi> <mo>+</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>γ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> <mi>sin</mi> <mo>⁡</mo> <mi>β</mi> <mi>sin</mi> <mo>⁡</mo> <mi>γ</mi> <mo>+</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>α</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>β</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>γ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>α</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>β</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>γ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>α</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>β</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>γ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>α</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>β</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>γ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>β</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>γ</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mn>2</mn> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>β</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>γ</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\tan \alpha +\tan \beta +\tan \gamma &=\tan \alpha \tan \beta \tan \gamma \\1&=\cot \beta \cot \gamma +\cot \gamma \cot \alpha +\cot \alpha \cot \beta \\\cot \left({\frac {\alpha }{2}}\right)+\cot \left({\frac {\beta }{2}}\right)+\cot \left({\frac {\gamma }{2}}\right)&=\cot \left({\frac {\alpha }{2}}\right)\cot \left({\frac {\beta }{2}}\right)\cot \left({\frac {\gamma }{2}}\right)\\1&=\tan \left({\frac {\beta }{2}}\right)\tan \left({\frac {\gamma }{2}}\right)+\tan \left({\frac {\gamma }{2}}\right)\tan \left({\frac {\alpha }{2}}\right)+\tan \left({\frac {\alpha }{2}}\right)\tan \left({\frac {\beta }{2}}\right)\\\sin \alpha +\sin \beta +\sin \gamma &=4\cos \left({\frac {\alpha }{2}}\right)\cos \left({\frac {\beta }{2}}\right)\cos \left({\frac {\gamma }{2}}\right)\\-\sin \alpha +\sin \beta +\sin \gamma &=4\cos \left({\frac {\alpha }{2}}\right)\sin \left({\frac {\beta }{2}}\right)\sin \left({\frac {\gamma }{2}}\right)\\\cos \alpha +\cos \beta +\cos \gamma &=4\sin \left({\frac {\alpha }{2}}\right)\sin \left({\frac {\beta }{2}}\right)\sin \left({\frac {\gamma }{2}}\right)+1\\-\cos \alpha +\cos \beta +\cos \gamma &=4\sin \left({\frac {\alpha }{2}}\right)\cos \left({\frac {\beta }{2}}\right)\cos \left({\frac {\gamma }{2}}\right)-1\\\sin(2\alpha )+\sin(2\beta )+\sin(2\gamma )&=4\sin \alpha \sin \beta \sin \gamma \\-\sin(2\alpha )+\sin(2\beta )+\sin(2\gamma )&=4\sin \alpha \cos \beta \cos \gamma \\\cos(2\alpha )+\cos(2\beta )+\cos(2\gamma )&=-4\cos \alpha \cos \beta \cos \gamma -1\\-\cos(2\alpha )+\cos(2\beta )+\cos(2\gamma )&=-4\cos \alpha \sin \beta \sin \gamma +1\\\sin ^{2}\alpha +\sin ^{2}\beta +\sin ^{2}\gamma &=2\cos \alpha \cos \beta \cos \gamma +2\\-\sin ^{2}\alpha +\sin ^{2}\beta +\sin ^{2}\gamma &=2\cos \alpha \sin \beta \sin \gamma \\\cos ^{2}\alpha +\cos ^{2}\beta +\cos ^{2}\gamma &=-2\cos \alpha \cos \beta \cos \gamma +1\\-\cos ^{2}\alpha +\cos ^{2}\beta +\cos ^{2}\gamma &=-2\cos \alpha \sin \beta \sin \gamma +1\\\sin ^{2}(2\alpha )+\sin ^{2}(2\beta )+\sin ^{2}(2\gamma )&=-2\cos(2\alpha )\cos(2\beta )\cos(2\gamma )+2\\\cos ^{2}(2\alpha )+\cos ^{2}(2\beta )+\cos ^{2}(2\gamma )&=2\cos(2\alpha )\,\cos(2\beta )\,\cos(2\gamma )+1\\1&=\sin ^{2}\left({\frac {\alpha }{2}}\right)+\sin ^{2}\left({\frac {\beta }{2}}\right)+\sin ^{2}\left({\frac {\gamma }{2}}\right)+2\sin \left({\frac {\alpha }{2}}\right)\,\sin \left({\frac {\beta }{2}}\right)\,\sin \left({\frac {\gamma }{2}}\right)\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ea204e4a845089ecd8c8c34b6caa9e041415969" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -40.505ex; width:98.065ex; height:82.176ex;" alt="{\displaystyle {\begin{aligned}\tan \alpha +\tan \beta +\tan \gamma &=\tan \alpha \tan \beta \tan \gamma \\1&=\cot \beta \cot \gamma +\cot \gamma \cot \alpha +\cot \alpha \cot \beta \\\cot \left({\frac {\alpha }{2}}\right)+\cot \left({\frac {\beta }{2}}\right)+\cot \left({\frac {\gamma }{2}}\right)&=\cot \left({\frac {\alpha }{2}}\right)\cot \left({\frac {\beta }{2}}\right)\cot \left({\frac {\gamma }{2}}\right)\\1&=\tan \left({\frac {\beta }{2}}\right)\tan \left({\frac {\gamma }{2}}\right)+\tan \left({\frac {\gamma }{2}}\right)\tan \left({\frac {\alpha }{2}}\right)+\tan \left({\frac {\alpha }{2}}\right)\tan \left({\frac {\beta }{2}}\right)\\\sin \alpha +\sin \beta +\sin \gamma &=4\cos \left({\frac {\alpha }{2}}\right)\cos \left({\frac {\beta }{2}}\right)\cos \left({\frac {\gamma }{2}}\right)\\-\sin \alpha +\sin \beta +\sin \gamma &=4\cos \left({\frac {\alpha }{2}}\right)\sin \left({\frac {\beta }{2}}\right)\sin \left({\frac {\gamma }{2}}\right)\\\cos \alpha +\cos \beta +\cos \gamma &=4\sin \left({\frac {\alpha }{2}}\right)\sin \left({\frac {\beta }{2}}\right)\sin \left({\frac {\gamma }{2}}\right)+1\\-\cos \alpha +\cos \beta +\cos \gamma &=4\sin \left({\frac {\alpha }{2}}\right)\cos \left({\frac {\beta }{2}}\right)\cos \left({\frac {\gamma }{2}}\right)-1\\\sin(2\alpha )+\sin(2\beta )+\sin(2\gamma )&=4\sin \alpha \sin \beta \sin \gamma \\-\sin(2\alpha )+\sin(2\beta )+\sin(2\gamma )&=4\sin \alpha \cos \beta \cos \gamma \\\cos(2\alpha )+\cos(2\beta )+\cos(2\gamma )&=-4\cos \alpha \cos \beta \cos \gamma -1\\-\cos(2\alpha )+\cos(2\beta )+\cos(2\gamma )&=-4\cos \alpha \sin \beta \sin \gamma +1\\\sin ^{2}\alpha +\sin ^{2}\beta +\sin ^{2}\gamma &=2\cos \alpha \cos \beta \cos \gamma +2\\-\sin ^{2}\alpha +\sin ^{2}\beta +\sin ^{2}\gamma &=2\cos \alpha \sin \beta \sin \gamma \\\cos ^{2}\alpha +\cos ^{2}\beta +\cos ^{2}\gamma &=-2\cos \alpha \cos \beta \cos \gamma +1\\-\cos ^{2}\alpha +\cos ^{2}\beta +\cos ^{2}\gamma &=-2\cos \alpha \sin \beta \sin \gamma +1\\\sin ^{2}(2\alpha )+\sin ^{2}(2\beta )+\sin ^{2}(2\gamma )&=-2\cos(2\alpha )\cos(2\beta )\cos(2\gamma )+2\\\cos ^{2}(2\alpha )+\cos ^{2}(2\beta )+\cos ^{2}(2\gamma )&=2\cos(2\alpha )\,\cos(2\beta )\,\cos(2\gamma )+1\\1&=\sin ^{2}\left({\frac {\alpha }{2}}\right)+\sin ^{2}\left({\frac {\beta }{2}}\right)+\sin ^{2}\left({\frac {\gamma }{2}}\right)+2\sin \left({\frac {\alpha }{2}}\right)\,\sin \left({\frac {\beta }{2}}\right)\,\sin \left({\frac {\gamma }{2}}\right)\end{aligned}}}"> <div class="mw-heading mw-heading2"><h2 id="Tarihsel_stenolar">Tarihsel stenolar</h2>[<a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&veaction=edit&section=40" title="Değiştirilen bölüm: Tarihsel stenolar" class="mw-editsection-visualeditor">değiştir</a> | <a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&action=edit&section=40" title="Bölümün kaynak kodunu değiştir: Tarihsel stenolar">kaynağı değiştir</a>]</div> <div role="note" class="hatnote navigation-not-searchable">Ana maddeler: <a href="/w/index.php?title=Versin%C3%BCs&action=edit&redlink=1" class="new" title="Versinüs (sayfa mevcut değil)">Versinüs</a> ve <a href="/w/index.php?title=Ekssekant&action=edit&redlink=1" class="new" title="Ekssekant (sayfa mevcut değil)">Ekssekant</a></div> <a href="/w/index.php?title=Versin%C3%BCs&action=edit&redlink=1" class="new" title="Versinüs (sayfa mevcut değil)">Versinüs</a>, <a href="/w/index.php?title=Koversin%C3%BCs&action=edit&redlink=1" class="new" title="Koversinüs (sayfa mevcut değil)">koversinüs</a>, <a href="/w/index.php?title=Haversin%C3%BCs&action=edit&redlink=1" class="new" title="Haversinüs (sayfa mevcut değil)">haversinüs</a> ve <a href="/w/index.php?title=Ekssekant&action=edit&redlink=1" class="new" title="Ekssekant (sayfa mevcut değil)">ekssekant</a> seyrüseferde kullanılmıştır. Örneğin, <a href="/w/index.php?title=Haversin%C3%BCs_form%C3%BCl%C3%BC&action=edit&redlink=1" class="new" title="Haversinüs formülü (sayfa mevcut değil)">haversinüs formülü</a> bir küre üzerindeki iki