Ters trigonometrik fonksiyonlar

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<a href="#" class="vector-toc-link"> <div class="vector-toc-text">Giriş</div> </a> </li> <li id="toc-Asıl_değerler" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Asıl_değerler"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Asıl değerler</span> </div> </a> <ul id="toc-Asıl_değerler-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ters_trigonometrik_fonksiyonların_ilişkisi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ters_trigonometrik_fonksiyonların_ilişkisi"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Ters trigonometrik fonksiyonların ilişkisi</span> </div> </a> <ul id="toc-Ters_trigonometrik_fonksiyonların_ilişkisi-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Trigonometrik_fonksiyonlar_ile_ters_trigonometrik_fonksiyonlar_arasındaki_ilişkiler" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Trigonometrik_fonksiyonlar_ile_ters_trigonometrik_fonksiyonlar_arasındaki_ilişkiler"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Trigonometrik fonksiyonlar ile ters trigonometrik fonksiyonlar arasındaki ilişkiler</span> </div> </a> <ul id="toc-Trigonometrik_fonksiyonlar_ile_ters_trigonometrik_fonksiyonlar_arasındaki_ilişkiler-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ters_trigonometrik_fonksiyonların_türevleri" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ters_trigonometrik_fonksiyonların_türevleri"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Ters trigonometrik fonksiyonların türevleri</span> </div> </a> <ul id="toc-Ters_trigonometrik_fonksiyonların_türevleri-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Belirli_integral_olarak_ifadesi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Belirli_integral_olarak_ifadesi"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Belirli integral olarak ifadesi</span> </div> </a> <ul id="toc-Belirli_integral_olarak_ifadesi-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sonsuz_seriler" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sonsuz_seriler"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Sonsuz seriler</span> </div> </a> <ul id="toc-Sonsuz_seriler-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Logaritmik_biçimler" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Logaritmik_biçimler"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Logaritmik biçimler</span> </div> </a> <button aria-controls="toc-Logaritmik_biçimler-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Logaritmik biçimler alt bölümünü aç/kapa</span> </button> <ul id="toc-Logaritmik_biçimler-sublist" class="vector-toc-list"> <li id="toc-Örnek_ispat" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Örnek_ispat"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Örnek ispat</span> </div> </a> <ul id="toc-Örnek_ispat-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Ayrıca_bakınız" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ayrıca_bakınız"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Ayrıca bakınız</span> </div> </a> <ul id="toc-Ayrıca_bakınız-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" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">İçindekiler tablosunu değiştir</span> </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"><span class="mw-page-title-main">Ters trigonometrik fonksiyonlar</span></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. 53 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-53" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">53 dil</span> </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/%D8%AF%D9%88%D8%A7%D9%84_%D9%85%D8%AB%D9%84%D8%AB%D9%8A%D8%A9_%D8%B9%D9%83%D8%B3%D9%8A%D8%A9" title="دوال مثلثية عكسية - Arapça" lang="ar" hreflang="ar" data-title="دوال مثلثية عكسية" data-language-autonym="العربية" data-language-local-name="Arapça" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/T%C9%99rs_triqonometrik_funksiyalar" title="Tərs triqonometrik funksiyalar - Azerbaycan dili" lang="az" hreflang="az" data-title="Tərs triqonometrik funksiyalar" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaycan dili" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%9A%D0%B8%D1%80%D0%B5_%D1%82%D1%80%D0%B8%D0%B3%D0%BE%D0%BD%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D0%BA_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F%D0%BB%D0%B0%D1%80" title="Кире тригонометрик функциялар - Başkırtça" lang="ba" hreflang="ba" data-title="Кире тригонометрик функциялар" data-language-autonym="Башҡортса" data-language-local-name="Başkırtça" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9E%D0%B1%D1%80%D0%B0%D1%82%D0%BD%D0%B8_%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%BD%D0%B8_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D0%B8" title="Обратни тригонометрични функции - Bulgarca" lang="bg" hreflang="bg" data-title="Обратни тригонометрични функции" data-language-autonym="Български" data-language-local-name="Bulgarca" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AC%E0%A6%BF%E0%A6%AA%E0%A6%B0%E0%A7%80%E0%A6%A4_%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%AA%E0%A7%87%E0%A6%95%E0%A7%8D%E0%A6%B7%E0%A6%95" title="বিপরীত ত্রিকোণমিতিক অপেক্ষক - Bengalce" lang="bn" hreflang="bn" data-title="বিপরীত ত্রিকোণমিতিক অপেক্ষক" data-language-autonym="বাংলা" data-language-local-name="Bengalce" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Inverses_de_les_funcions_trigonom%C3%A8triques" title="Inverses de les funcions trigonomètriques - Katalanca" lang="ca" hreflang="ca" data-title="Inverses de les funcions trigonomètriques" data-language-autonym="Català" data-language-local-name="Katalanca" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%81%D8%A7%D9%86%DA%A9%D8%B4%D9%86%DB%95_%D8%B3%DB%8E%DA%AF%DB%86%D8%B4%DB%95%DB%8C%DB%8C%DB%8C%DB%95_%DA%BE%DB%95%DA%B5%DA%AF%DB%95%DA%95%D8%A7%D9%88%DB%95%DA%A9%D8%A7%D9%86" title="فانکشنە سێگۆشەیییە ھەڵگەڕاوەکان - Orta Kürtçe" lang="ckb" hreflang="ckb" data-title="فانکشنە سێگۆشەیییە ھەڵگەڕاوەکان" data-language-autonym="کوردی" data-language-local-name="Orta Kürtçe" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Cyklometrick%C3%A1_funkce" title="Cyklometrická funkce - Çekçe" lang="cs" hreflang="cs" data-title="Cyklometrická funkce" data-language-autonym="Čeština" data-language-local-name="Çekçe" class="interlanguage-link-target"><span>Čeština</span></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%D0%BB%D0%BB%D0%B5_%D0%BA%D1%83%D1%82%C4%83%D0%BD%D0%BB%D0%B0_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%81%D0%B5%D0%BC" title="Тригонометрилле кутăнла функцисем - Çuvaşça" lang="cv" hreflang="cv" data-title="Тригонометрилле кутăнла функцисем" data-language-autonym="Чӑвашла" data-language-local-name="Çuvaşça" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Arcus-funktioner" title="Arcus-funktioner - Danca" lang="da" hreflang="da" data-title="Arcus-funktioner" data-language-autonym="Dansk" data-language-local-name="Danca" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Arkusfunktion" title="Arkusfunktion - Almanca" lang="de" hreflang="de" data-title="Arkusfunktion" data-language-autonym="Deutsch" data-language-local-name="Almanca" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%91%CE%BD%CF%84%CE%AF%CF%83%CF%84%CF%81%CE%BF%CF%86%CE%B5%CF%82_%CF%84%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%B9%CE%BA%CE%AD%CF%82_%CF%83%CF%85%CE%BD%CE%B1%CF%81%CF%84%CE%AE%CF%83%CE%B5%CE%B9%CF%82" title="Αντίστροφες τριγωνομετρικές συναρτήσεις - Yunanca" lang="el" hreflang="el" data-title="Αντίστροφες τριγωνομετρικές συναρτήσεις" data-language-autonym="Ελληνικά" data-language-local-name="Yunanca" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Inverse_trigonometric_functions" title="Inverse trigonometric functions - İngilizce" lang="en" hreflang="en" data-title="Inverse trigonometric functions" data-language-autonym="English" data-language-local-name="İngilizce" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Inversa_trigonometria_funkcio" title="Inversa trigonometria funkcio - Esperanto" lang="eo" hreflang="eo" data-title="Inversa trigonometria funkcio" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Funci%C3%B3n_trigonom%C3%A9trica_inversa" title="Función trigonométrica inversa - İspanyolca" lang="es" hreflang="es" data-title="Función trigonométrica inversa" data-language-autonym="Español" data-language-local-name="İspanyolca" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Arkusfunktsioonid" title="Arkusfunktsioonid - Estonca" lang="et" hreflang="et" data-title="Arkusfunktsioonid" data-language-autonym="Eesti" data-language-local-name="Estonca" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Alderantzizko_funtzio_trigonometriko" title="Alderantzizko funtzio trigonometriko - Baskça" lang="eu" hreflang="eu" data-title="Alderantzizko funtzio trigonometriko" data-language-autonym="Euskara" data-language-local-name="Baskça" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D9%88%D8%A7%D8%A8%D8%B9_%D9%85%D8%B9%DA%A9%D9%88%D8%B3_%D9%85%D8%AB%D9%84%D8%AB%D8%A7%D8%AA%DB%8C" title="توابع معکوس مثلثاتی - Farsça" lang="fa" hreflang="fa" data-title="توابع معکوس مثلثاتی" data-language-autonym="فارسی" data-language-local-name="Farsça" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Arkusfunktiot" title="Arkusfunktiot - Fince" lang="fi" hreflang="fi" data-title="Arkusfunktiot" data-language-autonym="Suomi" data-language-local-name="Fince" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Fonction_circulaire_r%C3%A9ciproque" title="Fonction circulaire réciproque - Fransızca" lang="fr" hreflang="fr" data-title="Fonction circulaire réciproque" data-language-autonym="Français" data-language-local-name="Fransızca" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Funci%C3%B3ns_trigonom%C3%A9tricas_inversas" title="Funcións trigonométricas inversas - Galiçyaca" lang="gl" hreflang="gl" data-title="Funcións trigonométricas inversas" data-language-autonym="Galego" data-language-local-name="Galiçyaca" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%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_%D7%94%D7%A4%D7%95%D7%9B%D7%95%D7%AA" title="פונקציות טריגונומטריות הפוכות - İbranice" lang="he" hreflang="he" data-title="פונקציות טריגונומטריות הפוכות" data-language-autonym="עברית" data-language-local-name="İbranice" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%A4%E0%A4%BF%E0%A4%B2%E0%A5%8B%E0%A4%AE_%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%AB%E0%A4%B2%E0%A4%A8" title="प्रतिलोम त्रिकोणमितीय फलन - Hintçe" lang="hi" hreflang="hi" data-title="प्रतिलोम त्रिकोणमितीय फलन" data-language-autonym="हिन्दी" data-language-local-name="Hintçe" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Arkus_funkcije" title="Arkus funkcije - Hırvatça" lang="hr" hreflang="hr" data-title="Arkus funkcije" data-language-autonym="Hrvatski" data-language-local-name="Hırvatça" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%80%D5%A1%D5%AF%D5%A1%D5%A4%D5%A1%D6%80%D5%B1_%D5%A5%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_%D6%86%D5%B8%D6%82%D5%B6%D5%AF%D6%81%D5%AB%D5%A1%D5%B6%D5%A5%D6%80" title="Հակադարձ եռանկյունաչափական ֆունկցիաներ - Ermenice" lang="hy" hreflang="hy" data-title="Հակադարձ եռանկյունաչափական ֆունկցիաներ" data-language-autonym="Հայերեն" data-language-local-name="Ermenice" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Fungsi_invers_trigonometri" title="Fungsi invers trigonometri - Endonezce" lang="id" hreflang="id" data-title="Fungsi invers trigonometri" data-language-autonym="Bahasa Indonesia" data-language-local-name="Endonezce" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Funzione_trigonometrica_inversa" title="Funzione trigonometrica inversa - İtalyanca" lang="it" hreflang="it" data-title="Funzione trigonometrica inversa" data-language-autonym="İtaliano" data-language-local-name="İtalyanca" class="interlanguage-link-target"><span>İtaliano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E9%80%86%E4%B8%89%E8%A7%92%E9%96%A2%E6%95%B0" title="逆三角関数 - Japonca" lang="ja" hreflang="ja" data-title="逆三角関数" data-language-autonym="日本語" data-language-local-name="Japonca" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9A%D0%B5%D1%80%D1%96_%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%8F%D0%BB%D1%8B%D2%9B_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F%D0%BB%D0%B0%D1%80" title="Кері тригонометриялық функциялар - Kazakça" lang="kk" hreflang="kk" data-title="Кері тригонометриялық функциялар" data-language-autonym="Қазақша" data-language-local-name="Kazakça" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%A2%E1%9E%93%E1%9E%BB%E1%9E%82%E1%9E%98%E1%9E%93%E1%9F%8D%E1%9E%8F%E1%9F%92%E1%9E%9A%E1%9E%B8%E1%9E%80%E1%9F%84%E1%9E%8E%E1%9E%98%E1%9E%B6%E1%9E%8F%E1%9F%92%E1%9E%9A%E1%9E%85%E1%9F%92%E1%9E%9A%E1%9E%B6%E1%9E%9F%E1%9F%8B" title="អនុគមន៍ត្រីកោណមាត្រច្រាស់ - Khmer dili" lang="km" hreflang="km" data-title="អនុគមន៍ត្រីកោណមាត្រច្រាស់" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="Khmer dili" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%97%AD%EC%82%BC%EA%B0%81_%ED%95%A8%EC%88%98" title="역삼각 함수 - Korece" lang="ko" hreflang="ko" data-title="역삼각 함수" data-language-autonym="한국어" data-language-local-name="Korece" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%90%D1%80%D0%BA%D1%84%D0%B5%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Аркфенкция - Kırgızca" lang="ky" hreflang="ky" data-title="Аркфенкция" data-language-autonym="Кыргызча" data-language-local-name="Kırgızca" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Functiones_trigonometricae_inversae" title="Functiones trigonometricae inversae - Latince" lang="la" hreflang="la" data-title="Functiones trigonometricae inversae" data-language-autonym="Latina" data-language-local-name="Latince" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Invers%C4%81s_trigonometrisk%C4%81s_funkcijas" title="Inversās trigonometriskās funkcijas - Letonca" lang="lv" hreflang="lv" data-title="Inversās trigonometriskās funkcijas" data-language-autonym="Latviešu" data-language-local-name="Letonca" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%98%D0%BD%D0%B2%D0%B5%D1%80%D0%B7%D0%BD%D0%B8_%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_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D0%B8" title="Инверзни тригонометриски функции - Makedonca" lang="mk" hreflang="mk" data-title="Инверзни тригонометриски функции" data-language-autonym="Македонски" data-language-local-name="Makedonca" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Cyclometrische_functie" title="Cyclometrische functie - Felemenkçe" lang="nl" hreflang="nl" data-title="Cyclometrische functie" data-language-autonym="Nederlands" data-language-local-name="Felemenkçe" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Arcus-funksjon" title="Arcus-funksjon - Norveççe Nynorsk" lang="nn" hreflang="nn" data-title="Arcus-funksjon" data-language-autonym="Norsk nynorsk" data-language-local-name="Norveççe Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Inverse_trigonometriske_funksjoner" title="Inverse trigonometriske funksjoner - Norveççe Bokmål" lang="nb" hreflang="nb" data-title="Inverse trigonometriske funksjoner" data-language-autonym="Norsk bokmål" data-language-local-name="Norveççe Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Funkcje_cyklometryczne" title="Funkcje cyklometryczne - Lehçe" lang="pl" hreflang="pl" data-title="Funkcje cyklometryczne" data-language-autonym="Polski" data-language-local-name="Lehçe" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Fun%C3%A7%C3%B5es_trigonom%C3%A9tricas_inversas" title="Funções trigonométricas inversas - Portekizce" lang="pt" hreflang="pt" data-title="Funções trigonométricas inversas" data-language-autonym="Português" data-language-local-name="Portekizce" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9E%D0%B1%D1%80%D0%B0%D1%82%D0%BD%D1%8B%D0%B5_%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%B8%D0%B5_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D0%B8" title="Обратные тригонометрические функции - Rusça" lang="ru" hreflang="ru" data-title="Обратные тригонометрические функции" data-language-autonym="Русский" data-language-local-name="Rusça" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Inverzne_trigonometrijske_funkcije" title="Inverzne trigonometrijske funkcije - Sırp-Hırvat Dili" lang="sh" hreflang="sh" data-title="Inverzne trigonometrijske funkcije" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Sırp-Hırvat Dili" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%B4%E0%B7%8A%E2%80%8D%E0%B6%BB%E0%B6%AD%E0%B7%92%E0%B6%BD%E0%B7%9D%E0%B6%B8_%E0%B6%AD%E0%B7%8A%E2%80%8D%E0%B6%BB%E0%B7%92%E0%B6%9A%E0%B7%9D%E0%B6%AB%E0%B6%B8%E0%B7%92%E0%B6%AD%E0%B7%92%E0%B6%9A_%E0%B7%81%E0%B7%8A%E2%80%8D%E0%B6%BB%E0%B7%92%E0%B6%AD" title="ප්‍රතිලෝම ත්‍රිකෝණමිතික ශ්‍රිත - Sinhali dili" lang="si" hreflang="si" data-title="ප්‍රතිලෝම ත්‍රිකෝණමිතික ශ්‍රිත" data-language-autonym="සිංහල" data-language-local-name="Sinhali dili" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Cyklometrick%C3%A1_funkcia" title="Cyklometrická funkcia - Slovakça" lang="sk" hreflang="sk" data-title="Cyklometrická funkcia" data-language-autonym="Slovenčina" data-language-local-name="Slovakça" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Kro%C5%BEna_funkcija" title="Krožna funkcija - Slovence" lang="sl" hreflang="sl" data-title="Krožna funkcija" data-language-autonym="Slovenščina" data-language-local-name="Slovence" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%98%D0%BD%D0%B2%D0%B5%D1%80%D0%B7%D0%BD%D0%B5_%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%98%D1%81%D0%BA%D0%B5_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%98%D0%B5" title="Инверзне тригонометријске функције - Sırpça" lang="sr" hreflang="sr" data-title="Инверзне тригонометријске функције" data-language-autonym="Српски / srpski" data-language-local-name="Sırpça" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%A8%E0%AF%87%E0%AE%B0%E0%AF%8D%E0%AE%AE%E0%AE%BE%E0%AE%B1%E0%AF%81_%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%9A%E0%AE%BE%E0%AE%B0%E0%AF%8D%E0%AE%AA%E0%AF%81%E0%AE%95%E0%AE%B3%E0%AF%8D" title="நேர்மாறு முக்கோணவியல் சார்புகள் - Tamilce" lang="ta" hreflang="ta" data-title="நேர்மாறு முக்கோணவியல் சார்புகள்" data-language-autonym="தமிழ்" data-language-local-name="Tamilce" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%9F%E0%B8%B1%E0%B8%87%E0%B8%81%E0%B9%8C%E0%B8%8A%E0%B8%B1%E0%B8%99%E0%B8%95%E0%B8%A3%E0%B8%B5%E0%B9%82%E0%B8%81%E0%B8%93%E0%B8%A1%E0%B8%B4%E0%B8%95%E0%B8%B4%E0%B8%9C%E0%B8%81%E0%B8%9C%E0%B8%B1%E0%B8%99" title="ฟังก์ชันตรีโกณมิติผกผัน - Tayca" lang="th" hreflang="th" data-title="ฟังก์ชันตรีโกณมิติผกผัน" data-language-autonym="ไทย" data-language-local-name="Tayca" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9E%D0%B1%D0%B5%D1%80%D0%BD%D0%B5%D0%BD%D1%96_%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%BD%D1%96_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%97" title="Обернені тригонометричні функції - Ukraynaca" lang="uk" hreflang="uk" data-title="Обернені тригонометричні функції" data-language-autonym="Українська" data-language-local-name="Ukraynaca" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/C%C3%A1c_h%C3%A0m_l%C6%B0%E1%BB%A3ng_gi%C3%A1c_ng%C6%B0%E1%BB%A3c" title="Các hàm lượng giác ngược - Vietnamca" lang="vi" hreflang="vi" data-title="Các hàm lượng giác ngược" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamca" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%8F%8D%E4%B8%89%E8%A7%92%E5%87%BD%E6%95%B0" title="反三角函数 - Çince" lang="zh" hreflang="zh" data-title="反三角函数" data-language-autonym="中文" data-language-local-name="Çince" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a 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kaynaklar ekleyerek</a> <a class="external text" href="https://tr.wikipedia.org/w/index.php?title=Ters_trigonometrik_fonksiyonlar&action=edit">madde içeriğinin geliştirilmesine</a> yardımcı olun. Kaynaksız içerik itiraz konusu olabilir ve <a href="/wiki/Vikipedi:Do%C4%9Frulanabilirlik#Kanıt_sorumluluğu" title="Vikipedi:Doğrulanabilirlik">kaldırılabilir</a>.<br /><small><span class="plainlinks"><i>Kaynak ara:</i> <a rel="nofollow" class="external text" href="//www.google.com/search?as_eq=wikipedia&q=%22Ters+trigonometrik+fonksiyonlar%22">"Ters trigonometrik fonksiyonlar"</a> – <a rel="nofollow" class="external text" href="//www.google.com/search?tbm=nws&q=%22Ters+trigonometrik+fonksiyonlar%22+-wikipedia">haber</a> · <a rel="nofollow" class="external text" href="//www.google.com/search?&q=%22Ters+trigonometrik+fonksiyonlar%22+site:news.google.com/newspapers&source=newspapers">gazete</a> · <a rel="nofollow" class="external text" href="//www.google.com/search?tbs=bks:1&q=%22Ters+trigonometrik+fonksiyonlar%22+-wikipedia">kitap</a> · <a rel="nofollow" class="external text" href="//scholar.google.com/scholar?q=%22Ters+trigonometrik+fonksiyonlar%22">akademik</a> · <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Ters+trigonometrik+fonksiyonlar%22&acc=on&wc=on">JSTOR</a></span></small></span> <small class="date-container"><i>(<span class="date">Şubat 2017</span>)</i></small><small class="hide-when-compact"><i> (<a href="/wiki/Yard%C4%B1m:Bak%C4%B1m_%C5%9Fablonunu_kald%C4%B1rmak" title="Yardım:Bakım şablonunu kaldırmak">Bu şablonun nasıl ve ne zaman kaldırılması gerektiğini öğrenin</a>)</i></small></div></td></tr></tbody></table> <p><a href="/wiki/Matematik" title="Matematik">Matematikte</a> <b>ters trigonometrik fonksiyonlar</b>, <a href="/wiki/Tan%C4%B1m_k%C3%BCmesi" title="Tanım kümesi">tanım kümesinde</a> bulunan <a href="/wiki/Trigonometrik_fonksiyonlar" title="Trigonometrik fonksiyonlar">trigonometrik fonksiyonların</a> <a href="/wiki/Ters_fonksiyon" title="Ters fonksiyon">ters fonksiyonudur</a>. </p><p>arcsin, arccos, arctan sırasıyla sin<sup>−1</sup>, cos<sup>−1</sup>, tan<sup>−1</sup> olarak gösterilir. Fakat bu dönüşüm, sin<sup>2</sup>(<i>x</i>) gibi yaygın kullanılan ifadelerde karmaşaya neden olabilir. Buradaki sayısal kuvvet, ters çarpan ile <a href="/wiki/Ters_fonksiyon" title="Ters fonksiyon">ters fonksiyon</a> arasında bir karmaşa meydana getirir. </p><p>Bilgisayar <a href="/wiki/Programlama_dili" title="Programlama dili">programlama dillerinde</a>, arcsin, arccos, arctan fonksiyonları genellikle asin, acos, atan olarak adlandırılır. Çoğu programlama dili de <i>atan2</i> fonksiyonunu iki argümanlı olarak kullanır ve <i>y</i> / <i>x'</i>in arctanjantını (−<span class="texhtml">π</span>, <span class="texhtml">π</span>] aralığında <i>y</i> ve <i>x</i> olarak ifade eder. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Asıl_değerler"><span id="As.C4.B1l_de.C4.9Ferler"></span>Asıl değerler</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ters_trigonometrik_fonksiyonlar&veaction=edit&section=1" title="Değiştirilen bölüm: Asıl değerler" class="mw-editsection-visualeditor"><span>değiştir</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Ters_trigonometrik_fonksiyonlar&action=edit&section=1" title="Bölümün kaynak kodunu değiştir: Asıl değerler"><span>kaynağı değiştir</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Altı trigonometrik fonksiyondan hiçbiri <a href="/wiki/Birebir_fonksiyon" title="Birebir fonksiyon">birebir fonksiyon</a> değildir, terslerinin alınmasında kısıtlamalar vardır. Bu yüzden ters fonksiyonların değerleri, asıl fonksiyonların tanım kümesinin <a href="/wiki/Alt_k%C3%BCme" title="Alt küme">alt kümesidir</a> </p><p>Örneğin çok değerli fonksiyonlarda, yalnızca <a href="/wiki/Karek%C3%B6k" title="Karekök">karekök</a> fonksiyonu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y={\sqrt {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y={\sqrt {x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58a6e259ae203e3c564ba35d228d13365f846a6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.52ex; height:3.009ex;" alt="{\displaystyle y={\sqrt {x}}}"></span>, <i>y</i><sup>2</sup> = <i>x</i> olarak tanımlanabilir. <i>y</i> = arcsin(<i>x</i>) fonksiyonu sin(<i>y</i>) = <i>x</i> olarak ifade edilebilir. sin(<i>y</i>) = <i>x'</i>yi ifade eden birçok <i>y</i> sayısı vardır. Örneğin sin(0) = 0, fakat sin(<span class="texhtml">π</span>) = 0, sin(2<span class="texhtml">π</span>) = 0, vb. arcsin fonksiyonu da çok değerlidir: arcsin(0) = 0, fakat arcsin(0) = <a href="/wiki/Pi" class="mw-disambig" title="Pi"><span class="texhtml">π</span></a>, arcsin(0) = 2<span class="texhtml">π</span>, vb. Yalnızca tek bir değer belirtildiğinde, fonksiyon kısıtlanır. Bu kısıtlama ile, tanım kümesindeki her bir <i>x</i> için arcsin(<i>x</i>) ifadesi yalnızca tek bir değere karşılık gelir, bu da <b>asıl değer</b> olarak adlandırılır. Bu özellikler tüm ters trigonometrik fonksiyonlarda uygulanır. </p><p>Aşağıdaki tabloda ters trigonometrik