Sinus dan kosinus - Wikipedia bahasa Indonesia, ensiklopedia bebas

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[o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Masuk log</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Halaman penyunting yang telah keluar log <a href="/wiki/Bantuan:Pengantar" aria-label="Pelajari lebih lanjut tentang menyunting"><span>pelajari lebih lanjut</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Istimewa:Kontribusi_saya" title="Daftar suntingan yang dibuat dari alamat IP ini [y]" accesskey="y"><span>Kontribusi</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Istimewa:Pembicaraan_saya" title="Pembicaraan tentang suntingan dari alamat IP ini [n]" accesskey="n"><span>Pembicaraan</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Situs"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Daftar isi" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Daftar isi</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">pindah ke bilah sisi</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">sembunyikan</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Awal</div> </a> </li> <li id="toc-Definisi_dengan_segitiga_siku-siku" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definisi_dengan_segitiga_siku-siku"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Definisi dengan segitiga siku-siku</span> </div> </a> <ul id="toc-Definisi_dengan_segitiga_siku-siku-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Definisi_dengan_lingkaran_satuan" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definisi_dengan_lingkaran_satuan"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Definisi dengan lingkaran satuan</span> </div> </a> <ul id="toc-Definisi_dengan_lingkaran_satuan-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Identitas" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Identitas"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Identitas</span> </div> </a> <button aria-controls="toc-Identitas-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>Gulingkan subbagian Identitas</span> </button> <ul id="toc-Identitas-sublist" class="vector-toc-list"> <li id="toc-Lawan_perkalian" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lawan_perkalian"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Lawan perkalian</span> </div> </a> <ul id="toc-Lawan_perkalian-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fungsi_invers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fungsi_invers"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Fungsi invers</span> </div> </a> <ul id="toc-Fungsi_invers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kalkulus" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Kalkulus"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Kalkulus</span> </div> </a> <ul id="toc-Kalkulus-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Identitas_trigonometri_Pythagoras" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Identitas_trigonometri_Pythagoras"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Identitas trigonometri Pythagoras</span> </div> </a> <ul id="toc-Identitas_trigonometri_Pythagoras-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rumus_sudut_ganda" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rumus_sudut_ganda"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Rumus sudut ganda</span> </div> </a> <ul id="toc-Rumus_sudut_ganda-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Sifat_yang_berhubungan_dengan_kuadran" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sifat_yang_berhubungan_dengan_kuadran"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Sifat yang berhubungan dengan kuadran</span> </div> </a> <ul id="toc-Sifat_yang_berhubungan_dengan_kuadran-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Definisi_dengan_deret" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definisi_dengan_deret"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Definisi dengan deret</span> </div> </a> <button aria-controls="toc-Definisi_dengan_deret-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>Gulingkan subbagian Definisi dengan deret</span> </button> <ul id="toc-Definisi_dengan_deret-sublist" class="vector-toc-list"> <li id="toc-Bentuk_pecahan_berulang" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bentuk_pecahan_berulang"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Bentuk pecahan berulang</span> </div> </a> <ul id="toc-Bentuk_pecahan_berulang-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Hukum_sinus" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Hukum_sinus"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Hukum sinus</span> </div> </a> <ul id="toc-Hukum_sinus-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hukum_kosinus" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Hukum_kosinus"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Hukum kosinus</span> </div> </a> <ul id="toc-Hukum_kosinus-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Nilai-nilai_spesial" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Nilai-nilai_spesial"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Nilai-nilai spesial</span> </div> </a> <ul id="toc-Nilai-nilai_spesial-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Titik_tetap" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Titik_tetap"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Titik tetap</span> </div> </a> <ul