nokta arasındaki mesafeyi hesaplamak için kullanılmıştır. Günümüzde nadiren kullanılmaktadırlar. <div class="mw-heading mw-heading2"><h2 id="Diğer">Diğer</h2>[<a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&veaction=edit&section=41" title="Değiştirilen bölüm: Diğer" class="mw-editsection-visualeditor">değiştir</a> | <a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&action=edit&section=41" title="Bölümün kaynak kodunu değiştir: Diğer">kaynağı değiştir</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Dirichlet_çekirdeği">Dirichlet çekirdeği</h3>[<a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&veaction=edit&section=42" title="Değiştirilen bölüm: Dirichlet çekirdeği" class="mw-editsection-visualeditor">değiştir</a> | <a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&action=edit&section=42" title="Bölümün kaynak kodunu değiştir: Dirichlet çekirdeği">kaynağı değiştir</a>]</div> <div role="note" class="hatnote navigation-not-searchable">Ana madde: <a href="/w/index.php?title=Dirichlet_%C3%A7ekirde%C4%9Fi&action=edit&redlink=1" class="new" title="Dirichlet çekirdeği (sayfa mevcut değil)">Dirichlet çekirdeği</a></div> <a href="/w/index.php?title=Dirichlet_%C3%A7ekirde%C4%9Fi&action=edit&redlink=1" class="new" title="Dirichlet çekirdeği (sayfa mevcut değil)">Dirichlet çekirdeği</a> Dn(x), bir sonraki özdeşliğin her iki tarafında meydana gelen fonksiyondur: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+2\cos x+2\cos(2x)+2\cos(3x)+\cdots +2\cos(nx)={\frac {\sin \left(\left(n+{\frac {1}{2}}\right)x\right)}{\sin \left({\frac {1}{2}}x\right)}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mn>2</mn> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+2\cos x+2\cos(2x)+2\cos(3x)+\cdots +2\cos(nx)={\frac {\sin \left(\left(n+{\frac {1}{2}}\right)x\right)}{\sin \left({\frac {1}{2}}x\right)}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c466bf6f250293684b6fdc210d8f54952ea1aaf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:70.89ex; height:7.843ex;" alt="{\displaystyle 1+2\cos x+2\cos(2x)+2\cos(3x)+\cdots +2\cos(nx)={\frac {\sin \left(\left(n+{\frac {1}{2}}\right)x\right)}{\sin \left({\frac {1}{2}}x\right)}}.}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73efd1f6493490b058097060a572606d2c550a06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }"> periyodundaki herhangi bir <a href="/wiki/%C4%B0ntegral#İntegrali_alınabilir_fonksiyon" title="İntegral">integrallenebilir fonksiyonun</a> Dirichlet çekirdeği ile <a href="/wiki/Konvol%C3%BCsyon" title="Konvolüsyon">konvolüsyonu</a>, fonksiyonun <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">inci derece Fourier yaklaşımı ile çakışır. Aynı durum herhangi bir <a href="/wiki/%C3%96l%C3%A7%C3%BC_(matematik)" title="Ölçü (matematik)">ölçü</a> veya <a href="/w/index.php?title=Da%C4%9F%C4%B1l%C4%B1m_(matematik)&action=edit&redlink=1" class="new" title="Dağılım (matematik) (sayfa mevcut değil)">genelleştirilmiş fonksiyon</a> için de geçerlidir. <div class="mw-heading mw-heading3"><h3 id="Tanjant_yarım_açı_ikamesi">Tanjant yarım açı ikamesi</h3>[<a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&veaction=edit&section=43" title="Değiştirilen bölüm: Tanjant yarım açı ikamesi" class="mw-editsection-visualeditor">değiştir</a> | <a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&action=edit&section=43" title="Bölümün kaynak kodunu değiştir: Tanjant yarım açı ikamesi">kaynağı değiştir</a>]</div> <div role="note" class="hatnote navigation-not-searchable">Ana madde: <a href="/w/index.php?title=Tanjant_yar%C4%B1m_a%C3%A7%C4%B1_ikamesi&action=edit&redlink=1" class="new" title="Tanjant yarım açı ikamesi (sayfa mevcut değil)">Tanjant yarım açı ikamesi</a></div> Eğer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t=\tan {\frac {x}{2}},"> <semantics> <mrow> <mi>t</mi> <mo>=</mo> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> </mrow> <annotation encoding="application/x-tex">t=\tan {\frac {x}{2}},</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b161f2b17fa5b9c9e644e5e9700e2231062f3ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.497ex; height:4.676ex;" alt="t=\tan {\frac {x}{2}},"> olarak alırsak<a href="#cite_note-54">[54]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin x={\frac {2t}{1+t^{2}}};\qquad \cos x={\frac {1-t^{2}}{1+t^{2}}};\qquad e^{ix}={\frac {1+it}{1-it}};\qquad dx={\frac {2\,dt}{1+t^{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>t</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>;</mo> <mspace width="2em" /> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>;</mo> <mspace width="2em" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>x</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mi>i</mi> <mi>t</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>i</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>;</mo> <mspace width="2em" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin x={\frac {2t}{1+t^{2}}};\qquad \cos x={\frac {1-t^{2}}{1+t^{2}}};\qquad e^{ix}={\frac {1+it}{1-it}};\qquad dx={\frac {2\,dt}{1+t^{2}}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2359ff0f7ed90e6485eef9340b9ba81b5379690e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:71.527ex; height:6.176ex;" alt="{\displaystyle \sin x={\frac {2t}{1+t^{2}}};\qquad \cos x={\frac {1-t^{2}}{1+t^{2}}};\qquad e^{ix}={\frac {1+it}{1-it}};\qquad dx={\frac {2\,dt}{1+t^{2}}},}"> burada <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{ix}=\cos x+i\sin x,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>x</mi> </mrow> </msup> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{ix}=\cos x+i\sin x,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aab1fcd1a6db5cc6678bb9cbd871580eeeb86eda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.999ex; height:3.009ex;" alt="{\displaystyle e^{ix}=\cos x+i\sin x,}"> bazen <a href="/w/index.php?title=Cis_(matematik)&action=edit&redlink=1" class="new" title="Cis (matematik) (sayfa mevcut değil)">cis</a> x olarak kısaltılır. <a href="/wiki/Kalk%C3%BCl%C3%BCs" title="Kalkülüs">Kalkülüste</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"> yerine tan x/2 kullanıldığında, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09b4b55580d6a821a07ad9fe35be88976917b10b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.572ex; height:2.176ex;" alt="{\displaystyle \sin x}"> yerine 2t/1 + t2, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/184ba70c3a71df25a25c09f34cd7f8175a9b5280" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.828ex; height:1.676ex;" alt="{\displaystyle \cos x}"> yerine 1 − t2/1 + t2 ve dx diferansiyeli yerine 2 dt/1 + t2 yazılır. Böylece <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09b4b55580d6a821a07ad9fe35be88976917b10b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.572ex; height:2.176ex;" alt="{\displaystyle \sin x}"> ve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/184ba70c3a71df25a25c09f34cd7f8175a9b5280" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.828ex; height:1.676ex;" alt="{\displaystyle \cos x}">'in rasyonel fonksiyonları, <a href="/wiki/%C4%B0lkel_fonksiyon" title="İlkel fonksiyon">antitürevlerini</a> bulmak için <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}">'nin rasyonel fonksiyonlarına dönüştürülür. <div class="mw-heading mw-heading3"><h3 id="Viète_sonsuz_çarpımı">Viète sonsuz çarpımı</h3>[<a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&veaction=edit&section=44" title="Değiştirilen bölüm: Viète sonsuz çarpımı" class="mw-editsection-visualeditor">değiştir</a> | <a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&action=edit&section=44" title="Bölümün kaynak kodunu değiştir: Viète sonsuz çarpımı">kaynağı değiştir</a>]</div> <div role="note" class="hatnote navigation-not-searchable">Ayrıca bakınız: <a href="/w/index.php?title=Vi%C3%A8te_form%C3%BCl%C3%BC&action=edit&redlink=1" class="new" title="Viète formülü (sayfa mevcut değil)">Viète formülü</a> ve <a href="/wiki/Sinc_fonksiyonu" title="Sinc fonksiyonu">Sinc fonksiyonu</a></div> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos {\frac {\theta }{2}}\cdot \cos {\frac {\theta }{4}}\cdot \cos {\frac {\theta }{8}}\cdots =\prod _{n=1}^{\infty }\cos {\frac {\theta }{2^{n}}}={\frac {\sin \theta }{\theta }}=\operatorname {sinc} \theta .