fonksiyonların asılları listelenmiştir. </p> <table class="wikitable" style="text-align:center"> <tbody><tr> <th>Fonksiyon </th> <th>Genel gösterim </th> <th>İfade </th> <th><i>x</i> değer aralığı </th> <th>Asıl değer aralığı <br /> (<a href="/wiki/Radyan" title="Radyan">radyan</a>) </th> <th>Asıl değer aralığı <br /> (<a href="/wiki/Derece_(birim)" title="Derece (birim)">derece</a>) </th></tr> <tr> <td><b>arcsinüs</b></td> <td><i>y</i> = arcsin <i>x</i></td> <td><i>x</i> = <a href="/wiki/Sin%C3%BCs_(matematik)" title="Sinüs (matematik)">sin</a> <i>y</i></td> <td>−1 ≤ <i>x</i> ≤ 1</td> <td>−<span class="texhtml">π</span>/2 ≤ <i>y</i> ≤ <span class="texhtml">π</span>/2</td> <td>−90° ≤ <i>y</i> ≤ 90° </td></tr> <tr> <td><b>arckosinüs</b></td> <td><i>y</i> = arccos <i>x</i></td> <td><i>x</i> = <a href="/wiki/Kosin%C3%BCs" title="Kosinüs">cos</a> <i>y</i></td> <td>−1 ≤ <i>x</i> ≤ 1</td> <td>0 ≤ <i>y</i> ≤ <span class="texhtml">π</span></td> <td>0° ≤ <i>y</i> ≤ 180° </td></tr> <tr> <td><b>arctanjant</b></td> <td><i>y</i> = arctan <i>x</i></td> <td><i>x</i> = <a href="/wiki/Tanjant" title="Tanjant">tan</a> <i>y</i></td> <td>tüm <a href="/wiki/Reel_say%C4%B1" class="mw-redirect" title="Reel sayı">reel sayılar</a></td> <td>−<span class="texhtml">π</span>/2 < <i>y</i> < <span class="texhtml">π</span>/2</td> <td>−90° < <i>y</i> < 90° </td></tr> <tr> <td><b>arckotanjant</b></td> <td><i>y</i> = arccot <i>x</i></td> <td><i>x</i> = <a href="/wiki/Kotanjant" title="Kotanjant">cot</a> <i>y</i></td> <td>tüm reel sayılar </td> <td>0 < <i>y</i> < <span class="texhtml">π</span></td> <td>0° < <i>y</i> < 180° </td></tr> <tr> <td><b>arcsekant</b></td> <td><i>y</i> = arcsec <i>x</i></td> <td><i>x</i> = <a href="/wiki/Sekant" title="Sekant">sec</a> <i>y</i></td> <td><i>x</i> ≤ −1 or 1 ≤ <i>x</i></td> <td>0 ≤ <i>y</i> < <span class="texhtml">π</span>/2 or <span class="texhtml">π</span>/2 < <i>y</i> ≤ <span class="texhtml">π</span></td> <td>0° ≤ <i>y</i> < 90° or 90° < <i>y</i> ≤ 180° </td></tr> <tr> <td><b>arckosekant</b></td> <td><i>y</i> = arccsc <i>x</i></td> <td><i>x</i> = <a href="/wiki/Kosekant" title="Kosekant">csc</a> <i>y</i></td> <td><i>x</i> ≤ −1 or 1 ≤ <i>x</i></td> <td>−<span class="texhtml">π</span>/2 ≤ <i>y</i> < 0 or 0 < <i>y</i> ≤ <span class="texhtml">π</span>/2</td> <td>-90° ≤ <i>y</i> < 0° or 0° < <i>y</i> ≤ 90° </td></tr> </tbody></table> <p>Eğer <i>x</i> bir <a href="/wiki/Karma%C5%9F%C4%B1k_say%C4%B1" title="Karmaşık sayı">karmaşık sayı</a> olursa, <i>y</i> değer aralığı yalnızca <a href="/wiki/Ger%C3%A7el_k%C4%B1s%C4%B1m" title="Gerçel kısım">gerçel kısımda</a> olur. </p> <div class="mw-heading mw-heading2"><h2 id="Ters_trigonometrik_fonksiyonların_ilişkisi"><span id="Ters_trigonometrik_fonksiyonlar.C4.B1n_ili.C5.9Fkisi"></span>Ters trigonometrik fonksiyonların ilişkisi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ters_trigonometrik_fonksiyonlar&veaction=edit&section=2" title="Değiştirilen bölüm: Ters trigonometrik fonksiyonların ilişkisi" class="mw-editsection-visualeditor"><span>değiştir</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Ters_trigonometrik_fonksiyonlar&action=edit&section=2" title="Bölümün kaynak kodunu değiştir: Ters trigonometrik fonksiyonların ilişkisi"><span>kaynağı değiştir</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Dosya:Arcsine_Arccosine.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Arcsine_Arccosine.svg/170px-Arcsine_Arccosine.svg.png" decoding="async" width="170" height="312" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Arcsine_Arccosine.svg/255px-Arcsine_Arccosine.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Arcsine_Arccosine.svg/340px-Arcsine_Arccosine.svg.png 2x" data-file-width="240" data-file-height="440" /></a><figcaption>arcsin(<i>x</i>) (kırmızı), arccos(<i>x</i>) (mavi) fonksiyonlarının asıl değerleri nin <a href="/wiki/Kartezyen_koordinat_sistemi" title="Kartezyen koordinat sistemi">kartezyen koordinatındaki</a> grafiği.</figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Dosya:Arctangent_Arccotangent.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Arctangent_Arccotangent.svg/290px-Arctangent_Arccotangent.svg.png" decoding="async" width="290" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Arctangent_Arccotangent.svg/435px-Arctangent_Arccotangent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Arctangent_Arccotangent.svg/580px-Arctangent_Arccotangent.svg.png 2x" data-file-width="420" data-file-height="260" /></a><figcaption>arctan(<i>x</i>) ve arccot(<i>x</i>) fonksiyonlarının kartezyen düzlemindeki asıl değerleri.</figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Dosya:Arcsecant_Arccosecant.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/56/Arcsecant_Arccosecant.svg/290px-Arcsecant_Arccosecant.svg.png" decoding="async" width="290" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/56/Arcsecant_Arccosecant.svg/435px-Arcsecant_Arccosecant.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/56/Arcsecant_Arccosecant.svg/580px-Arcsecant_Arccosecant.svg.png 2x" data-file-width="420" data-file-height="260" /></a><figcaption>arcsec(<i>x</i>) ve arccsc(<i>x</i>) fonksiyonlarının kartezyen düzlemindeki grafikleri.</figcaption></figure> <p><a href="/wiki/T%C3%BCmler_a%C3%A7%C4%B1lar" title="Tümler açılar">Tümler açılar</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arccos x={\frac {\pi }{2}}-\arcsin x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccos</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>arcsin</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arccos x={\frac {\pi }{2}}-\arcsin x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dce0f6afbdf08944616f77cda3f75a0534acde27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.72ex; height:4.676ex;" alt="{\displaystyle \arccos x={\frac {\pi }{2}}-\arcsin x}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {arccot} x={\frac {\pi }{2}}-\arctan x}"> <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>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>arctan</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {arccot} x={\frac {\pi }{2}}-\arctan x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9f6c81df598388b903746b528bcf1656a9a9350" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:24.213ex; height:4.676ex;" alt="{\displaystyle \operatorname {arccot} x={\frac {\pi }{2}}-\arctan x}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {arccsc} x={\frac {\pi }{2}}-\operatorname {arcsec} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccsc</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>arcsec</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {arccsc} x={\frac {\pi }{2}}-\operatorname {arcsec} x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e05daba0d4ae5d3629691036aa01cfa7d636d915" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.716ex; height:4.676ex;" alt="{\displaystyle \operatorname {arccsc} x={\frac {\pi }{2}}-\operatorname {arcsec} x}"></span></dd></dl> <p>Negatif argümanlar: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arcsin(-x)=-\arcsin x\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>arcsin</mi> <mo>⁡</mo> <mi>x</mi> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arcsin(-x)=-\arcsin x\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f951d2474af7cca97000aa9aeeb8a7ea9d85a1ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.271ex; width:23.766ex; height:2.843ex;" alt="{\displaystyle \arcsin(-x)=-\arcsin x\!}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arccos(-x)=\pi -\arccos x\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>π</mi> <mo>−</mo> <mi>arccos</mi> <mo>⁡</mo> <mi>x</mi> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arccos(-x)=\pi -\arccos x\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6feea81f20d37018f0861bd01b5cbea0dbf55ef0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.271ex; width:26.254ex; height:2.843ex;" alt="{\displaystyle \arccos(-x)=\pi -\arccos x\!}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arctan(-x)=-\arctan x\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>arctan</mi> <mo>⁡</mo> <mi>x</mi> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arctan(-x)=-\arctan x\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/739cd22add73cdb235f2aff9fa3f8800ffd553a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.271ex; width:24.774ex; height:2.843ex;" alt="{\displaystyle \arctan(-x)=-\arctan x\!}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {arccot}(-x)=\pi -\operatorname {arccot} x\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>π</mi> <mo>−</mo> <mi>arccot</mi> <mo>⁡</mo> <mi>x</mi> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {arccot}(-x)=\pi -\operatorname {arccot} x\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57bc05d8aee05c539f81b017e0b8cfe9acc86418" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.271ex; width:26.231ex; height:2.843ex;" alt="{\displaystyle \operatorname {arccot}(-x)=\pi -\operatorname {arccot} x\!}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {arcsec}(-x)=\pi -\operatorname {arcsec} x\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arcsec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>π</mi> <mo>−</mo> <mi>arcsec</mi> <mo>⁡</mo> <mi>x</mi> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {arcsec}(-x)=\pi -\operatorname {arcsec} x\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe26c12ec7a899df3d946a682cc1a8cb867a7bd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.271ex; width:25.994ex; height:2.843ex;" alt="{\displaystyle \operatorname {arcsec}(-x)=\pi -\operatorname {arcsec} x\!}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {arccsc}(-x)=-\operatorname {arccsc} x\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccsc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>arccsc</mi> <mo>⁡</mo> <mi>x</mi> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {arccsc}(-x)=-\operatorname {arccsc} x\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20d871a3f1ae3ed9714cccd4014a09de697e8e29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.271ex; width:24.016ex; height:2.843ex;" alt="{\displaystyle \operatorname {arccsc}(-x)=-\operatorname {arccsc} x\!}"></span></dd></dl> <p>Karşıt argümanlar: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arccos(1/x)\,=\operatorname {arcsec} x\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>=</mo> <mi>arcsec</mi> <mo>⁡</mo> <mi>x</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arccos(1/x)\,=\operatorname {arcsec} x\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1eb0a4a8140758f4fba43903a5e9c949ed42e4f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.358ex; height:2.843ex;" alt="{\displaystyle \arccos(1/x)\,=\operatorname {arcsec} x\,}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arcsin(1/x)\,=\operatorname {arccsc} x\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>=</mo> <mi>arccsc</mi> <mo>⁡</mo> <mi>x</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arcsin(1/x)\,=\operatorname {arccsc} x\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb68d4ecd6526668df58ad4203e16b908c2d0924" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.103ex; height:2.843ex;" alt="{\displaystyle \arcsin(1/x)\,=\operatorname {arccsc} x\,}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arctan(1/x)={\tfrac {1}{2}}\pi -\arctan x=\operatorname {arccot} x,{\text{ eğer }}x>0\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>x</mi> <mo stretchy="false">)</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> <mo>−</mo> <mi>arctan</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mi>arccot</mi> <mo>⁡</mo> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> eğer </mtext> </mrow> <mi>x</mi> <mo>></mo> <mn>0</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arctan(1/x)={\tfrac {1}{2}}\pi -\arctan x=\operatorname {arccot} x,{\text{ eğer }}x>0\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec97665982ec7382823936005d5130c5a026632d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:52.629ex; height:3.509ex;" alt="{\displaystyle \arctan(1/x)={\tfrac {1}{2}}\pi -\arctan x=\operatorname {arccot} x,{\text{ eğer }}x>0\,}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arctan(1/x)=-{\tfrac {1}{2}}\pi -\arctan x=-\pi +\operatorname {arccot} x,{\text{ eğer }}x<0\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>x</mi> <mo