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>4</mn> </mfrac> </mrow> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>8</mn> </mfrac> </mrow> <mo>⋯</mo> <mo>=</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mi>θ</mi> </mfrac> </mrow> <mo>=</mo> <mi>sinc</mi> <mo>⁡</mo> <mi>θ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos {\frac {\theta }{2}}\cdot \cos {\frac {\theta }{4}}\cdot \cos {\frac {\theta }{8}}\cdots =\prod _{n=1}^{\infty }\cos {\frac {\theta }{2^{n}}}={\frac {\sin \theta }{\theta }}=\operatorname {sinc} \theta .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bb946a8da936c5f1d3c9a568a8fd933fb3d5523" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:53.625ex; height:6.843ex;" alt="{\displaystyle \cos {\frac {\theta }{2}}\cdot \cos {\frac {\theta }{4}}\cdot \cos {\frac {\theta }{8}}\cdots =\prod _{n=1}^{\infty }\cos {\frac {\theta }{2^{n}}}={\frac {\sin \theta }{\theta }}=\operatorname {sinc} \theta .}"> <div class="mw-heading mw-heading2"><h2 id="Ayrıca_bakınız">Ayrıca bakınız</h2>[<a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&veaction=edit&section=45" title="Değiştirilen bölüm: Ayrıca bakınız" class="mw-editsection-visualeditor">değiştir</a> | <a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&action=edit&section=45" title="Bölümün kaynak kodunu değiştir: Ayrıca bakınız">kaynağı değiştir</a>]</div> <style data-mw-deduplicate="TemplateStyles:r25961922">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col"> <ul><li><a href="/wiki/Aristarkus_e%C5%9Fitsizli%C4%9Fi" title="Aristarkus eşitsizliği">Aristarkus eşitsizliği</a></li> <li><a href="/wiki/T%C3%BCrev_alma_kurallar%C4%B1#Trigonometrik_fonksiyonların_türevleri" title="Türev alma kuralları">Trigonometrik fonksiyonların türevleri</a></li> <li><a href="/wiki/Tam_trigonometrik_de%C4%9Ferler" title="Tam trigonometrik değerler">Tam trigonometrik değerler</a> (sinüs ve kosinüsün rasyonel olmayan sayı -surd- cinsinden ifade edilen değerleri)</li> <li><a href="/w/index.php?title=Ekssekant&action=edit&redlink=1" class="new" title="Ekssekant (sayfa mevcut değil)">Ekssekant</a></li> <li><a href="/w/index.php?title=Yar%C4%B1m_kenar_form%C3%BCl%C3%BC&action=edit&redlink=1" class="new" title="Yarım kenar formülü (sayfa mevcut değil)">Yarım kenar formülü</a></li> <li><a href="/wiki/Hiperbolik_fonksiyon" title="Hiperbolik fonksiyon">Hiperbolik fonksiyon</a></li> <li>Üçgenlerin çözümü için yasalar: <ul><li><a href="/wiki/Kosin%C3%BCs_teoremi" title="Kosinüs teoremi">Kosinüs teoremi</a> <ul><li><a href="/w/index.php?title=K%C3%BCresel_kosin%C3%BCs_yasas%C4%B1&action=edit&redlink=1" class="new" title="Küresel kosinüs yasası (sayfa mevcut değil)">Küresel kosinüs yasası</a></li></ul></li> <li><a href="/wiki/Sin%C3%BCs_teoremi" title="Sinüs teoremi">Sinüs teoremi</a></li> <li><a href="/wiki/Tanjant_teoremi" title="Tanjant teoremi">Tanjant teoremi</a></li> <li><a href="/wiki/Kotanjant_teoremi" title="Kotanjant teoremi">Kotanjant teoremi</a></li> <li><a href="/wiki/Mollweide_form%C3%BCl%C3%BC" title="Mollweide formülü">Mollweide formülü</a></li></ul></li> <li><a href="/wiki/Trigonometrik_fonksiyonlar%C4%B1n_integralleri" title="Trigonometrik fonksiyonların integralleri">Trigonometrik fonksiyonların integralleri</a></li> <li><a href="/w/index.php?title=Trigonometride_an%C4%B1msat%C4%B1c%C4%B1lar&action=edit&redlink=1" class="new" title="Trigonometride anımsatıcılar (sayfa mevcut değil)">Trigonometride anımsatıcılar</a></li> <li><a href="/w/index.php?title=Pentagramma_mirificum&action=edit&redlink=1" class="new" title="Pentagramma mirificum (sayfa mevcut değil)">Pentagramma mirificum</a></li> <li><a href="/wiki/Trigonometrik_%C3%B6zde%C5%9Fliklerin_ispatlar%C4%B1" title="Trigonometrik özdeşliklerin ispatları">Trigonometrik özdeşliklerin ispatları</a></li> <li><a href="/w/index.php?title=Prosthaphaeresis&action=edit&redlink=1" class="new" title="Prosthaphaeresis (sayfa mevcut değil)">Prosthaphaeresis</a></li> <li><a href="/wiki/Pisagor_teoremi" title="Pisagor teoremi">Pisagor teoremi</a></li> <li><a href="/w/index.php?title=Tanjant_yar%C4%B1m_a%C3%A7%C4%B1_form%C3%BCl%C3%BC&action=edit&redlink=1" class="new" title="Tanjant yarım açı formülü (sayfa mevcut değil)">Tanjant yarım açı formülü</a></li> <li><a href="/wiki/Tam_trigonometrik_de%C4%9Ferler" title="Tam trigonometrik değerler">Trigonometrik sayı</a></li> <li><a href="/wiki/Trigonometri" title="Trigonometri">Trigonometri</a></li> <li><a href="/wiki/Trigonometrinin_kullan%C4%B1m_alanlar%C4%B1" title="Trigonometrinin kullanım alanları">Trigonometrinin kullanım alanları</a></li> <li><a href="/w/index.php?title=Versin%C3%BCs&action=edit&redlink=1" class="new" title="Versinüs (sayfa mevcut değil)">Versinüs</a> ve <a href="/w/index.php?title=Haversin%C3%BCs&action=edit&redlink=1" class="new" title="Haversinüs (sayfa mevcut değil)">haversinüs</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Kaynakça">Kaynakça</h2>[<a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&veaction=edit&section=46" title="Değiştirilen bölüm: Kaynakça" class="mw-editsection-visualeditor">değiştir</a> | <a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&action=edit&section=46" title="Bölümün kaynak kodunu değiştir: Kaynakça">kaynağı değiştir</a>]</div> <style data-mw-deduplicate="TemplateStyles:r32805677">.mw-parser-output .reflist{font-size:90%;margin-bottom:0.5em;list-style-type:decimal}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-count:2}.mw-parser-output .reflist-columns-3{column-count:3}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-AS4345-1"><a href="#cite_ref-AS4345_1-0">^</a> <cite class="kaynak kitap"><a href="/w/index.php?title=Milton_Abramowitz&action=edit&redlink=1" class="new" title="Milton Abramowitz (sayfa mevcut değil)">Abramowitz, Milton</a>; <a href="/w/index.php?title=Irene_Stegun&action=edit&redlink=1" class="new" title="Irene Stegun (sayfa mevcut değil)">Stegun, Irene Ann</a>, (Ed.) (1983) [Haziran 1964]. <a rel="nofollow" class="external text" href="https://personal.math.ubc.ca/~cbm/aands/page_73.htm">"Chapter 4, eqn 4.3.45"</a>. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. 55 (Düzeltmelerle birlikte 10. orijinal baskının ek düzeltmelerle birlikte 9. yeniden baskısı (Aralık 1972); 1. bas.). Washington D.C., USA; New York, USA: <a href="/wiki/Ticaret_Bakanl%C4%B1%C4%9F%C4%B1_(Amerika_Birle%C5%9Fik_Devletleri)" title="Ticaret Bakanlığı (Amerika Birleşik Devletleri)">United States Department of Commerce</a>, <a href="/wiki/Ulusal_Standartlar_ve_Teknoloji_Enstit%C3%BCs%C3%BC" title="Ulusal Standartlar ve Teknoloji Enstitüsü">Ulusal Standartlar ve Teknoloji Enstitüsü</a>; <a href="/wiki/Dover_Publications" title="Dover Publications">Dover Publications</a>. s. 73. <a href="/wiki/Uluslararas%C4%B1_Standart_Kitap_Numaras%C4%B1" title="Uluslararası Standart Kitap Numarası">ISBN</a> <a href="/wiki/%C3%96zel:KitapKaynaklar%C4%B1/0-486-61272-4" title="Özel:KitapKaynakları/0-486-61272-4">0-486-61272-4</a>. <a href="/wiki/Kongre_K%C3%BCt%C3%BCphanesi_Kontrol_Numaras%C4%B1" title="Kongre Kütüphanesi Kontrol Numarası">LCCN</a> <a rel="nofollow" class="external text" href="//lccn.loc.gov/64-60036">64-60036</a>. <a href="/wiki/Mathematical_Reviews" title="Mathematical Reviews">MR</a> <a rel="nofollow" class="external text" href="//www.ams.org/mathscinet-getitem?mr=0167642">0167642</a>. <a href="/wiki/%C3%96zel:KitapKaynaklar%C4%B1/9780486612720" class="internal mw-magiclink-isbn">ISBN 978-0-486-61272-0</a>. <a href="/wiki/Kongre_K%C3%BCt%C3%BCphanesi_Kontrol_Numaras%C4%B1" title="Kongre Kütüphanesi Kontrol Numarası">LCCN</a>-<a rel="nofollow" class="external text" href="http://lccn.loc.gov/6512253">6512253-{{{3}}}</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+4%2C+eqn+4.3.45&rft.btitle=Handbook+of+Mathematical+Functions+with+Formulas%2C+Graphs%2C+and+Mathematical+Tables&rft.place=Washington+D.C.%2C+USA%3B+New+York%2C+USA&rft.series=Applied+Mathematics+Series&rft.pages=73&rft.edition=D%C3%BCzeltmelerle+birlikte+10.+orijinal+bask%C4%B1n%C4%B1n+ek+d%C3%BCzeltmelerle+birlikte+9.+yeniden+bask%C4%B1s%C4%B1+%28Aral%C4%B1k+1972%29%3B+1.&rft.pub=United+States+Department+of+Commerce%2C+Ulusal+Standartlar+ve+Teknoloji+Enstit%C3%BCs%C3%BC%3B+Dover+Publications&rft.date=1983&rft_id=%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D0167642&rft_id=info%3Alccn%2F64-60036&rft.isbn=0-486-61272-4&rft_id=https%3A%2F%2Fpersonal.math.ubc.ca%2F~cbm%2Faands%2Fpage_73.htm&rfr_id=info%3Asid%2Ftr.wikipedia.org%3ATrigonometrik+%C3%B6zde%C5%9Flikler+listesi" class="Z3988">  </li> <li id="cite_note-2"><a href="#cite_ref-2">^</a> <a href="#CITEREFSelby1970">Selby 1970</a>, p. 188 </li> <li id="cite_note-3"><a href="#cite_ref-3">^</a> Abramowitz and Stegun, p. 72, 4.3.13–15 </li> <li id="cite_note-4"><a href="#cite_ref-4">^</a> Abramowitz and Stegun, p. 72, 4.3.7–9 </li> <li id="cite_note-5"><a href="#cite_ref-5">^</a> Abramowitz and Stegun, p. 72, 4.3.16 </li> <li id="cite_note-mathworld_addition-6">^ <a href="#cite_ref-mathworld_addition_6-0">a</a> <a href="#cite_ref-mathworld_addition_6-1">b</a> <a href="#cite_ref-mathworld_addition_6-2">c</a> <a href="#cite_ref-mathworld_addition_6-3">d</a> <cite id="Reference-Mathworld-Trigonometric_Addition_Formulas"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Eric W. Weisstein</a>, <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/TrigonometricAdditionFormulas.html">Trigonometric Addition Formulas</a> (<a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>)</cite> </li> <li id="cite_note-7"><a href="#cite_ref-7">^</a> Abramowitz and Stegun, p. 72, 4.3.17 </li> <li id="cite_note-8"><a href="#cite_ref-8">^</a> Abramowitz and Stegun, p. 72, 4.3.18 </li> <li id="cite_note-:0-9">^ <a href="#cite_ref-:0_9-0">a</a> <a href="#cite_ref-:0_9-1">b</a> <cite class="kaynak web"><a rel="nofollow" class="external text" href="http://www.milefoot.com/math/trig/22anglesumidentities.htm">"Angle Sum and Difference Identities"</a>. www.milefoot.com. 3 Nisan 2023 tarihinde kaynağından <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230403215053/http://www.milefoot.com/math/trig/22anglesumidentities.htm">arşivlendi</a>. Erişim tarihi: 12 Ekim 2019.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.milefoot.com&rft.atitle=Angle+Sum+and+Difference+Identities&rft_id=http%3A%2F%2Fwww.milefoot.com%2Fmath%2Ftrig%2F22anglesumidentities.htm&rfr_id=info%3Asid%2Ftr.wikipedia.org%3ATrigonometrik+%C3%B6zde%C5%9Flikler+listesi" class="Z3988">  </li> <li id="cite_note-10"><a href="#cite_ref-10">^</a> Abramowitz and Stegun, p. 72, 4.3.19 </li> <li id="cite_note-11"><a href="#cite_ref-11">^</a> Abramowitz and Stegun, p. 80, 4.4.32 </li> <li id="cite_note-12"><a href="#cite_ref-12">^</a> Abramowitz and Stegun, p. 80, 4.4.33 </li> <li id="cite_note-13"><a href="#cite_ref-13">^</a> Abramowitz and Stegun, p. 80, 4.4.34 </li> <li id="cite_note-14"><a href="#cite_ref-14">^</a> <cite class="kaynak konferans">Bronstein, Manuel (1989). "Simplification of real elementary functions". Gonnet, G. H. (Ed.). Proceedings of the ACM-<a href="/w/index.php?title=SIGSAM&action=edit&redlink=1" class="new" title="SIGSAM (sayfa mevcut değil)">SIGSAM</a> 1989 International Symposium on Symbolic and Algebraic Computation. ISSAC '89 (Portland US-OR, 1989-07). New York: <a href="/wiki/Association_for_Computing_Machinery" title="Association for Computing Machinery">ACM</a>. ss. 207-211. <a href="/wiki/Say%C4%B1sal_nesne_tan%C4%B1mlay%C4%B1c%C4%B1s%C4%B1" title="Sayısal nesne tanımlayıcısı">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145/74540.74566">10.1145/74540.74566</a>. <a href="/wiki/Uluslararas%C4%B1_Standart_Kitap_Numaras%C4%B1" title="Uluslararası Standart Kitap Numarası">ISBN</a> <a href="/wiki/%C3%96zel:KitapKaynaklar%C4%B1/0-89791-325-6" title="Özel:KitapKaynakları/0-89791-325-6">0-89791-325-6</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=konferans&rft.atitle=Simplification+of+real+elementary+functions&rft.btitle=Proceedings+of+the+ACM-SIGSAM+1989+International+Symposium+on+Symbolic+and+Algebraic+Computation&rft.place=New+York&rft.pages=207-211&rft.pub=ACM&rft.date=1989&rft_id=info%3Adoi%2F10.1145%2F74540.74566&rft.isbn=0-89791-325-6&rft.aulast=Bronstein&rft.aufirst=Manuel&rfr_id=info%3Asid%2Ftr.wikipedia.org%3ATrigonometrik+%C3%B6zde%C5%9Flikler+listesi" class="Z3988">  </li> <li id="cite_note-15"><a href="#cite_ref-15">^</a> Michael Hardy. (2016). "On Tangents and Secants of Infinite Sums." The American Mathematical Monthly, volume 123, number 7, 701–703. <a rel="nofollow" class="external free" href="https://doi.org/10.4169/amer.math.monthly.123.7.701">https://doi.org/10.4169/amer.math.monthly.123.7.701</a> </li> <li id="cite_note-16"><a href="#cite_ref-16">^</a> <cite class="kaynak dergi">Hardy, Michael (2016). <a rel="nofollow" class="external text" href="https://zenodo.org/record/1000408">"On Tangents and Secants of Infinite Sums"</a>. American Mathematical Monthly. 123 (7). ss. 701-703. <a href="/wiki/Say%C4%B1sal_nesne_tan%C4%B1mlay%C4%B1c%C4%B1s%C4%B1" title="Sayısal nesne tanımlayıcısı">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4169/amer.math.monthly.123.7.701">10.4169/amer.math.monthly.123.7.701</a>. 13 Ekim 2023 tarihinde kaynağından <a rel="nofollow" class="external text" href="https://web.archive.org/web/20231013042917/https://zenodo.org/record/1000408">arşivlendi</a>. Erişim tarihi: 24 Eylül 2024.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=On+Tangents+and+Secants+of+Infinite+Sums&rft.pages=701-703&rft.date=2016&rft_id=info%3Adoi%2F10.4169%2Famer.math.monthly.123.7.701&rft.aulast=Hardy&rft.aufirst=Michael&rft_id=https%3A%2F%2Fzenodo.org%2Frecord%2F1000408&rfr_id=info%3Asid%2Ftr.wikipedia.org%3ATrigonometrik+%C3%B6zde%C5%9Flikler+listesi" class="Z3988">  </li> <li id="cite_note-cut-the-knot.org-17">^ <a href="#cite_ref-cut-the-knot.org_17-0">a</a> <a href="#cite_ref-cut-the-knot.org_17-1">b</a> <cite class="kaynak web"><a rel="nofollow" class="external text" href="https://www.cut-the-knot.org/proofs/sine_cosine.shtml">"Sine, Cosine, and Ptolemy's Theorem"</a>. 25 Eylül 2024 tarihinde kaynağından <a rel="nofollow" class="external text" href="https://web.archive.org/web/20240925030921/https://www.cut-the-knot.org/proofs/sine_cosine.shtml">arşivlendi</a>. Erişim tarihi: 24 Eylül 2024.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Sine%2C+Cosine%2C+and+Ptolemy%27s+Theorem&rft_id=https%3A%2F%2Fwww.cut-the-knot.org%2Fproofs%2Fsine_cosine.shtml&rfr_id=info%3Asid%2Ftr.wikipedia.org%3ATrigonometrik+%C3%B6zde%C5%9Flikler+listesi" class="Z3988">  </li> <li id="cite_note-mathworld_multiple_angle-18">^ <a href="#cite_ref-mathworld_multiple_angle_18-0">a</a> <a href="#cite_ref-mathworld_multiple_angle_18-1">b</a> <cite id="Reference-Mathworld-Multiple-Angle_Formulas"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Eric W. Weisstein</a>, <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Multiple-AngleFormulas.html">Multiple-Angle Formulas</a> (<a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>)</cite> </li> <li id="cite_note-19"><a href="#cite_ref-19">^</a> Abramowitz and Stegun, p. 74, 4.3.48 </li> <li id="cite_note-STM1-20">^ <a href="#cite_ref-STM1_20-0">a</a> <a href="#cite_ref-STM1_20-1">b</a> <a href="#CITEREFSelby1970">Selby 1970</a>, pg. 190 </li> <li id="cite_note-21"><a href="#cite_ref-21">^</a> <cite class="kaynak web">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/">"Multiple-Angle Formulas"</a>. mathworld.wolfram.com (İngilizce). 