stretchy="false">)</mo> <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> <mo>−</mo> <mi>arctan</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mi>π</mi> <mo>+</mo> <mi>arccot</mi> <mo>⁡</mo> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> eğer </mtext> </mrow> <mi>x</mi> <mo><</mo> <mn>0</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arctan(1/x)=-{\tfrac {1}{2}}\pi -\arctan x=-\pi +\operatorname {arccot} x,{\text{ eğer }}x<0\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76c4e7bf76f3979e2c181fa08d3452d763fee487" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:60.418ex; height:3.509ex;" alt="{\displaystyle \arctan(1/x)=-{\tfrac {1}{2}}\pi -\arctan x=-\pi +\operatorname {arccot} x,{\text{ eğer }}x<0\,}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {arccot}(1/x)={\tfrac {1}{2}}\pi -\operatorname {arccot} x=\arctan x,{\text{ eğer }}x>0\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>x</mi> <mo stretchy="false">)</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> <mo>−</mo> <mi>arccot</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> eğer </mtext> </mrow> <mi>x</mi> <mo>></mo> <mn>0</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {arccot}(1/x)={\tfrac {1}{2}}\pi -\operatorname {arccot} x=\arctan x,{\text{ eğer }}x>0\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39cf86c8706b810dbf38658923c9623006afc807" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:52.369ex; height:3.509ex;" alt="{\displaystyle \operatorname {arccot}(1/x)={\tfrac {1}{2}}\pi -\operatorname {arccot} x=\arctan x,{\text{ eğer }}x>0\,}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {arccot}(1/x)={\tfrac {3}{2}}\pi -\operatorname {arccot} x=\pi +\arctan x,{\text{ eğer }}x<0\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>π</mi> <mo>−</mo> <mi>arccot</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mi>π</mi> <mo>+</mo> <mi>arctan</mi> <mo>⁡</mo> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> eğer </mtext> </mrow> <mi>x</mi> <mo><</mo> <mn>0</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {arccot}(1/x)={\tfrac {3}{2}}\pi -\operatorname {arccot} x=\pi +\arctan x,{\text{ eğer }}x<0\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8891f37e11179a7744684c6b2832c0ac69d51cb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:56.541ex; height:3.509ex;" alt="{\displaystyle \operatorname {arccot}(1/x)={\tfrac {3}{2}}\pi -\operatorname {arccot} x=\pi +\arctan x,{\text{ eğer }}x<0\,}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {arcsec}(1/x)=\arccos x\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arcsec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>arccos</mi> <mo>⁡</mo> <mi>x</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {arcsec}(1/x)=\arccos x\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de7bb212931f11471dd9361a80c9445d9d30a85f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.971ex; height:2.843ex;" alt="{\displaystyle \operatorname {arcsec}(1/x)=\arccos x\,}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {arccsc}(1/x)=\arcsin x\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccsc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>arcsin</mi> <mo>⁡</mo> <mi>x</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {arccsc}(1/x)=\arcsin x\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66820165c4130cb416694b6e3155836da27b5413" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.716ex; height:2.843ex;" alt="{\displaystyle \operatorname {arccsc}(1/x)=\arcsin x\,}"></span></dd></dl> <p>Eğer yalnızca bir sinüs tablosu varsa: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arccos x=\arcsin {\sqrt {1-x^{2}}},{\text{ eğer }}0\leq x\leq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccos</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mi>arcsin</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> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> eğer </mtext> </mrow> <mn>0</mn> <mo>≤</mo> <mi>x</mi> <mo>≤</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arccos x=\arcsin {\sqrt {1-x^{2}}},{\text{ eğer }}0\leq x\leq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ac13493fa3aae7e8f4b16213869d503c81767b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:42.532ex; height:3.843ex;" alt="{\displaystyle \arccos x=\arcsin {\sqrt {1-x^{2}}},{\text{ eğer }}0\leq x\leq 1}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arctan x=\arcsin {\frac {x}{\sqrt {x^{2}+1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arctan</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mi>arcsin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arctan x=\arcsin {\frac {x}{\sqrt {x^{2}+1}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b962fc4081962df7662806fde8638347da211d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:27.177ex; height:6.009ex;" alt="{\displaystyle \arctan x=\arcsin {\frac {x}{\sqrt {x^{2}+1}}}}"></span></dd></dl> <p>Burada bir karmaşık sayının karekökü kullanılırsa, bunun pozitif gerçel kısmı (veya kare negatif gerçel ise <a href="/wiki/Sanal_k%C4%B1s%C4%B1m" title="Sanal kısım">sanal kısım</a>) seçilir. </p><p><i>Tanjant yarım açı formülü</i>nden, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan {\frac {\theta }{2}}={\frac {\sin \theta }{1+\cos \theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <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> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan {\frac {\theta }{2}}={\frac {\sin \theta }{1+\cos \theta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9d226878a7bd45ce5d55ee6f4a14b51a5f5247f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:18.271ex; height:5.676ex;" alt="{\displaystyle \tan {\frac {\theta }{2}}={\frac {\sin \theta }{1+\cos \theta }}}"></span>, aşağıdakiler elde edilebilir; </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arcsin x=2\arctan {\frac {x}{1+{\sqrt {1-x^{2}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arcsin</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mn>2</mn> <mi>arctan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mrow> <mn>1</mn> <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> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arcsin x=2\arctan {\frac {x}{1+{\sqrt {1-x^{2}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f173cb904ea159768ce374a2ec6235352a2421c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:32.73ex; height:6.009ex;" alt="{\displaystyle \arcsin x=2\arctan {\frac {x}{1+{\sqrt {1-x^{2}}}}}}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arccos x=2\arctan {\frac {\sqrt {1-x^{2}}}{1+x}},{\text{ eğer }}-1<x\leq +1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccos</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mn>2</mn> <mi>arctan</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> <mrow> <mn>1</mn> <mo>+</mo> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> eğer </mtext> </mrow> <mo>−</mo> <mn>1</mn> <mo><</mo> <mi>x</mi> <mo>≤</mo> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arccos x=2\arctan {\frac {\sqrt {1-x^{2}}}{1+x}},{\text{ eğer }}-1<x\leq +1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f171df9d8c62e9a4cbc379e4cb7202aea8436345" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:50.071ex; height:6.343ex;" alt="{\displaystyle \arccos x=2\arctan {\frac {\sqrt {1-x^{2}}}{1+x}},{\text{ eğer }}-1<x\leq +1}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arctan x=2\arctan {\frac {x}{1+{\sqrt {1+x^{2}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arctan</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mn>2</mn> <mi>arctan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mrow> <mn>1</mn> <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> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arctan x=2\arctan {\frac {x}{1+{\sqrt {1+x^{2}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11c68e7819b46b8054c8235e4ac6f11658851039" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:33.234ex; height:6.009ex;" alt="{\displaystyle \arctan x=2\arctan {\frac {x}{1+{\sqrt {1+x^{2}}}}}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Trigonometrik_fonksiyonlar_ile_ters_trigonometrik_fonksiyonlar_arasındaki_ilişkiler"><span id="Trigonometrik_fonksiyonlar_ile_ters_trigonometrik_fonksiyonlar_aras.C4.B1ndaki_ili.C5.9Fkiler"></span>Trigonometrik fonksiyonlar ile ters trigonometrik fonksiyonlar arasındaki ilişkiler</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ters_trigonometrik_fonksiyonlar&veaction=edit&section=3" title="Değiştirilen bölüm: Trigonometrik fonksiyonlar ile ters trigonometrik fonksiyonlar arasındaki ilişkiler" class="mw-editsection-visualeditor"><span>değiştir</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Ters_trigonometrik_fonksiyonlar&action=edit&section=3" title="Bölümün kaynak kodunu değiştir: Trigonometrik fonksiyonlar ile ters trigonometrik fonksiyonlar arasındaki ilişkiler"><span>kaynağı değiştir</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(\arccos x)=\cos(\arcsin x)={\sqrt {1-x^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arccos</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>cos</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"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(\arccos x)=\cos(\arcsin x)={\sqrt {1-x^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb3d925c7516a151e19c1214aeb9b27c83447d44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.106ex; height:3.509ex;" alt="{\displaystyle \sin(\arccos x)=\cos(\arcsin x)={\sqrt {1-x^{2}}}}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(\arctan x)={\frac {x}{\sqrt {1+x^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arctan</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(\arctan x)={\frac {x}{\sqrt {1+x^{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b854d778a5f62dedba78862e04de221f67e2be5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:25.493ex; height:6.009ex;" alt="{\displaystyle \sin(\arctan x)={\frac {x}{\sqrt {1+x^{2}}}}}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(\arctan x)={\frac {1}{\sqrt {1+x^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arctan</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(\arctan x)={\frac {1}{\sqrt {1+x^{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d77575bd81f9948b9a8d4ca2a322cc05bb19cb13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:25.748ex; height:6.509ex;" alt="{\displaystyle \cos(\arctan x)={\frac {1}{\sqrt {1+x^{2}}}}}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan(\arcsin x)={\frac {x}{\sqrt {1-x^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</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> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan(\arcsin x)={\frac {x}{\sqrt {1-x^{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e0ce58103c9104aaae4150ca852c10ac9d42d85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:25.493ex; height:6.009ex;" alt="{\displaystyle \tan(\arcsin x)={\frac {x}{\sqrt {1-x^{2}}}}}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan(\arccos x)={\frac {\sqrt {1-x^{2}}}{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</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> <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 \tan(\arccos x)={\frac {\sqrt {1-x^{2}}}{x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e49f6502c657657e03d833007397394c65aa6c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.748ex; height:6.176ex;" alt="{\displaystyle \tan(\arccos x)={\frac {\sqrt {1-x^{2}}}{x}}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Ters_trigonometrik_fonksiyonların_türevleri"><span id="Ters_trigonometrik_fonksiyonlar.C4.B1n_t.C3.BCrevleri"></span>Ters trigonometrik fonksiyonların türevleri</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ters_trigonometrik_fonksiyonlar&veaction=edit&section=4" title="Değiştirilen bölüm: Ters trigonometrik fonksiyonların türevleri" class="mw-editsection-visualeditor"><span>değiştir</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Ters_trigonometrik_fonksiyonlar&action=edit&section=4" title="Bölümün kaynak kodunu değiştir: Ters trigonometrik fonksiyonların türevleri"><span>kaynağı değiştir</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>x</i> in reel ve karmaşık değerlerinin <a href="/wiki/T%C3%BCrev" title="Türev">türevleri</a> şöyledir: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {d}{dx}}\arcsin