25 Mart 2022 tarihinde kaynağından <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220325080937/https://mathworld.wolfram.com/">arşivlendi</a>. Erişim tarihi: 6 Şubat 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=mathworld.wolfram.com&rft.atitle=Multiple-Angle+Formulas&rft.aulast=Weisstein&rft.aufirst=Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2F&rfr_id=info%3Asid%2Ftr.wikipedia.org%3ATrigonometrik+%C3%B6zde%C5%9Flikler+listesi" class="Z3988">  </li> <li id="cite_note-22"><a href="#cite_ref-22">^</a> <cite class="kaynak web">Ward, Ken. <a rel="nofollow" class="external text" href="http://trans4mind.com/personal_development/mathematics/trigonometry/multipleAnglesRecursiveFormula.htm">"Multiple angles recursive formula"</a>. Ken Ward's Mathematics Pages. 19 Mayıs 2024 tarihinde kaynağından <a rel="nofollow" class="external text" href="https://web.archive.org/web/20240519195037/https://www.trans4mind.com/personal_development/mathematics/trigonometry/multipleAnglesRecursiveFormula.htm">arşivlendi</a>. Erişim tarihi: 24 Eylül 2024.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Ken+Ward%27s+Mathematics+Pages&rft.atitle=Multiple+angles+recursive+formula&rft.aulast=Ward&rft.aufirst=Ken&rft_id=http%3A%2F%2Ftrans4mind.com%2Fpersonal_development%2Fmathematics%2Ftrigonometry%2FmultipleAnglesRecursiveFormula.htm&rfr_id=info%3Asid%2Ftr.wikipedia.org%3ATrigonometrik+%C3%B6zde%C5%9Flikler+listesi" class="Z3988">  </li> <li id="cite_note-ReferenceA-23">^ <a href="#cite_ref-ReferenceA_23-0">a</a> <a href="#cite_ref-ReferenceA_23-1">b</a> <cite class="kaynak kitap"><a href="/w/index.php?title=Milton_Abramowitz&action=edit&redlink=1" class="new" title="Milton Abramowitz (sayfa mevcut değil)">Abramowitz, Milton</a>; <a href="/w/index.php?title=Irene_Stegun&action=edit&redlink=1" class="new" title="Irene Stegun (sayfa mevcut değil)">Stegun, Irene Ann</a>, (Ed.) (1983) [Haziran 1964]. <a rel="nofollow" class="external text" href="https://personal.math.ubc.ca/~cbm/aands/page_72.htm">"Chapter 4, eqn 4.3.20-22"</a>. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. 55 (Düzeltmelerle birlikte 10. orijinal baskının ek düzeltmelerle birlikte 9. yeniden baskısı (Aralık 1972); 1. bas.). Washington D.C., USA; New York, USA: <a href="/wiki/Ticaret_Bakanl%C4%B1%C4%9F%C4%B1_(Amerika_Birle%C5%9Fik_Devletleri)" title="Ticaret Bakanlığı (Amerika Birleşik Devletleri)">United States Department of Commerce</a>, <a href="/wiki/Ulusal_Standartlar_ve_Teknoloji_Enstit%C3%BCs%C3%BC" title="Ulusal Standartlar ve Teknoloji Enstitüsü">Ulusal Standartlar ve Teknoloji Enstitüsü</a>; <a href="/wiki/Dover_Publications" title="Dover Publications">Dover Publications</a>. s. 72. <a href="/wiki/Uluslararas%C4%B1_Standart_Kitap_Numaras%C4%B1" title="Uluslararası Standart Kitap Numarası">ISBN</a> <a href="/wiki/%C3%96zel:KitapKaynaklar%C4%B1/0-486-61272-4" title="Özel:KitapKaynakları/0-486-61272-4">0-486-61272-4</a>. <a href="/wiki/Kongre_K%C3%BCt%C3%BCphanesi_Kontrol_Numaras%C4%B1" title="Kongre Kütüphanesi Kontrol Numarası">LCCN</a> <a rel="nofollow" class="external text" href="//lccn.loc.gov/64-60036">64-60036</a>. <a href="/wiki/Mathematical_Reviews" title="Mathematical Reviews">MR</a> <a rel="nofollow" class="external text" href="//www.ams.org/mathscinet-getitem?mr=0167642">0167642</a>. <a href="/wiki/%C3%96zel:KitapKaynaklar%C4%B1/9780486612720" class="internal mw-magiclink-isbn">ISBN 978-0-486-61272-0</a>. <a href="/wiki/Kongre_K%C3%BCt%C3%BCphanesi_Kontrol_Numaras%C4%B1" title="Kongre Kütüphanesi Kontrol Numarası">LCCN</a>-<a rel="nofollow" class="external text" href="http://lccn.loc.gov/6512253">6512253-{{{3}}}</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+4%2C+eqn+4.3.20-22&rft.btitle=Handbook+of+Mathematical+Functions+with+Formulas%2C+Graphs%2C+and+Mathematical+Tables&rft.place=Washington+D.C.%2C+USA%3B+New+York%2C+USA&rft.series=Applied+Mathematics+Series&rft.pages=72&rft.edition=D%C3%BCzeltmelerle+birlikte+10.+orijinal+bask%C4%B1n%C4%B1n+ek+d%C3%BCzeltmelerle+birlikte+9.+yeniden+bask%C4%B1s%C4%B1+%28Aral%C4%B1k+1972%29%3B+1.&rft.pub=United+States+Department+of+Commerce%2C+Ulusal+Standartlar+ve+Teknoloji+Enstit%C3%BCs%C3%BC%3B+Dover+Publications&rft.date=1983&rft_id=%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D0167642&rft_id=info%3Alccn%2F64-60036&rft.isbn=0-486-61272-4&rft_id=https%3A%2F%2Fpersonal.math.ubc.ca%2F~cbm%2Faands%2Fpage_72.htm&rfr_id=info%3Asid%2Ftr.wikipedia.org%3ATrigonometrik+%C3%B6zde%C5%9Flikler+listesi" class="Z3988">  </li> <li id="cite_note-mathworld_half_angle-24">^ <a href="#cite_ref-mathworld_half_angle_24-0">a</a> <a href="#cite_ref-mathworld_half_angle_24-1">b</a> <cite id="Reference-Mathworld-Half-Angle_Formulas"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Eric W. Weisstein</a>, <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Half-AngleFormulas.html">Half-Angle Formulas</a> (<a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>)</cite> </li> <li id="cite_note-25"><a href="#cite_ref-25">^</a> Abramowitz and Stegun, p. 72, 4.3.24–26 </li> <li id="cite_note-mathworld_double_angle-26"><a href="#cite_ref-mathworld_double_angle_26-0">^</a> <cite id="Reference-Mathworld-Double-Angle_Formulas"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Eric W. Weisstein</a>, <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Double-AngleFormulas.html">Double-Angle Formulas</a> (<a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>)</cite> </li> <li id="cite_note-Stegun_p._72,_4-27"><a href="#cite_ref-Stegun_p._72,_4_27-0">^</a> Abramowitz and Stegun, p. 72, 4.3.27–28 </li> <li id="cite_note-28"><a href="#cite_ref-28">^</a> Abramowitz ve Stegun, s. 72, 4.3.31-33 </li> <li id="cite_note-29"><a href="#cite_ref-29">^</a> <cite class="kaynak kitap">Eves, Howard (1990). <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontohi0000eves">An introduction to the history of mathematics</a>. 6. Philadelphia: Saunders College Pub. s. <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontohi0000eves/page/309">309</a>. <a href="/wiki/Uluslararas%C4%B1_Standart_Kitap_Numaras%C4%B1" title="Uluslararası Standart Kitap Numarası">ISBN</a> <a href="/wiki/%C3%96zel:KitapKaynaklar%C4%B1/0-03-029558-0" title="Özel:KitapKaynakları/0-03-029558-0">0-03-029558-0</a>. <a href="/wiki/OCLC" title="OCLC">OCLC</a> <a rel="nofollow" class="external text" href="//www.worldcat.org/oclc/20842510">20842510</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+introduction+to+the+history+of+mathematics&rft.place=Philadelphia&rft.series=6.&rft.pages=309&rft.pub=Saunders+College+Pub&rft.date=1990&rft_id=info%3Aoclcnum%2F20842510&rft.isbn=0-03-029558-0&rft.aulast=Eves&rft.aufirst=Howard&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontohi0000eves&rfr_id=info%3Asid%2Ftr.wikipedia.org%3ATrigonometrik+%C3%B6zde%C5%9Flikler+listesi" class="Z3988">  </li> <li id="cite_note-30"><a href="#cite_ref-30">^</a> Abramowitz and Stegun, p. 72, 4.3.34–39 </li> <li id="cite_note-31"><a href="#cite_ref-31">^</a> <cite class="kaynak dergi">Johnson, Warren P. (Apr 2010). <a rel="nofollow" class="external text" href="https://archive.org/details/sim_american-mathematical-monthly_2010-04_117_4/page/311">"Trigonometric Identities à la Hermite"</a>. <a href="/w/index.php?title=American_Mathematical_Monthly&action=edit&redlink=1" class="new" title="American Mathematical Monthly (sayfa mevcut değil)">American Mathematical Monthly</a>. 117 (4). ss. 311-327. <a href="/wiki/Say%C4%B1sal_nesne_tan%C4%B1mlay%C4%B1c%C4%B1s%C4%B1" title="Sayısal nesne tanımlayıcısı">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4169/000298910x480784">10.4169/000298910x480784</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Trigonometric+Identities+%C3%A0+la+Hermite&rft.pages=311-327&rft.date=2010-04&rft_id=info%3Adoi%2F10.4169%2F000298910x480784&rft.aulast=Johnson&rft.aufirst=Warren+P.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fsim_american-mathematical-monthly_2010-04_117_4%2Fpage%2F311&rfr_id=info%3Asid%2Ftr.wikipedia.org%3ATrigonometrik+%C3%B6zde%C5%9Flikler+listesi" class="Z3988">  </li> <li id="cite_note-32"><a href="#cite_ref-32">^</a> <cite class="kaynak web"><a rel="nofollow" class="external text" href="https://math.stackexchange.com/q/2095330">"Product Identity Multiple Angle"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Product+Identity+Multiple+Angle&rft_id=https%3A%2F%2Fmath.stackexchange.com%2Fq%2F2095330&rfr_id=info%3Asid%2Ftr.wikipedia.org%3ATrigonometrik+%C3%B6zde%C5%9Flikler+listesi" class="Z3988"> Arşivlenmesi gereken bağlantıya sahip kaynak şablonu içeren maddeler (<a href="/wiki/Kategori:Ar%C5%9Fivlenmesi_gereken_ba%C4%9Flant%C4%B1ya_sahip_kaynak_%C5%9Fablonu_i%C3%A7eren_maddeler" title="Kategori:Arşivlenmesi gereken bağlantıya sahip kaynak şablonu içeren maddeler">link</a>) </li> <li id="cite_note-33"><a href="#cite_ref-33">^</a> Apostol, T.M. (1967) Calculus. 2nd edition. New York, NY, Wiley. Pp 334-335. </li> <li id="cite_note-ReferenceB-34">^ <a href="#cite_ref-ReferenceB_34-0">a</a> <a href="#cite_ref-ReferenceB_34-1">b</a> <cite id="Reference-Mathworld-Harmonic_Addition_Theorem"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Eric W. Weisstein</a>, <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/.html">Harmonic Addition Theorem</a> (<a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>)</cite> </li> <li id="cite_note-Muniz-35"><a href="#cite_ref-Muniz_35-0">^</a> <cite class="kaynak dergi">Ortiz Muñiz, Eddie (Feb 1953). <a rel="nofollow" class="external text" href="https://archive.org/details/sim_american-journal-of-physics_1953-02_21_2/page/140">"A Method for Deriving Various Formulas in Electrostatics and Electromagnetism Using Lagrange's Trigonometric Identities"</a>. American Journal of Physics. 21 (2). s. 140. <a href="/wiki/Bibcode" title="Bibcode">Bibcode</a>:<a rel="nofollow" class="external text" href="http://adsabs.harvard.edu/abs/1953AmJPh..21..140M">1953AmJPh..21..140M</a>. <a href="/wiki/Say%C4%B1sal_nesne_tan%C4%B1mlay%C4%B1c%C4%B1s%C4%B1" title="Sayısal nesne tanımlayıcısı">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1119/1.1933371">10.1119/1.1933371</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=A+Method+for+Deriving+Various+Formulas+in+Electrostatics+and+Electromagnetism+Using+Lagrange%27s+Trigonometric+Identities&rft.pages=140&rft.date=1953-02&rft_id=info%3Adoi%2F10.1119%2F1.1933371&rft_id=info%3Abibcode%2F1953AmJPh..21..140M&rft.aulast=Ortiz+Mu%C3%B1iz&rft.aufirst=Eddie&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fsim_american-journal-of-physics_1953-02_21_2%2Fpage%2F140&rfr_id=info%3Asid%2Ftr.wikipedia.org%3ATrigonometrik+%C3%B6zde%C5%9Flikler+listesi" class="Z3988">  </li> <li id="cite_note-36"><a href="#cite_ref-36">^</a> <cite class="kaynak kitap">Agarwal, Ravi P.; O'Regan, Donal (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=jWvAfcNnphIC">Ordinary and Partial Differential Equations: With Special Functions, Fourier Series, and Boundary Value Problems</a>. illustrated. Springer Science & Business Media. s. 185. <a href="/wiki/Uluslararas%C4%B1_Standart_Kitap_Numaras%C4%B1" title="Uluslararası Standart Kitap Numarası">ISBN</a> <a href="/wiki/%C3%96zel:KitapKaynaklar%C4%B1/978-0-387-79146-3" title="Özel:KitapKaynakları/978-0-387-79146-3">978-0-387-79146-3</a>. 25 Eylül 2024 tarihinde kaynağından <a rel="nofollow" class="external text" href="https://web.archive.org/web/20240925000626/https://books.google.com/books?id=jWvAfcNnphIC">arşivlendi</a>. Erişim tarihi: 24 Eylül 2024.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ordinary+and+Partial+Differential+Equations%3A+With+Special+Functions%2C+Fourier+Series%2C+and+Boundary+Value+Problems&rft.series=illustrated&rft.pages=185&rft.pub=Springer+Science+%26+Business+Media&rft.date=2008&rft.isbn=978-0-387-79146-3&rft.aulast=Agarwal&rft.aufirst=Ravi+P.&rft.au=O%27Regan%2C+Donal&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DjWvAfcNnphIC&rfr_id=info%3Asid%2Ftr.wikipedia.org%3ATrigonometrik+%C3%B6zde%C5%9Flikler+listesi" class="Z3988">  <a rel="nofollow" class="external text" href="https://books.google.com/books?id=jWvAfcNnphIC&pg=PA185">Extract of page 185</a> 25 Eylül 2024 tarihinde <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> sitesinde <a rel="nofollow" class="external text" href="https://web.archive.org/web/20240925023808/https://books.google.com/books?id=jWvAfcNnphIC&pg=PA185">arşivlendi</a>. </li> <li id="cite_note-37"><a href="#cite_ref-37">^</a> <cite class="kaynak kitap">Jeffrey, Alan; Dai, Hui-hui (2008). "Section 2.4.1.6". Handbook of Mathematical Formulas and Integrals. 4th. Academic Press. <a href="/wiki/Uluslararas%C4%B1_Standart_Kitap_Numaras%C4%B1" title="Uluslararası Standart Kitap Numarası">ISBN</a> <a href="/wiki/%C3%96zel:KitapKaynaklar%C4%B1/978-0-12-374288-9" title="Özel:KitapKaynakları/978-0-12-374288-9">978-0-12-374288-9</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Section+2.4.1.6&rft.btitle=Handbook+of+Mathematical+Formulas+and+Integrals&rft.series=4th&rft.pub=Academic+Press&rft.date=2008&rft.isbn=978-0-12-374288-9&rft.aulast=Jeffrey&rft.aufirst=Alan&rft.au=Dai%2C+Hui-hui&rfr_id=info%3Asid%2Ftr.wikipedia.org%3ATrigonometrik+%C3%B6zde%C5%9Flikler+listesi" class="Z3988">  </li> <li id="cite_note-38"><a href="#cite_ref-38">^</a> <cite class="kaynak dergi">Fay, Temple H.; Kloppers, P. Hendrik (2001). <a rel="nofollow" class="external text" href="https://dx.doi.org/10.1080/00207390117151">"The Gibbs' phenomenon"</a>. International Journal of Mathematical Education in Science and Technology. 32 (1). ss. 73-89. <a href="/wiki/Say%C4%B1sal_nesne_tan%C4%B1mlay%C4%B1c%C4%B1s%C4%B1" title="Sayısal nesne tanımlayıcısı">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080/00207390117151">10.1080/00207390117151</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+Gibbs%27+phenomenon&rft.pages=73-89&rft.date=2001&rft_id=info%3Adoi%2F10.1080%2F00207390117151&rft.aulast=Fay&rft.aufirst=Temple+H.&rft.au=Kloppers%2C+P.+Hendrik&rft_id=http%3A%2F%2Fdx.doi.org%2F10.1080%2F00207390117151&rfr_id=info%3Asid%2Ftr.wikipedia.org%3ATrigonometrik+%C3%B6zde%C5%9Flikler+listesi" class="Z3988">  </li> <li id="cite_note-39"><a href="#cite_ref-39">^</a> Abramowitz and Stegun, p. 74, 4.3.47 </li> <li id="cite_note-40"><a href="#cite_ref-40">^</a> Abramowitz and Stegun, p. 71, 4.3.2 </li> <li id="cite_note-41"><a href="#cite_ref-41">^</a> Abramowitz and Stegun, p. 71, 4.3.1 </li> <li id="cite_note-42"><a href="#cite_ref-42">^</a> Abramowitz and Stegun, p. 80, 4.4.26–31 </li> <li id="cite_note-43"><a href="#cite_ref-43">^</a> Abramowitz and Stegun, p. 74, 4.3.65–66 </li> <li id="cite_note-44"><a href="#cite_ref-44">^</a> Abramowitz and Stegun, p. 75, 4.3.89–90 </li> <li id="cite_note-45"><a href="#cite_ref-45">^</a> Abramowitz and Stegun, p. 85, 4.5.68–69 </li> <li id="cite_note-46"><a href="#cite_ref-46">^</a> <a href="#CITEREFAbramowitzStegun1972">Abramowitz & Stegun 1972</a>, p. 73, 4.3.45 </li> <li id="cite_note-Wu-47">^ <a href="#cite_ref-Wu_47-0">a</a> <a href="#cite_ref-Wu_47-1">b</a> <a href="#cite_ref-Wu_47-2">c</a> Wu, Rex H. "Proof Without Words: Euler's Arctangent Identity", Mathematics Magazine 77(3), June 