x&{}={\frac {1}{\sqrt {1-x^{2}}}}\\{\frac {d}{dx}}\arccos x&{}={\frac {-1}{\sqrt {1-x^{2}}}}\\{\frac {d}{dx}}\arctan x&{}={\frac {1}{1+x^{2}}}\\{\frac {d}{dx}}\operatorname {arccot} x&{}={\frac {-1}{1+x^{2}}}\\{\frac {d}{dx}}\operatorname {arcsec} x&{}={\frac {1}{x\,{\sqrt {x^{2}-1}}}}\\{\frac {d}{dx}}\operatorname {arccsc} x&{}={\frac {-1}{x\,{\sqrt {x^{2}-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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>arcsin</mi> <mo>⁡</mo> <mi>x</mi> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </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> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>arccos</mi> <mo>⁡</mo> <mi>x</mi> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>arctan</mi> <mo>⁡</mo> <mi>x</mi> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>arccot</mi> <mo>⁡</mo> <mi>x</mi> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>arcsec</mi> <mo>⁡</mo> <mi>x</mi> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>x</mi> <mspace width="thinmathspace" /> <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> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>arccsc</mi> <mo>⁡</mo> <mi>x</mi> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mi>x</mi> <mspace width="thinmathspace" /> <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> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {d}{dx}}\arcsin x&{}={\frac {1}{\sqrt {1-x^{2}}}}\\{\frac {d}{dx}}\arccos x&{}={\frac {-1}{\sqrt {1-x^{2}}}}\\{\frac {d}{dx}}\arctan x&{}={\frac {1}{1+x^{2}}}\\{\frac {d}{dx}}\operatorname {arccot} x&{}={\frac {-1}{1+x^{2}}}\\{\frac {d}{dx}}\operatorname {arcsec} x&{}={\frac {1}{x\,{\sqrt {x^{2}-1}}}}\\{\frac {d}{dx}}\operatorname {arccsc} x&{}={\frac {-1}{x\,{\sqrt {x^{2}-1}}}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a4d8ba92d7eeae23301b272cdf0a55a1298562c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -19.005ex; width:27.065ex; height:39.176ex;" alt="{\displaystyle {\begin{aligned}{\frac {d}{dx}}\arcsin x&{}={\frac {1}{\sqrt {1-x^{2}}}}\\{\frac {d}{dx}}\arccos x&{}={\frac {-1}{\sqrt {1-x^{2}}}}\\{\frac {d}{dx}}\arctan x&{}={\frac {1}{1+x^{2}}}\\{\frac {d}{dx}}\operatorname {arccot} x&{}={\frac {-1}{1+x^{2}}}\\{\frac {d}{dx}}\operatorname {arcsec} x&{}={\frac {1}{x\,{\sqrt {x^{2}-1}}}}\\{\frac {d}{dx}}\operatorname {arccsc} x&{}={\frac {-1}{x\,{\sqrt {x^{2}-1}}}}\end{aligned}}}"></span></dd></dl> <p><i>x</i> in yalnızca reel değerleri şöyledir: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {d}{dx}}\operatorname {arcsec} x&{}={\frac {1}{|x|\,{\sqrt {x^{2}-1}}}};\qquad |x|>1\\{\frac {d}{dx}}\operatorname {arccsc} x&{}={\frac {-1}{|x|\,{\sqrt {x^{2}-1}}}};\qquad |x|>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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>arcsec</mi> <mo>⁡</mo> <mi>x</mi> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thinmathspace" /> <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> </mfrac> </mrow> <mo>;</mo> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>></mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>arccsc</mi> <mo>⁡</mo> <mi>x</mi> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thinmathspace" /> <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> </mfrac> </mrow> <mo>;</mo> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>></mo> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {d}{dx}}\operatorname {arcsec} x&{}={\frac {1}{|x|\,{\sqrt {x^{2}-1}}}};\qquad |x|>1\\{\frac {d}{dx}}\operatorname {arccsc} x&{}={\frac {-1}{|x|\,{\sqrt {x^{2}-1}}}};\qquad |x|>1\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32758476b278cc56782c7e2bdb2c4cb08d122762" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.171ex; width:40.544ex; height:13.509ex;" alt="{\displaystyle {\begin{aligned}{\frac {d}{dx}}\operatorname {arcsec} x&{}={\frac {1}{|x|\,{\sqrt {x^{2}-1}}}};\qquad |x|>1\\{\frac {d}{dx}}\operatorname {arccsc} x&{}={\frac {-1}{|x|\,{\sqrt {x^{2}-1}}}};\qquad |x|>1\end{aligned}}}"></span></dd></dl> <p>Örnek bir türev: eğer <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =\arcsin x\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mi>arcsin</mi> <mo>⁡</mo> <mi>x</mi> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =\arcsin x\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b5448f0cadfc3cfa394d5d0f2bffb20c7f641e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.271ex; width:11.752ex; height:2.176ex;" alt="{\displaystyle \theta =\arcsin x\!}"></span> ise; </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d\arcsin x}{dx}}={\frac {d\theta }{d\sin \theta }}={\frac {d\theta }{\cos \theta d\theta }}={\frac {1}{\cos \theta }}={\frac {1}{\sqrt {1-\sin ^{2}\theta }}}={\frac {1}{\sqrt {1-x^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>arcsin</mi> <mo>⁡</mo> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>θ</mi> </mrow> <mrow> <mi>d</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>θ</mi> </mrow> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mi>d</mi> <mi>θ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> <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> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d\arcsin x}{dx}}={\frac {d\theta }{d\sin \theta }}={\frac {d\theta }{\cos \theta d\theta }}={\frac {1}{\cos \theta }}={\frac {1}{\sqrt {1-\sin ^{2}\theta }}}={\frac {1}{\sqrt {1-x^{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ea8abe9d2a9925be64df5e06e19533c2cfd1942" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:67.635ex; height:6.676ex;" alt="{\displaystyle {\frac {d\arcsin x}{dx}}={\frac {d\theta }{d\sin \theta }}={\frac {d\theta }{\cos \theta d\theta }}={\frac {1}{\cos \theta }}={\frac {1}{\sqrt {1-\sin ^{2}\theta }}}={\frac {1}{\sqrt {1-x^{2}}}}}"></span> olur.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Belirli_integral_olarak_ifadesi">Belirli integral olarak ifadesi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ters_trigonometrik_fonksiyonlar&veaction=edit&section=5" title="Değiştirilen bölüm: Belirli integral olarak ifadesi" class="mw-editsection-visualeditor"><span>değiştir</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Ters_trigonometrik_fonksiyonlar&action=edit&section=5" title="Bölümün kaynak kodunu değiştir: Belirli integral olarak ifadesi"><span>kaynağı değiştir</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Bir noktadaki türevin integrali ve sabit değeri, ters trigonometrik fonksiyonların belirli integrallarinin ifadesini verir: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\arcsin x&{}=\int _{0}^{x}{\frac {1}{\sqrt {1-z^{2}}}}\,dz,\qquad |x|\leq 1\\\arccos x&{}=\int _{x}^{1}{\frac {1}{\sqrt {1-z^{2}}}}\,dz,\qquad |x|\leq 1\\\arctan x&{}=\int _{0}^{x}{\frac {1}{z^{2}+1}}\,dz,\\\operatorname {arccot} x&{}=\int _{x}^{\infty }{\frac {1}{z^{2}+1}}\,dz,\\\operatorname {arcsec} x&{}=\int _{1}^{x}{\frac {1}{z{\sqrt {z^{2}-1}}}}\,dz,\qquad x\geq 1\\\operatorname {arcsec} x&{}=\pi +\int _{x}^{-1}{\frac {1}{z{\sqrt {z^{2}-1}}}}\,dz,\qquad x\leq -1\\\operatorname {arccsc} x&{}=\int _{x}^{\infty }{\frac {1}{z{\sqrt {z^{2}-1}}}}\,dz,\qquad x\geq 1\\\operatorname {arccsc} x&{}=\int _{-\infty }^{x}{\frac {1}{z{\sqrt {z^{2}-1}}}}\,dz,\qquad x\leq -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> <mi>arcsin</mi> <mo>⁡</mo> <mi>x</mi> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>z</mi> <mo>,</mo> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mi>arccos</mi> <mo>⁡</mo> <mi>x</mi> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>z</mi> <mo>,</mo> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mi>arctan</mi> <mo>⁡</mo> <mi>x</mi> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>z</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>arccot</mi> <mo>⁡</mo> <mi>x</mi> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>z</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>arcsec</mi> <mo>⁡</mo> <mi>x</mi> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>z</mi> <mo>,</mo> <mspace width="2em" /> <mi>x</mi> <mo>≥</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mi>arcsec</mi> <mo>⁡</mo> <mi>x</mi> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mi>π</mi> <mo>+</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>z</mi> <mo>,</mo> <mspace width="2em" /> <mi>x</mi> <mo>≤</mo> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mi>arccsc</mi> <mo>⁡</mo> <mi>x</mi> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>z</mi> <mo>,</mo> <mspace width="2em" /> <mi>x</mi> <mo>≥</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mi>arccsc</mi> <mo>⁡</mo> <mi>x</mi> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>z</mi> <mo>,</mo> <mspace width="2em" /> <mi>x</mi> <mo>≤</mo> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\arcsin x&{}=\int _{0}^{x}{\frac {1}{\sqrt {1-z^{2}}}}\,dz,\qquad |x|\leq 1\\\arccos x&{}=\int _{x}^{1}{\frac {1}{\sqrt {1-z^{2}}}}\,dz,\qquad |x|\leq 1\\\arctan x&{}=\int _{0}^{x}{\frac {1}{z^{2}+1}}\,dz,\\\operatorname {arccot} x&{}=\int _{x}^{\infty }{\frac {1}{z^{2}+1}}\,dz,\\\operatorname {arcsec} x&{}=\int _{1}^{x}{\frac {1}{z{\sqrt {z^{2}-1}}}}\,dz,\qquad x\geq 1\\\operatorname {arcsec} x&{}=\pi +\int _{x}^{-1}{\frac {1}{z{\sqrt {z^{2}-1}}}}\,dz,\qquad x\leq -1\\\operatorname {arccsc} x&{}=\int _{x}^{\infty }{\frac {1}{z{\sqrt {z^{2}-1}}}}\,dz,\qquad x\geq 1\\\operatorname {arccsc} x&{}=\int _{-\infty }^{x}{\frac {1}{z{\sqrt {z^{2}-1}}}}\,dz,\qquad x\leq -1\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f48035e73e4bd013d96029c96d295f4f9fa5160a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -26.505ex; margin-top: -0.21ex; width:47.554ex; height:54.176ex;" alt="{\displaystyle {\begin{aligned}\arcsin x&{}=\int _{0}^{x}{\frac {1}{\sqrt {1-z^{2}}}}\,dz,\qquad |x|\leq 1\\\arccos x&{}=\int _{x}^{1}{\frac {1}{\sqrt {1-z^{2}}}}\,dz,\qquad |x|\leq 1\\\arctan x&{}=\int _{0}^{x}{\frac {1}{z^{2}+1}}\,dz,\\\operatorname {arccot} x&{}=\int _{x}^{\infty }{\frac {1}{z^{2}+1}}\,dz,\\\operatorname {arcsec} x&{}=\int _{1}^{x}{\frac {1}{z{\sqrt {z^{2}-1}}}}\,dz,\qquad x\geq 1\\\operatorname {arcsec} x&{}=\pi +\int _{x}^{-1}{\frac {1}{z{\sqrt {z^{2}-1}}}}\,dz,\qquad x\leq -1\\\operatorname {arccsc} x&{}=\int _{x}^{\infty }{\frac {1}{z{\sqrt {z^{2}-1}}}}\,dz,\qquad x\geq 1\\\operatorname {arccsc} x&{}=\int _{-\infty }^{x}{\frac {1}{z{\sqrt {z^{2}-1}}}}\,dz,\qquad x\leq -1\end{aligned}}}"></span></dd></dl> <p><i>x</i> 1'e eşit olduğunda, integraller tanım kümesini <a href="/wiki/Belirsiz_integral" title="Belirsiz integral">belirsiz integral</a> ile kısıtlar, fakat yine de iyi tanımlıdırlar. </p> <div class="mw-heading mw-heading2"><h2 id="Sonsuz_seriler">Sonsuz seriler</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ters_trigonometrik_fonksiyonlar&veaction=edit&section=6" title="Değiştirilen bölüm: Sonsuz seriler" class="mw-editsection-visualeditor"><span>değiştir</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Ters_trigonometrik_fonksiyonlar&action=edit&section=6" title="Bölümün kaynak kodunu değiştir: Sonsuz seriler"><span>kaynağı değiştir</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Sinüs ve kosinüs gibi fonksiyonların ters trigonometrik fonksiyonları sonsuz <a href="/wiki/Seri" title="Seri">seriler</a> kullanılarak hesaplanabilir, şöyle ki: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arcsin z=z+\left({\frac {1}{2}}\right){\frac {z^{3}}{3}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {z^{5}}{5}}+\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right){\frac {z^{7}}{7}}+\cdots \ =\sum _{n=0}^{\infty }{\frac {{\binom {2n}{n}}z^{2n+1}}{4^{n}(2n+1)}};\qquad |z|\leq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arcsin</mi> <mo>⁡</mo> <mi>z</mi> <mo>=</mo> <mi>z</mi> <mo>+</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>⋅</mo> <mn>3</mn> </mrow> <mrow> <mn>2</mn> <mo>⋅</mo> <mn>4</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mn>5</mn> </mfrac> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>⋅</mo> <mn>3</mn> <mo>⋅</mo> <mn>5</mn> </mrow> <mrow> <mn>2</mn> <mo>⋅</mo> <mn>4</mn> <mo>⋅</mo> <mn>6</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mn>7</mn> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mtext> </mtext> <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> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mn>2</mn> <mi>n</mi> </mrow> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>;</mo> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arcsin