2004, p. 189. </li> <li id="cite_note-48"><a href="#cite_ref-48">^</a> <cite id="CITEREFS._M._Abrarov,_R._K._Jagpal,_R._Siddiqui_and_B._M._Quine2021" class="kaynak">S. M. Abrarov, R. K. Jagpal, R. Siddiqui and B. M. Quine (2021), "Algorithmic determination of a large integer in the two-term Machin-like formula for π", Mathematics, 2162, 9 (17), <a href="/wiki/ArXiv" title="ArXiv">arXiv</a>:<a rel="nofollow" class="external text" href="//arxiv.org/abs/2107.01027">2107.01027</a> $2, <a href="/wiki/Say%C4%B1sal_nesne_tan%C4%B1mlay%C4%B1c%C4%B1s%C4%B1" title="Sayısal nesne tanımlayıcısı">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.3390/math9172162">10.3390/math9172162</a> <img alt="Özgürce erişilebilir" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png" decoding="async" width="9" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/14px-Lock-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/18px-Lock-green.svg.png 2x" data-file-width="512" data-file-height="813" /></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics&rft.atitle=Algorithmic+determination+of+a+large+integer+in+the+two-term+Machin-like+formula+for+%CF%80&rft.volume=9&rft.issue=17&rft.date=2021&rft_id=info%3Aarxiv%2F2107.01027&rft_id=info%3Adoi%2F10.3390%2Fmath9172162&rft.au=S.+M.+Abrarov%2C+R.+K.+Jagpal%2C+R.+Siddiqui+and+B.+M.+Quine&rfr_id=info%3Asid%2Ftr.wikipedia.org%3ATrigonometrik+%C3%B6zde%C5%9Flikler+listesi" class="Z3988"> KB1 bakım: Birden fazla ad: yazar listesi (<a href="/wiki/Kategori:KB1_bak%C4%B1m:_Birden_fazla_ad:_yazar_listesi" title="Kategori:KB1 bakım: Birden fazla ad: yazar listesi">link</a>) </li> <li id="cite_note-49"><a href="#cite_ref-49">^</a> <cite class="kaynak dergi">Humble, Steve (Nov 2004). <a rel="nofollow" class="external text" href="https://archive.org/details/sim_mathematical-gazette_2004-11_88_513/page/524">"Grandma's identity"</a>. Mathematical Gazette. Cilt 88. ss. 524-525. <a href="/wiki/Say%C4%B1sal_nesne_tan%C4%B1mlay%C4%B1c%C4%B1s%C4%B1" title="Sayısal nesne tanımlayıcısı">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017/s0025557200176223">10.1017/s0025557200176223</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Grandma%27s+identity&rft.pages=524-525&rft.date=2004-11&rft_id=info%3Adoi%2F10.1017%2Fs0025557200176223&rft.aulast=Humble&rft.aufirst=Steve&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fsim_mathematical-gazette_2004-11_88_513%2Fpage%2F524&rfr_id=info%3Asid%2Ftr.wikipedia.org%3ATrigonometrik+%C3%B6zde%C5%9Flikler+listesi" class="Z3988">  </li> <li id="cite_note-50"><a href="#cite_ref-50">^</a> <cite id="Reference-Mathworld-Sine"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Eric W. Weisstein</a>, <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/.html">Sine</a> (<a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>)</cite> </li> <li id="cite_note-Harris-51"><a href="#cite_ref-Harris_51-0">^</a> Harris, Edward M. "Sums of Arctangents", in Roger B. Nelson, Proofs Without Words (1993, Mathematical Association of America), p. 39. </li> <li id="cite_note-52"><a href="#cite_ref-52">^</a> <a href="/w/index.php?title=Abramowitz_and_Stegun&action=edit&redlink=1" class="new" title="Abramowitz and Stegun (sayfa mevcut değil)">Milton Abramowitz and Irene Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables</a>, <a href="/wiki/Dover_Publications" title="Dover Publications">Dover Publications</a>, New York, 1972, formulae 9.1.42–9.1.45 </li> <li id="cite_note-53"><a href="#cite_ref-53">^</a> Er. K. C. Joshi, Krishna's IIT MATHEMATIKA. Krishna Prakashan Media. Meerut, India. page 636. </li> <li id="cite_note-54"><a href="#cite_ref-54">^</a> Abramowitz and Stegun, p. 72, 4.3.23 </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Bibliyografya">Bibliyografya</h2>[<a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&veaction=edit&section=47" title="Değiştirilen bölüm: Bibliyografya" class="mw-editsection-visualeditor">değiştir</a> | <a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&action=edit&section=47" title="Bölümün kaynak kodunu değiştir: Bibliyografya">kaynağı değiştir</a>]</div> <style data-mw-deduplicate="TemplateStyles:r24952096">.mw-parser-output .refbegin{font-size:90%;margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-100{font-size:100%}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns dl,.mw-parser-output .refbegin-columns ol,.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li,.mw-parser-output .refbegin-columns dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="refbegin reflist" style=""> <ul><li><cite class="kaynak kitap"><a href="/w/index.php?title=Milton_Abramowitz&action=edit&redlink=1" class="new" title="Milton Abramowitz (sayfa mevcut değil)">Abramowitz, Milton</a>; <a href="/w/index.php?title=Irene_Stegun&action=edit&redlink=1" class="new" title="Irene Stegun (sayfa mevcut değil)">Stegun, Irene A.</a>, (Ed.) (1972). <a rel="nofollow" class="external text" href="https://archive.org/details/handbookofmathe000abra">Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables</a>. New York: <a href="/wiki/Dover_Publications" title="Dover Publications">Dover Publications</a>. <a href="/wiki/Uluslararas%C4%B1_Standart_Kitap_Numaras%C4%B1" title="Uluslararası Standart Kitap Numarası">ISBN</a> <a href="/wiki/%C3%96zel:KitapKaynaklar%C4%B1/978-0-486-61272-0" title="Özel:KitapKaynakları/978-0-486-61272-0">978-0-486-61272-0</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+Mathematical+Functions+with+Formulas%2C+Graphs%2C+and+Mathematical+Tables&rft.place=New+York&rft.pub=Dover+Publications&rft.date=1972&rft.isbn=978-0-486-61272-0&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fhandbookofmathe000abra&rfr_id=info%3Asid%2Ftr.wikipedia.org%3ATrigonometrik+%C3%B6zde%C5%9Flikler+listesi" class="Z3988"> </li> <li><cite id="CITEREFNielsen1966" class="kaynak">Nielsen, Kaj L. (1966), Logarithmic and Trigonometric Tables to Five Places, 2., New York: <a href="/wiki/Barnes_%26_Noble" title="Barnes & Noble">Barnes & Noble</a>, <a href="/wiki/Kongre_K%C3%BCt%C3%BCphanesi_Kontrol_Numaras%C4%B1" title="Kongre Kütüphanesi Kontrol Numarası">LCCN</a> <a rel="nofollow" class="external text" href="//lccn.loc.gov/61-9103">61-9103</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Logarithmic+and+Trigonometric+Tables+to+Five+Places&rft.place=New+York&rft.series=2.&rft.pub=Barnes+%26+Noble&rft.date=1966&rft_id=info%3Alccn%2F61-9103&rft.aulast=Nielsen&rft.aufirst=Kaj+L.&rfr_id=info%3Asid%2Ftr.wikipedia.org%3ATrigonometrik+%C3%B6zde%C5%9Flikler+listesi" class="Z3988"> </li> <li><cite id="CITEREFSelby1970" class="kaynak">Selby, Samuel M., (Ed.) (1970), Standard Mathematical Tables, 18., The Chemical Rubber Co.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Standard+Mathematical+Tables&rft.series=18.&rft.pub=The+Chemical+Rubber+Co.&rft.date=1970&rfr_id=info%3Asid%2Ftr.wikipedia.org%3ATrigonometrik+%C3%B6zde%C5%9Flikler+listesi" class="Z3988"> </li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Dış_bağlantılar">Dış bağlantılar</h2>[<a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&veaction=edit&section=48" title="Değiştirilen bölüm: Dış bağlantılar" class="mw-editsection-visualeditor">değiştir</a> | <a href="/w/index.php?title=Trigonometrik_%C3%B6zde%C5%9Flikler_listesi&action=edit&section=48" title="Bölümün kaynak kodunu değiştir: Dış bağlantılar">kaynağı değiştir</a>]</div> <ul><li><a rel="nofollow" class="external text" href="http://www.jdawiseman.com/papers/easymath/surds_sin_cos.html">Values of sin and cos, expressed in surds, for integer multiples of 3° and of 5 5/8°</a>, and for the same angles <a rel="nofollow" class="external text" href="http://www.jdawiseman.com/papers/easymath/surds_csc_sec.html">csc and sec</a> and <a rel="nofollow" class="external text" href="http://www.jdawiseman.com/papers/easymath/surds_tan.html">tan</a></li> <li><cite class="kaynak web"><a rel="nofollow" class="external text" href="https://inspiria.edu.in/trigonometry-formula">"Complete List of Trigonometric Formulas"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Complete+List+of+Trigonometric+Formulas&rft_id=https%3A%2F%2Finspiria.edu.in%2Ftrigonometry-formula&rfr_id=info%3Asid%2Ftr.wikipedia.org%3ATrigonometrik+%C3%B6zde%C5%9Flikler+listesi" class="Z3988"> Arşivlenmesi gereken bağlantıya sahip kaynak şablonu içeren maddeler (<a href="/wiki/Kategori:Ar%C5%9Fivlenmesi_gereken_ba%C4%9Flant%C4%B1ya_sahip_kaynak_%C5%9Fablonu_i%C3%A7eren_maddeler" title="Kategori:Arşivlenmesi gereken bağlantıya sahip kaynak şablonu içeren maddeler">link</a>) </li></ul> <div role="navigation" class="navbox" aria-labelledby="Trigonometri" style="padding:3px"><table class="nowraplinks collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><style data-mw-deduplicate="TemplateStyles:r25548259">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar-mini 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trigonometri">Genelleştirilmiş</a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Açı ölçü birimleri</th><td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/Devir_(birim)" title="Devir (birim)">Devir</a></li> <li><a href="/wiki/Derece_(birim)" title="Derece (birim)">Derece</a></li> <li><a href="/wiki/Radyan" title="Radyan">Radyan</a></li> <li><a href="/wiki/Grad" title="Grad">Grad</a></li></ul> </div></td><td class="navbox-image" rowspan="8" style="width:1px;padding:0px 0px 0px 2px"><div><a href="/wiki/Dosya:Sinxoverx.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Sinxoverx.svg/100px-Sinxoverx.svg.png" decoding="async" width="100" height="99" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Sinxoverx.svg/150px-Sinxoverx.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Sinxoverx.svg/200px-Sinxoverx.svg.png 2x" data-file-width="429" data-file-height="425" /></a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Trigonometrik_fonksiyonlar" title="Trigonometrik fonksiyonlar">Trigonometrik fonksiyonlar</a> & <a href="/wiki/Ters_trigonometrik_fonksiyonlar" title="Ters trigonometrik fonksiyonlar">Ters trigonometrik fonksiyonlar</a></th><td class="navbox-list navbox-even hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/Sin%C3%BCs_(matematik)" title="Sinüs (matematik)">Sinüs</a> (sin)</li> <li><a href="/wiki/Kosin%C3%BCs" title="Kosinüs">Kosinüs</a> (cos)</li> <li><a href="/wiki/Tanjant" title="Tanjant">Tanjant</a> (tan)</li> <li><a href="/wiki/Kotanjant" title="Kotanjant">Kotanjant</a> (cot)</li> <li><a href="/wiki/Sekant" title="Sekant">Sekant</a> (sec)</li> <li><a href="/wiki/Kosekant" title="Kosekant">Kosekant</a> (csc)</li> <li><a href="/w/index.php?title=Versin%C3%BCs&action=edit&redlink=1" class="new" title="Versinüs (sayfa mevcut değil)">Versinüs</a> (versin)</li> <li><a href="/w/index.php?title=Verkosin%C3%BCs&action=edit&redlink=1" class="new" title="Verkosinüs (sayfa mevcut değil)">Verkosinüs</a> (vercosin)</li> <li><a href="/w/index.php?title=Koversin%C3%BCs&action=edit&redlink=1" class="new" title="Koversinüs (sayfa mevcut değil)">Koversinüs</a> (coversin)</li> <li><a href="/w/index.php?title=Koverkosin%C3%BCs&action=edit&redlink=1" class="new" title="Koverkosinüs (sayfa mevcut değil)">Koverkosinüs</a> (covercosin)</li> <li><a href="/w/index.php?title=Haversin%C3%BCs&action=edit&redlink=1" class="new" title="Haversinüs (sayfa mevcut değil)">Haversinüs</a> (haversin)</li> <li><a href="/w/index.php?title=Haverkosin%C3%BCs&action=edit&redlink=1" class="new" title="Haverkosinüs (sayfa mevcut değil)">Haverkosinüs</a> (havercosin)</li> <li><a href="/w/index.php?title=Hakoversin%C3%BCs&action=edit&redlink=1" class="new" title="Hakoversinüs (sayfa mevcut değil)">Hakoversinüs</a> (hacoversin)</li> <li><a href="/w/index.php?title=Hakoverkosin%C3%BCs&action=edit&redlink=1" class="new" title="Hakoverkosinüs (sayfa mevcut değil)">Hakoverkosinüs</a> (hacovercosin)</li> <li><a href="/w/index.php?title=Ekssekant&action=edit&redlink=1" class="new" title="Ekssekant (sayfa mevcut değil)">Ekssekant</a> (exsec)</li> <li><a href="/w/index.php?title=Ekskosekant&action=edit&redlink=1" class="new" title="Ekskosekant (sayfa mevcut değil)">Ekskosekant</a> (excsc)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Referans</th><td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a class="mw-selflink selflink">Özdeşlikler</a></li> <li><a href="/wiki/Tam_trigonometrik_de%C4%9Ferler" title="Tam trigonometrik değerler">Tam sabitler</a></li> <li><a href="/wiki/Trigonometrik_tablolar" title="Trigonometrik tablolar">Tablolar</a></li> <li><a href="/wiki/Birim_%C3%A7ember" title="Birim çember">Birim çember</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Yasalar ve teoremler</th><td class="navbox-list navbox-even hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/Kosin%C3%BCs_teoremi" title="Kosinüs teoremi">Kosinüs teoremi</a></li> <li><a href="/wiki/Sin%C3%BCs_teoremi" title="Sinüs teoremi">Sinüs teoremi</a></li> <li><a href="/wiki/Tanjant_teoremi" title="Tanjant teoremi">Tanjant teoremi</a></li> <li><a href="/wiki/Kotanjant_teoremi" title="Kotanjant teoremi">Kotanjant teoremi</a></li> <li><a href="/wiki/Pisagor_teoremi" title="Pisagor teoremi">Pisagor teoremi</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Kalk%C3%BCl%C3%BCs" title="Kalkülüs">Kalkülüs</a></th><td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/Trigonometrik_yerine_koyma" title="Trigonometrik yerine koyma">Trigonometrik yerine koyma</a></li> <li><a href="/wiki/Trigonometrik_fonksiyonlar%C4%B1n_integralleri" title="Trigonometrik fonksiyonların integralleri">İntegraller</a> (<a href="/wiki/Ters_trigonometrik_fonksiyonlar%C4%B1n_integralleri_listesi" title="Ters trigonometrik fonksiyonların integralleri listesi">Ters fonksiyonlar</a>)</li> <li><a href="/wiki/Trigonometrik_fonksiyonlar%C4%B1n_t%C3%BCrevleri" title="Trigonometrik fonksiyonların türevleri">Türevler</a></li> <li><a href="/wiki/Trigonometrik_seri" title="Trigonometrik seri">Trigonometrik seri</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">İlgili konular</th><td class="navbox-list navbox-even hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%C3%9C%C3%A7gen" title="Üçgen">Üçgen</a></li> <li><a href="/wiki/%C3%87ember" title="Çember">Çember</a></li> <li><a href="/wiki/Geometri" title="Geometri">Geometri</a></li> <li><a href="/wiki/A%C3%A7%C4%B1" title="Açı">Açı</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Kullanıldığı dallar</th><td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/Matematik" title="Matematik">Matematik</a></li> <li><a href="/wiki/Geometri" title="Geometri">Geometri</a></li> <li><a href="/wiki/Fizik" title="Fizik">Fizik</a></li> <li><a href="/wiki/M%C3%BChendislik" title="Mühendislik">Mühendislik</a></li> <li><a href="/wiki/Astronomi" title="Astronomi">Astronomi</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Katkı sağlayan matematikçiler</th><td class="navbox-list navbox-even hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/Hipparkos" title="Hipparkos">Hipparchus</a></li> <li><a href="/wiki/Batlamyus" title="Batlamyus">Ptolemy</a></li> <li><a href="/wiki/Brahmagupta" title="Brahmagupta">Brahmagupta</a></li> <li><a href="/wiki/Batt%C3%A2n%C3%AE" title="Battânî">Battânî</a></li> <li><a href="/wiki/Regiomontanus" title="Regiomontanus">Regiomontanus</a></li> <li><a href="/wiki/Fran%C3%A7ois_Vi%C3%A8te" title="François Viète">Viète</a></li> <li><a href="/wiki/Abraham_de_Moivre" title="Abraham de Moivre">de Moivre</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euler</a></li> <li><a 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Trigonometrik özdeşlikler listesi - Vikipedi