z=z+\left({\frac {1}{2}}\right){\frac {z^{3}}{3}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {z^{5}}{5}}+\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right){\frac {z^{7}}{7}}+\cdots \ =\sum _{n=0}^{\infty }{\frac {{\binom {2n}{n}}z^{2n+1}}{4^{n}(2n+1)}};\qquad |z|\leq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3aca52fd0581fb7c17e8189c7e1529878004320" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:92.313ex; height:7.343ex;" alt="{\displaystyle \arcsin z=z+\left({\frac {1}{2}}\right){\frac {z^{3}}{3}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {z^{5}}{5}}+\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right){\frac {z^{7}}{7}}+\cdots \ =\sum _{n=0}^{\infty }{\frac {{\binom {2n}{n}}z^{2n+1}}{4^{n}(2n+1)}};\qquad |z|\leq 1}"></span></dd></dl> <p><br /> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arccos z={\frac {\pi }{2}}-\arcsin z={\frac {\pi }{2}}-\left(z+\left({\frac {1}{2}}\right){\frac {z^{3}}{3}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {z^{5}}{5}}+\cdots \ \right)={\frac {\pi }{2}}-\sum _{n=0}^{\infty }{\frac {{\binom {2n}{n}}z^{2n+1}}{4^{n}(2n+1)}};\qquad |z|\leq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccos</mi> <mo>⁡</mo> <mi>z</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>arcsin</mi> <mo>⁡</mo> <mi>z</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mi>z</mi> <mo>+</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>⋅</mo> <mn>3</mn> </mrow> <mrow> <mn>2</mn> <mo>⋅</mo> <mn>4</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mn>5</mn> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mtext> </mtext> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <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> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mn>2</mn> <mi>n</mi> </mrow> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>;</mo> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arccos z={\frac {\pi }{2}}-\arcsin z={\frac {\pi }{2}}-\left(z+\left({\frac {1}{2}}\right){\frac {z^{3}}{3}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {z^{5}}{5}}+\cdots \ \right)={\frac {\pi }{2}}-\sum _{n=0}^{\infty }{\frac {{\binom {2n}{n}}z^{2n+1}}{4^{n}(2n+1)}};\qquad |z|\leq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/464c95d33ab2ab424ff08bf40d0196a04abf0185" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:104.24ex; height:7.343ex;" alt="{\displaystyle \arccos z={\frac {\pi }{2}}-\arcsin z={\frac {\pi }{2}}-\left(z+\left({\frac {1}{2}}\right){\frac {z^{3}}{3}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {z^{5}}{5}}+\cdots \ \right)={\frac {\pi }{2}}-\sum _{n=0}^{\infty }{\frac {{\binom {2n}{n}}z^{2n+1}}{4^{n}(2n+1)}};\qquad |z|\leq 1}"></span></dd></dl> <p><br /> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arctan z=z-{\frac {z^{3}}{3}}+{\frac {z^{5}}{5}}-{\frac {z^{7}}{7}}+\cdots \ =\sum _{n=0}^{\infty }{\frac {(-1)^{n}z^{2n+1}}{2n+1}};\qquad |z|\leq 1\qquad z\neq i,-i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arctan</mi> <mo>⁡</mo> <mi>z</mi> <mo>=</mo> <mi>z</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mn>5</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mn>7</mn> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mtext> </mtext> <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>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>;</mo> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mn>1</mn> <mspace width="2em" /> <mi>z</mi> <mo>≠</mo> <mi>i</mi> <mo>,</mo> <mo>−</mo> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arctan z=z-{\frac {z^{3}}{3}}+{\frac {z^{5}}{5}}-{\frac {z^{7}}{7}}+\cdots \ =\sum _{n=0}^{\infty }{\frac {(-1)^{n}z^{2n+1}}{2n+1}};\qquad |z|\leq 1\qquad z\neq i,-i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9f2296588de015d857fc5e1fe21bfd715fd40a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:80.63ex; height:7.009ex;" alt="{\displaystyle \arctan z=z-{\frac {z^{3}}{3}}+{\frac {z^{5}}{5}}-{\frac {z^{7}}{7}}+\cdots \ =\sum _{n=0}^{\infty }{\frac {(-1)^{n}z^{2n+1}}{2n+1}};\qquad |z|\leq 1\qquad z\neq i,-i}"></span></dd></dl> <p><br /> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {arccot} z={\frac {\pi }{2}}-\arctan z\ ={\frac {\pi }{2}}-\left(z-{\frac {z^{3}}{3}}+{\frac {z^{5}}{5}}-{\frac {z^{7}}{7}}+\cdots \ \right)={\frac {\pi }{2}}-\sum _{n=0}^{\infty }{\frac {(-1)^{n}z^{2n+1}}{2n+1}};\qquad |z|\leq 1\qquad z\neq i,-i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccot</mi> <mo>⁡</mo> <mi>z</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>arctan</mi> <mo>⁡</mo> <mi>z</mi> <mtext> </mtext> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mi>z</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mn>5</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mn>7</mn> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mtext> </mtext> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <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>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>;</mo> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mn>1</mn> <mspace width="2em" /> <mi>z</mi> <mo>≠</mo> <mi>i</mi> <mo>,</mo> <mo>−</mo> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {arccot} z={\frac {\pi }{2}}-\arctan z\ ={\frac {\pi }{2}}-\left(z-{\frac {z^{3}}{3}}+{\frac {z^{5}}{5}}-{\frac {z^{7}}{7}}+\cdots \ \right)={\frac {\pi }{2}}-\sum _{n=0}^{\infty }{\frac {(-1)^{n}z^{2n+1}}{2n+1}};\qquad |z|\leq 1\qquad z\neq i,-i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/079b30dd3191904636858a9830f22d1334f84cbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:110.437ex; height:7.009ex;" alt="{\displaystyle \operatorname {arccot} z={\frac {\pi }{2}}-\arctan z\ ={\frac {\pi }{2}}-\left(z-{\frac {z^{3}}{3}}+{\frac {z^{5}}{5}}-{\frac {z^{7}}{7}}+\cdots \ \right)={\frac {\pi }{2}}-\sum _{n=0}^{\infty }{\frac {(-1)^{n}z^{2n+1}}{2n+1}};\qquad |z|\leq 1\qquad z\neq i,-i}"></span></dd></dl> <p><br /> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {arcsec} z=\arccos {(1/z)}={\frac {\pi }{2}}-\left(z^{-1}+\left({\frac {1}{2}}\right){\frac {z^{-3}}{3}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {z^{-5}}{5}}+\cdots \ \right)={\frac {\pi }{2}}-\sum _{n=0}^{\infty }{\frac {{\binom {2n}{n}}z^{-(2n+1)}}{4^{n}(2n+1)}};\qquad |z|\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arcsec</mi> <mo>⁡</mo> <mi>z</mi> <mo>=</mo> <mi>arccos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> </msup> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>⋅</mo> <mn>3</mn> </mrow> <mrow> <mn>2</mn> <mo>⋅</mo> <mn>4</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>5</mn> </mrow> </msup> <mn>5</mn> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mtext> </mtext> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <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> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mn>2</mn> <mi>n</mi> </mrow> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> <mrow> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>;</mo> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {arcsec} z=\arccos {(1/z)}={\frac {\pi }{2}}-\left(z^{-1}+\left({\frac {1}{2}}\right){\frac {z^{-3}}{3}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {z^{-5}}{5}}+\cdots \ \right)={\frac {\pi }{2}}-\sum _{n=0}^{\infty }{\frac {{\binom {2n}{n}}z^{-(2n+1)}}{4^{n}(2n+1)}};\qquad |z|\geq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad09ebc982558c772674e535a4d51451f3552083" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:109.36ex; height:7.343ex;" alt="{\displaystyle \operatorname {arcsec} z=\arccos {(1/z)}={\frac {\pi }{2}}-\left(z^{-1}+\left({\frac {1}{2}}\right){\frac {z^{-3}}{3}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {z^{-5}}{5}}+\cdots \ \right)={\frac {\pi }{2}}-\sum _{n=0}^{\infty }{\frac {{\binom {2n}{n}}z^{-(2n+1)}}{4^{n}(2n+1)}};\qquad |z|\geq 1}"></span></dd></dl> <p><br /> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {arccsc} z=\arcsin {(1/z)}=z^{-1}+\left({\frac {1}{2}}\right){\frac {z^{-3}}{3}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {z^{-5}}{5}}+\cdots \ =\sum _{n=0}^{\infty }{\frac {{\binom {2n}{n}}z^{-(2n+1)}}{4^{n}(2n+1)}};\qquad |z|\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccsc</mi> <mo>⁡</mo> <mi>z</mi> <mo>=</mo> <mi>arcsin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> </msup> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>⋅</mo> <mn>3</mn> </mrow> <mrow> <mn>2</mn> <mo>⋅</mo> <mn>4</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>5</mn> </mrow> </msup> <mn>5</mn> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mtext> </mtext> <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> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mn>2</mn> <mi>n</mi> </mrow> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> <mrow> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>;</mo> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {arccsc} z=\arcsin {(1/z)}=z^{-1}+\left({\frac {1}{2}}\right){\frac {z^{-3}}{3}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {z^{-5}}{5}}+\cdots \ =\sum _{n=0}^{\infty }{\frac {{\binom {2n}{n}}z^{-(2n+1)}}{4^{n}(2n+1)}};\qquad |z|\geq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6a05bf179bbcbbd731729eb8e754a7282affa4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:95.666ex; height:7.343ex;" alt="{\displaystyle \operatorname {arccsc} z=\arcsin {(1/z)}=z^{-1}+\left({\frac {1}{2}}\right){\frac {z^{-3}}{3}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {z^{-5}}{5}}+\cdots \ =\sum _{n=0}^{\infty }{\frac {{\binom {2n}{n}}z^{-(2n+1)}}{4^{n}(2n+1)}};\qquad |z|\geq 1}"></span></dd></dl> <p><br /> <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a>, arctanjant için daha kullanışlı bir seri buldu: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arctan z={\frac {z}{1+z^{2}}}\sum _{n=0}^{\infty }\prod _{k=1}^{n}{\frac {2kz^{2}}{(2k+1)(1+z^{2})}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arctan</mi> <mo>⁡</mo> <mi>z</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>z</mi> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <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> <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 class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>k</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arctan z={\frac {z}{1+z^{2}}}\sum _{n=0}^{\infty }\prod _{k=1}^{n}{\frac {2kz^{2}}{(2k+1)(1+z^{2})}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6af26b72e8f91c5b056d084e9032f80ba6ec68ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:43.135ex; height:6.843ex;" alt="{\displaystyle \arctan z={\frac {z}{1+z^{2}}}\sum _{n=0}^{\infty }\prod _{k=1}^{n}{\frac {2kz^{2}}{(2k+1)(1+z^{2})}}.}"></span></dd></dl> <p>(<i>n</i> = 0 için toplamdaki terimin <b>boş çarpım</b> (ki bu 1'dir) olduğuna dikkat edin.) </p><p><br /> Alternatif olarak bu şöyle de ifade edilebilir; </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arctan z=\sum _{n=0}^{\infty }{\frac {2^{\,2n}\,(n!)^{2}}{\left(2n+1\right)!}}\;{\frac {z^{\,2n+1}}{\left(1+z^{2}\right)^{n+1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arctan</mi> <mo>⁡</mo> <mi>z</mi> <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> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="thinmathspace" /> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>n</mi> <mo>!</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>!</mo> </mrow> </mfrac> </mrow> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mspace width="thinmathspace" /> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <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 \arctan z=\sum _{n=0}^{\infty }{\frac {2^{\,2n}\,(n!)^{2}}{\left(2n+1\right)!}}\;{\frac {z^{\,2n+1}}{\left(1+z^{2}\right)^{n+1}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adf1d57b5cb7bb7b3c66bb8c3edf32c788e52ec2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:37.391ex; height:7.176ex;" alt="{\displaystyle \arctan z=\sum _{n=0}^{\infty }{\frac {2^{\,2n}\,(n!)^{2}}{\left(2n+1\right)!}}\;{\frac {z^{\,2n+1}}{\left(1+z^{2}\right)^{n+1}}}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Logaritmik_biçimler"><span id="Logaritmik_bi.C3.A7imler"></span>Logaritmik biçimler</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ters_trigonometrik_fonksiyonlar&veaction=edit&section=7" title="Değiştirilen bölüm: Logaritmik biçimler" class="mw-editsection-visualeditor"><span>değiştir</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Ters_trigonometrik_fonksiyonlar&action=edit&section=7" title="Bölümün kaynak kodunu değiştir: Logaritmik biçimler"><span>kaynağı değiştir</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Bu logaritmik biçimler <a href="/wiki/Karma%C5%9F%C4%B1k_d%C3%BCzlem" title="Karmaşık düzlem">karmaşık düzlemde</a> bulunur. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\arcsin x&{}=-i\,\ln \left(i\,x+{\sqrt {1-x^{2}}}\right)&{}=\operatorname {arccsc} {\frac {1}{x}}\\[10pt]\arccos x&{}=-i\,\ln \left(x+i\,{\sqrt {1-x^{2}}}\right)={\frac {\pi }{2}}\,+i\ln \left(i\,x+{\sqrt {1-x^{2}}}\right)={\frac {\pi }{2}}-\arcsin x&{}=\operatorname {arcsec} {\frac {1}{x}}\\[10pt]\arctan x&{}={\tfrac {1}{2}}i\left(\ln \left(1-i\,x\right)-\ln \left(1+i\,x\right)\right)&{}=\operatorname {arccot} {\frac {1}{x}}\\[10pt]\operatorname {arccot} x&{}={\tfrac {1}{2}}i\left(\ln \left(1-{\frac {i}{x}}\right)-\ln \left(1+{\frac {i}{x}}\right)\right)&{}=\arctan {\frac {1}{x}}\\[10pt]\operatorname {arcsec} x&{}=-i\,\ln \left(i\,{\sqrt {1-{\frac {1}{x^{2}}}}}+{\frac {1}{x}}\right)=i\,\ln \left({\sqrt {1-{\frac {1}{x^{2}}}}}+{\frac {i}{x}}\right)+{\frac {\pi }{2}}={\frac {\pi }{2}}-\operatorname {arccsc} x&{}=\arccos {\frac {1}{x}}\\[10pt]\operatorname {arccsc} x&{}=-i\,\ln \left({\sqrt {1-{\frac {1}{x^{2}}}}}+{\frac {i}{x}}\right)&{}=\arcsin {\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="1.3em 1.3em 1.3em 1.3em 1.3em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>arcsin</mi> <mo>⁡</mo> <mi>x</mi> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mo>−</mo> <mi>i</mi> <mspace width="thinmathspace" /> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mspace width="thinmathspace" /> <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> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mi>arccsc</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>arccos</mi> <mo>⁡</mo> <mi>x</mi> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mo>−</mo> <mi>i</mi> <mspace width="thinmathspace" /> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mspace width="thinmathspace" /> <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> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>+</mo> <mi>i</mi> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mspace width="thinmathspace" /> <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> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>arcsin</mi> <mo>⁡</mo> <mi>x</mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mi>arcsec</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>arctan</mi> <mo>⁡</mo> <mi>x</mi> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>i</mi> <mrow> <mo>(</mo> <mrow> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mi>i</mi> <mspace width="thinmathspace" /> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>i</mi> <mspace width="thinmathspace" /> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mi>arccot</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>arccot</mi> <mo>⁡</mo> <mi>x</mi> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>i</mi> <mrow> <mo>(</mo> <mrow> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mi>x</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mi>x</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>arcsec</mi> <mo>⁡</mo> <mi>x</mi> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mo>−</mo> <mi>i</mi> <mspace width="thinmathspace" /> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mspace width="thinmathspace" /> <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> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>i</mi> <mspace width="thinmathspace" /> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <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> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mi>x</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>arccsc</mi> <mo>⁡</mo> <mi>x</mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mi>arccos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>arccsc</mi> <mo>⁡</mo> <mi>x</mi> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mo>−</mo> <mi>i</mi> <mspace width="thinmathspace" /> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <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> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mi>x</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mi>arcsin</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}\arcsin x&{}=-i\,\ln \left(i\,x+{\sqrt {1-x^{2}}}\right)&{}=\operatorname {arccsc} {\frac {1}{x}}\\[10pt]\arccos x&{}=-i\,\ln \left(x+i\,{\sqrt {1-x^{2}}}\right)={\frac {\pi }{2}}\,+i\ln \left(i\,x+{\sqrt {1-x^{2}}}\right)={\frac {\pi }{2}}-\arcsin x&{}=\operatorname {arcsec} {\frac {1}{x}}\\[10pt]\arctan x&{}={\tfrac {1}{2}}i\left(\ln \left(1-i\,x\right)-\ln \left(1+i\,x\right)\right)&{}=\operatorname {arccot} {\frac {1}{x}}\\[10pt]\operatorname {arccot} x&{}={\tfrac {1}{2}}i\left(\ln \left(1-{\frac {i}{x}}\right)-\ln \left(1+{\frac {i}{x}}\right)\right)&{}=\arctan {\frac {1}{x}}\\[10pt]\operatorname {arcsec} x&{}=-i\,\ln \left(i\,{\sqrt {1-{\frac {1}{x^{2}}}}}+{\frac {1}{x}}\right)=i\,\ln \left({\sqrt {1-{\frac {1}{x^{2}}}}}+{\frac {i}{x}}\right)+{\frac {\pi }{2}}={\frac {\pi }{2}}-\operatorname {arccsc} x&{}=\arccos {\frac {1}{x}}\\[10pt]\operatorname {arccsc} x&{}=-i\,\ln \left({\sqrt {1-{\frac {1}{x^{2}}}}}+{\frac {i}{x}}\right)&{}=\arcsin {\frac {1}{x}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20f53c0f477f8830d52de7a3c008b4bac7fa8adf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -22.838ex; width:98.792ex; height:46.843ex;" alt="{\displaystyle {\begin{aligned}\arcsin x&{}=-i\,\ln \left(i\,x+{\sqrt {1-x^{2}}}\right)&{}=\operatorname {arccsc} {\frac {1}{x}}\\[10pt]\arccos x&{}=-i\,\ln \left(x+i\,{\sqrt {1-x^{2}}}\right)={\frac {\pi }{2}}\,+i\ln \left(i\,x+{\sqrt {1-x^{2}}}\right)={\frac {\pi }{2}}-\arcsin x&{}=\operatorname {arcsec} {\frac {1}{x}}\\[10pt]\arctan x&{}={\tfrac {1}{2}}i\left(\ln \left(1-i\,x\right)-\ln \left(1+i\,x\right)\right)&{}=\operatorname {arccot} {\frac {1}{x}}\\[10pt]\operatorname {arccot} x&{}={\tfrac {1}{2}}i\left(\ln \left(1-{\frac {i}{x}}\right)-\ln \left(1+{\frac {i}{x}}\right)\right)&{}=\arctan {\frac {1}{x}}\\[10pt]\operatorname {arcsec} x&{}=-i\,\ln \left(i\,{\sqrt {1-{\frac {1}{x^{2}}}}}+{\frac {1}{x}}\right)=i\,\ln \left({\sqrt {1-{\frac {1}{x^{2}}}}}+{\frac {i}{x}}\right)+{\frac {\pi }{2}}={\frac {\pi }{2}}-\operatorname {arccsc} x&{}=\arccos {\frac {1}{x}}\\[10pt]\operatorname {arccsc} x&{}=-i\,\ln \left({\sqrt {1-{\frac {1}{x^{2}}}}}+{\frac {i}{x}}\right)&{}=\arcsin {\frac {1}{x}}\end{aligned}}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Örnek_ispat"><span id=".C3.96rnek_ispat"></span>Örnek ispat</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ters_trigonometrik_fonksiyonlar&veaction=edit&section=8" title="Değiştirilen bölüm: Örnek ispat" class="mw-editsection-visualeditor"><span>değiştir</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Ters_trigonometrik_fonksiyonlar&action=edit&section=8" title="Bölümün kaynak kodunu değiştir: Örnek ispat"><span>kaynağı değiştir</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =\arcsin x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mi>arcsin</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =\arcsin x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af6d3cb79a9989a50c0b4a64a5546df04ed6b15c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.868ex; height:2.176ex;" alt="{\displaystyle \theta =\arcsin x}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(\theta )=\sin(\arcsin x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> <mi>arcsin</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(\theta )=\sin(\arcsin x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc0cc02c1de95e46df106f36a7d699c96e3565a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.198ex; height:2.843ex;" alt="{\displaystyle \sin(\theta )=\sin(\arcsin x)}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(\theta )=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(\theta )=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a87c9de9bd12c386921b827631d03bd8f6aa90dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.184ex; height:2.843ex;" alt="{\displaystyle \sin(\theta )=x}"></span></dd></dl> <p><a href="/wiki/Trigonometrik_fonksiyonlar#Üstel_fonksiyonlar_ve_karmaşık_sayılarla_İlişkisi" title="Trigonometrik fonksiyonlar">Sinüsün üstel biçimi</a> şöyledir; </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {e^{i\phi }-e^{-i\phi }}{2i}}=\sin(\phi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ϕ</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>ϕ</mi> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>i</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {e^{i\phi }-e^{-i\phi }}{2i}}=\sin(\phi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5dc3ca04a97d0689b0f80cba9766c221636b864" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.83ex; height:5.676ex;" alt="{\displaystyle {\frac {e^{i\phi }-e^{-i\phi }}{2i}}=\sin(\phi )}"></span></dd></dl> <p>Böylece ifade şöyle olur: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {e^{i\theta }-e^{-i\theta }}{2i}}=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {e^{i\theta }-e^{-i\theta }}{2i}}=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4759dcd9c801a5153bb5fe74abaee1e7d8c4ea81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.692ex; height:5.676ex;" alt="{\displaystyle {\frac {e^{i\theta }-e^{-i\theta }}{2i}}=x}"></span></dd></dl> <p>Burada aşağıdaki gibi bir <a href="/wiki/De%C4%9Fi%C5%9Fken_de%C4%9Fi%C5%9Ftirme" title="Değişken değiştirme">değişken değiştirme</a> uygulanırsa; </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=e^{i\,\theta }.\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mspace width="thinmathspace" /> <mi>θ</mi> </mrow> </msup> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=e^{i\,\theta }.\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4003faeae7d1b88ba43144284f6a1c281f26721e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.385ex; height:2.676ex;" alt="{\displaystyle k=e^{i\,\theta }.\,}"></span></dd></dl> <p>Eşitlik şöyle olur; </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {k-{\frac {1}{k}}}{2i}}=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>k</mi> </mfrac> </mrow> </mrow> <mrow> <mn>2</mn> <mi>i</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {k-{\frac {1}{k}}}{2i}}=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b74692851f22563117292e5f7bd17b86ca2120dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.009ex; height:6.509ex;" alt="{\displaystyle {\frac {k-{\frac {1}{k}}}{2i}}=x}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {k-{\frac {1}{k}}}=2ix}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>k</mi> </mfrac> </mrow> </mrow> <mo>=</mo> <mn>2</mn> <mi>i</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {k-{\frac {1}{k}}}=2ix}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f83355801f8aa5b4a32727cd8fdf4c7f9dbd2b7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.492ex; height:5.343ex;" alt="{\displaystyle {k-{\frac {1}{k}}}=2ix}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {k-2ix-{\frac {1}{k}}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>2</mn> <mi>i</mi> <mi>x</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>k</mi> </mfrac> </mrow> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {k-2ix-{\frac {1}{k}}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9295a87e6e08b92d234b5ec48d137dc5a7b559d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.495ex; height:5.343ex;" alt="{\displaystyle {k-2ix-{\frac {1}{k}}}=0}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k^{2}-2\,i\,k\,x-1\,=\,0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mspace width="thinmathspace" /> <mi>i</mi> <mspace width="thinmathspace" /> <mi>k</mi> <mspace width="thinmathspace" /> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k^{2}-2\,i\,k\,x-1\,=\,0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/919c8f0f2da792dc3f431d194028811f1e42d583" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:19.811ex; height:2.843ex;" alt="{\displaystyle k^{2}-2\,i\,k\,x-1\,=\,0}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=ix\pm {\sqrt {1-x^{2}}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <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> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=ix\pm {\sqrt {1-x^{2}}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ad461d85b385e69246690a28c728a33651d9f8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.38ex; height:3.509ex;" alt="{\displaystyle k=ix\pm {\sqrt {1-x^{2}}}\,}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{i\theta }=ix\pm {\sqrt {1-x^{2}}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> <mo>=</mo> <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> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{i\theta }=ix\pm {\sqrt {1-x^{2}}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1a4fba728523390f731a945ebaf55efc71e64f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.823ex; height:3.509ex;" alt="{\displaystyle e^{i\theta }=ix\pm {\sqrt {1-x^{2}}}\,}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\theta =\ln \left(ix\pm {\sqrt {1-x^{2}}}\right)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mi>θ</mi> <mo>=</mo> <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> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\theta =\ln \left(ix\pm {\sqrt {1-x^{2}}}\right)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/771ed41f6b433a5d1cc61ef253a69620c4845fc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.776ex; height:4.843ex;" alt="{\displaystyle i\theta =\ln \left(ix\pm {\sqrt {1-x^{2}}}\right)\,}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =-i\ln \left(ix\pm {\sqrt {1-x^{2}}}\right)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mo>−</mo> <mi>i</mi> <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> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =-i\ln \left(ix\pm {\sqrt {1-x^{2}}}\right)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51c2601ae46bfe704b5ba7d007c81b400ded28bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.972ex; height:4.843ex;" alt="{\displaystyle \theta =-i\ln \left(ix\pm {\sqrt {1-x^{2}}}\right)\,}"></span></dd></dl> <p>(yukarıdaki eşitliğin pozitif kısmı alınırsa) </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =\arcsin x=-i\ln \left(ix+{\sqrt {1-x^{2}}}\right)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mi>arcsin</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mi>i</mi> <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> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =\arcsin x=-i\ln \left(ix+{\sqrt {1-x^{2}}}\right)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17e860d10f219add4b95fe5fd62b5af20cc7d2c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:36.749ex; height:4.843ex;" alt="{\displaystyle \theta =\arcsin x=-i\ln \left(ix+{\sqrt {1-x^{2}}}\right)\,}"></span></dd></dl> <table style="text-align:center"> <caption><b><a href="/wiki/Karma%C5%9F%C4%B1k_d%C3%BCzlem" title="Karmaşık düzlem">Karmaşık düzlemdeki</a> ters trigonometrik fonksiyonlar</b> </caption> <tbody><tr> <td><figure class="mw-halign-none" typeof="mw:File"><a href="/wiki/Dosya:Complex_arcsin.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/be/Complex_arcsin.jpg/140px-Complex_arcsin.jpg" decoding="async" width="140" height="140" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/be/Complex_arcsin.jpg/210px-Complex_arcsin.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/be/Complex_arcsin.jpg/280px-Complex_arcsin.jpg 2x" data-file-width="820" data-file-height="820" /></a><figcaption></figcaption></figure> </td> <td><figure class="mw-halign-none" typeof="mw:File"><a href="/wiki/Dosya:Complex_arccos.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Complex_arccos.jpg/141px-Complex_arccos.jpg" decoding="async" width="141" height="140" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Complex_arccos.jpg/211px-Complex_arccos.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Complex_arccos.jpg/282px-Complex_arccos.jpg 2x" data-file-width="836" data-file-height="831" /></a><figcaption></figcaption></figure> </td> <td><figure class="mw-halign-none" typeof="mw:File"><a href="/wiki/Dosya:Complex_arctan.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Complex_arctan.jpg/141px-Complex_arctan.jpg" decoding="async" width="141" height="140" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Complex_arctan.jpg/211px-Complex_arctan.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Complex_arctan.jpg/281px-Complex_arctan.jpg 2x" data-file-width="831" data-file-height="827" /></a><figcaption></figcaption></figure> </td> <td><figure class="mw-halign-none" typeof="mw:File"><a href="/wiki/Dosya:Complex_ArcCot.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/Complex_ArcCot.jpg/140px-Complex_ArcCot.jpg" decoding="async" width="140" height="140" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/Complex_ArcCot.jpg/210px-Complex_ArcCot.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/Complex_ArcCot.jpg/280px-Complex_ArcCot.jpg 2x" data-file-width="1044" data-file-height="1044" /></a><figcaption></figcaption></figure> </td> <td><figure class="mw-halign-none" typeof="mw:File"><a href="/wiki/Dosya:Complex_ArcSec.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Complex_ArcSec.jpg/140px-Complex_ArcSec.jpg" decoding="async" width="140" height="140" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Complex_ArcSec.jpg/210px-Complex_ArcSec.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Complex_ArcSec.jpg/280px-Complex_ArcSec.jpg 2x" data-file-width="833" data-file-height="833" /></a><figcaption></figcaption></figure> </td> <td><figure class="mw-halign-none" typeof="mw:File"><a href="/wiki/Dosya:Complex_ArcCsc.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Complex_ArcCsc.jpg/140px-Complex_ArcCsc.jpg" decoding="async" width="140" height="140" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Complex_ArcCsc.jpg/210px-Complex_ArcCsc.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Complex_ArcCsc.jpg/280px-Complex_ArcCsc.jpg 2x" data-file-width="1039" data-file-height="1039" /></a><figcaption></figcaption></figure> </td></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arcsin(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arcsin(z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dee6682ca9597bbe4c2a4288c5cc07a85f6dc9f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.86ex; height:2.843ex;" alt="{\displaystyle \arcsin(z)}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arccos(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arccos(z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc22b22fad84588b35c004301571a76763b69fde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.115ex; height:2.843ex;" alt="{\displaystyle \arccos(z)}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arctan(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arctan(z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5639d7225fe861c9dc7b9d17420d53b474cbe9dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.364ex; height:2.843ex;" alt="{\displaystyle \arctan(z)}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {arccot}(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {arccot}(z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33429aa4786f03503d41646358f0d7dc63d9aed5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.103ex; height:2.843ex;" alt="{\displaystyle \operatorname {arccot}(z)}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {arcsec}(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arcsec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {arcsec}(z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a82a7fce4bcb85464efa221ca744a6352c412eee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.985ex; height:2.843ex;" alt="{\displaystyle \operatorname {arcsec}(z)}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {arccsc}(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccsc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {arccsc}(z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03e7565a043ebf7a7ddfa814cfea67edf0ab28f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.985ex; height:2.843ex;" alt="{\displaystyle \operatorname {arccsc}(z)}"></span> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Ayrıca_bakınız"><span id="Ayr.C4.B1ca_bak.C4.B1n.C4.B1z"></span>Ayrıca bakınız</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ters_trigonometrik_fonksiyonlar&veaction=edit&section=9" title="Değiştirilen bölüm: Ayrıca bakınız" class="mw-editsection-visualeditor"><span>değiştir</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Ters_trigonometrik_fonksiyonlar&action=edit&section=9" title="Bölümün kaynak kodunu değiştir: Ayrıca bakınız"><span>kaynağı değiştir</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Trigonometrik_fonksiyonlar" title="Trigonometrik fonksiyonlar">Trigonometrik fonksiyonlar</a></li> <li><a href="/wiki/Karek%C3%B6k" title="Karekök">Karekök</a></li> <li><a href="/w/index.php?title=Gauss_s%C3%BCrekli_kesri&action=edit&redlink=1" class="new" title="Gauss sürekli kesri (sayfa mevcut değil)">Gauss sürekli kesri</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:" 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href="/wiki/Genelle%C5%9Ftirilmi%C5%9F_trigonometri" title="Genelleştirilmiş 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><span typeof="mw:File"><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></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Trigonometrik_fonksiyonlar" title="Trigonometrik fonksiyonlar">Trigonometrik fonksiyonlar</a> &<br /><a class="mw-selflink selflink">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> (<i>sin</i>)</li> <li><a href="/wiki/Kosin%C3%BCs" title="Kosinüs">Kosinüs</a> (<i>cos</i>)</li> <li><a href="/wiki/Tanjant" title="Tanjant">Tanjant</a> (<i>tan</i>)</li> <li><a href="/wiki/Kotanjant" title="Kotanjant">Kotanjant</a> (<i>cot</i>)</li> <li><a href="/wiki/Sekant" title="Sekant">Sekant</a> (<i>sec</i>)</li> <li><a href="/wiki/Kosekant" title="Kosekant">Kosekant</a> (<i>csc</i>)</li> <li><a href="/w/index.php?title=Versin%C3%BCs&action=edit&redlink=1" class="new" title="Versinüs (sayfa mevcut değil)">Versinüs</a> (<i>versin</i>)</li> <li><a href="/w/index.php?title=Verkosin%C3%BCs&action=edit&redlink=1" class="new" title="Verkosinüs (sayfa mevcut değil)">Verkosinüs</a> (<i>vercosin</i>)</li> <li><a href="/w/index.php?title=Koversin%C3%BCs&action=edit&redlink=1" class="new" title="Koversinüs (sayfa mevcut değil)">Koversinüs</a> (<i>coversin</i>)</li> <li><a href="/w/index.php?title=Koverkosin%C3%BCs&action=edit&redlink=1" class="new" title="Koverkosinüs (sayfa mevcut değil)">Koverkosinüs</a> (<i>covercosin</i>)</li> <li><a href="/w/index.php?title=Haversin%C3%BCs&action=edit&redlink=1" class="new" title="Haversinüs (sayfa mevcut değil)">Haversinüs</a> (<i>haversin</i>)</li> <li><a href="/w/index.php?title=Haverkosin%C3%BCs&action=edit&redlink=1" class="new" title="Haverkosinüs (sayfa mevcut değil)">Haverkosinüs</a> (<i>havercosin</i>)</li> <li><a href="/w/index.php?title=Hakoversin%C3%BCs&action=edit&redlink=1" class="new" title="Hakoversinüs (sayfa mevcut değil)">Hakoversinüs</a> (<i>hacoversin</i>)</li> <li><a href="/w/index.php?title=Hakoverkosin%C3%BCs&action=edit&redlink=1" class="new" title="Hakoverkosinüs (sayfa mevcut değil)">Hakoverkosinüs</a> (<i>hacovercosin</i>)</li> <li><a href="/w/index.php?title=Ekssekant&action=edit&redlink=1" class="new" title="Ekssekant (sayfa mevcut değil)">Ekssekant</a> (<i>exsec</i>)</li> <li><a href="/w/index.php?title=Ekskosekant&action=edit&redlink=1" class="new" title="Ekskosekant (sayfa mevcut değil)">Ekskosekant</a> (<i>excsc</i>)</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 href="/wiki/Trigonometrik_%C3%B6zde%C5%9Flikler_listesi" title="Trigonometrik özdeşlikler listesi">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 href="/wiki/Jean-Baptiste_Joseph_Fourier" title="Jean-Baptiste Joseph Fourier">Fourier</a></li></ul> </div></td></tr></tbody></table></div></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">"<a dir="ltr" href="https://tr.wikipedia.org/w/index.php?title=Ters_trigonometrik_fonksiyonlar&oldid=30475926">https://tr.wikipedia.org/w/index.php?title=Ters_trigonometrik_fonksiyonlar&oldid=30475926</a>" sayfasından alınmıştır</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" 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