id="toc-Titik_tetap-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Panjang_busur" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Panjang_busur"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Panjang busur</span> </div> </a> <ul id="toc-Panjang_busur-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hubungan_dengan_bilangan_kompleks" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Hubungan_dengan_bilangan_kompleks"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Hubungan dengan bilangan kompleks</span> </div> </a> <button aria-controls="toc-Hubungan_dengan_bilangan_kompleks-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>Gulingkan subbagian Hubungan dengan bilangan kompleks</span> </button> <ul id="toc-Hubungan_dengan_bilangan_kompleks-sublist" class="vector-toc-list"> <li id="toc-Argumen_kompleks" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Argumen_kompleks"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.1</span> <span>Argumen kompleks</span> </div> </a> <ul id="toc-Argumen_kompleks-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Grafik_kompleks" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Grafik_kompleks"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.2</span> <span>Grafik kompleks</span> </div> </a> <ul id="toc-Grafik_kompleks-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Sejarah" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sejarah"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>Sejarah</span> </div> </a> <button aria-controls="toc-Sejarah-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>Gulingkan subbagian Sejarah</span> </button> <ul id="toc-Sejarah-sublist" class="vector-toc-list"> <li id="toc-Etimologi" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Etimologi"> <div class="vector-toc-text"> <span class="vector-toc-numb">12.1</span> <span>Etimologi</span> </div> </a> <ul id="toc-Etimologi-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Implementasi_perangkat_lunak" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Implementasi_perangkat_lunak"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>Implementasi perangkat lunak</span> </div> </a> <button aria-controls="toc-Implementasi_perangkat_lunak-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>Gulingkan subbagian Implementasi perangkat lunak</span> </button> <ul id="toc-Implementasi_perangkat_lunak-sublist" class="vector-toc-list"> <li id="toc-Implementasi_berdasarkan_satuan_putaran" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Implementasi_berdasarkan_satuan_putaran"> <div class="vector-toc-text"> <span class="vector-toc-numb">13.1</span> <span>Implementasi berdasarkan satuan putaran</span> </div> </a> <ul id="toc-Implementasi_berdasarkan_satuan_putaran-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Bacaan_lebih_lanjut" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bacaan_lebih_lanjut"> <div class="vector-toc-text"> <span class="vector-toc-numb">14</span> <span>Bacaan lebih lanjut</span> </div> </a> <ul id="toc-Bacaan_lebih_lanjut-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Referensi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Referensi"> <div class="vector-toc-text"> <span class="vector-toc-numb">15</span> <span>Referensi</span> </div> </a> <ul id="toc-Referensi-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="Daftar isi" 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="Gulingkan daftar isi" > <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">Gulingkan daftar isi</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">Sinus dan kosinus</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="Pergi ke artikel dalam bahasa lain. Terdapat 7 bahasa" > <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-7" 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">7 bahasa</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B8%E0%A6%BE%E0%A6%87%E0%A6%A8_%E0%A6%93_%E0%A6%95%E0%A7%8B%E0%A6%B8%E0%A6%BE%E0%A6%87%E0%A6%A8" title="সাইন ও কোসাইন – Bengali" lang="bn" hreflang="bn" data-title="সাইন ও কোসাইন" data-language-autonym="বাংলা" data-language-local-name="Bengali" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Sinus_und_Kosinus" title="Sinus und Kosinus – Jerman" lang="de" hreflang="de" data-title="Sinus und Kosinus" data-language-autonym="Deutsch" data-language-local-name="Jerman" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Sine_and_cosine" title="Sine and cosine – Inggris" lang="en" hreflang="en" data-title="Sine and cosine" data-language-autonym="English" data-language-local-name="Inggris" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%A1%E1%83%98%E1%83%9C%E1%83%A3%E1%83%A1%E1%83%98_%E1%83%93%E1%83%90_%E1%83%99%E1%83%9D%E1%83%A1%E1%83%98%E1%83%9C%E1%83%A3%E1%83%A1%E1%83%98" title="სინუსი და კოსინუსი – Georgia" lang="ka" hreflang="ka" data-title="სინუსი და კოსინუსი" data-language-autonym="ქართული" data-language-local-name="Georgia" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Sinus_en_cosinus" title="Sinus en cosinus – Belanda" lang="nl" hreflang="nl" data-title="Sinus en cosinus" data-language-autonym="Nederlands" data-language-local-name="Belanda" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-uk badge-Q70893996 mw-list-item" title=""><a href="https://uk.wikipedia.org/wiki/%D0%A1%D0%B8%D0%BD%D1%83%D1%81_%D1%82%D0%B0_%D0%BA%D0%BE%D1%81%D0%B8%D0%BD%D1%83%D1%81" title="Синус та косинус – Ukraina" lang="uk" hreflang="uk" data-title="Синус та косинус" data-language-autonym="Українська" data-language-local-name="Ukraina" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh badge-Q70893996 mw-list-item" title=""><a href="https://zh.wikipedia.org/wiki/%E6%AD%A3%E5%BC%A6%E5%92%8C%E4%BD%99%E5%BC%A6" title="正弦和余弦 – Tionghoa" lang="zh" hreflang="zh" data-title="正弦和余弦" data-language-autonym="中文" data-language-local-name="Tionghoa" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q13647261#sitelinks-wikipedia" title="Sunting pranala interwiki" class="wbc-editpage">Sunting pranala</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Ruang nama"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Sinus_dan_kosinus" title="Lihat halaman isi [c]" accesskey="c"><span>Halaman</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Pembicaraan:Sinus_dan_kosinus" rel="discussion" title="Pembicaraan halaman isi [t]" accesskey="t"><span>Pembicaraan</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Ubah varian bahasa" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Bahasa Indonesia</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Tampilan"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Sinus_dan_kosinus"><span>Baca</span></a></li><li id="ca-ve-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Sinus_dan_kosinus&veaction=edit" title="Sunting halaman ini [v]" accesskey="v"><span>Sunting</span></a></li><li id="ca-edit" class="collapsible vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Sinus_dan_kosinus&action=edit" title="Sunting kode sumber halaman ini [e]" accesskey="e"><span>Sunting sumber</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Sinus_dan_kosinus&action=history" title="Revisi sebelumnya dari halaman ini. 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class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">pindah ke bilah sisi</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">sembunyikan</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Dari Wikipedia bahasa Indonesia, ensiklopedia bebas</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">(Dialihkan dari <a href="/w/index.php?title=Cosinus&redirect=no" class="mw-redirect" title="Cosinus">Cosinus</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="id" dir="ltr"><style data-mw-deduplicate="TemplateStyles:r18844875">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}</style><div role="note" class="hatnote navigation-not-searchable">"Sinus" dan "Kosinus" dialihkan ke halaman ini. Untuk penggunaan lain, lihat <a href="/wiki/Sinus_(disambiguasi)" class="mw-disambig" title="Sinus (disambiguasi)">Sinus (disambiguasi)</a>.</div> <style data-mw-deduplicate="TemplateStyles:r22657712">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}body.skin-minerva .mw-parser-output .infobox-header,body.skin-minerva .mw-parser-output .infobox-subheader,body.skin-minerva .mw-parser-output .infobox-above,body.skin-minerva .mw-parser-output .infobox-title,body.skin-minerva .mw-parser-output .infobox-image,body.skin-minerva .mw-parser-output .infobox-full-data,body.skin-minerva .mw-parser-output .infobox-below{text-align:center}</style><table class="infobox"><tbody><tr><th colspan="2" class="infobox-above" style="background:#e0e0e0;padding:0.15em 0.5em 0.25em;font-weight:bold;">Sinus dan kosinus</th></tr><tr><td colspan="2" class="infobox-image" style="padding-bottom:0.4em;"><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/Berkas:Sine_cosine_one_period.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Sine_cosine_one_period.svg/220px-Sine_cosine_one_period.svg.png" decoding="async" width="220" height="88" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Sine_cosine_one_period.svg/330px-Sine_cosine_one_period.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/71/Sine_cosine_one_period.svg/440px-Sine_cosine_one_period.svg.png 2x" data-file-width="600" data-file-height="240" /></a></span></td></tr><tr><th colspan="2" class="infobox-header" style="background:#e0e0e0;padding-bottom:0.2em;">Informasi umum</th></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;">Definisi umum</th><td class="infobox-data"><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}&\sin(\alpha )={\frac {\text{sisi tegak}}{\text{hipotenusa}}}\\[8pt]&\cos(\alpha )={\frac {\text{sisi alas}}{\text{hipotenusa}}}\\[8pt]\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="1.1em 1.1em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mtext>sisi tegak</mtext> <mtext>hipotenusa</mtext> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mtext>sisi alas</mtext> <mtext>hipotenusa</mtext> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&\sin(\alpha )={\frac {\text{sisi tegak}}{\text{hipotenusa}}}\\[8pt]&\cos(\alpha )={\frac {\text{sisi alas}}{\text{hipotenusa}}}\\[8pt]\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67630670294b220b032c32f81e866f046af500b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:22.089ex; height:13.843ex;" alt="{\displaystyle {\begin{aligned}&\sin(\alpha )={\frac {\text{sisi tegak}}{\text{hipotenusa}}}\\[8pt]&\cos(\alpha )={\frac {\text{sisi alas}}{\text{hipotenusa}}}\\[8pt]\end{aligned}}}"></span></td></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;">Tujuan diciptakan</th><td class="infobox-data"><a href="/wiki/Astronomi_India" title="Astronomi India">Astronomi India</a></td></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;">Tanggal penemuan solusi</th><td class="infobox-data"><a href="/wiki/Kemaharajaan_Gupta" title="Kemaharajaan Gupta">Kemaharajaan Gupta</a></td></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;">Bidang penerapan</th><td class="infobox-data"><a href="/wiki/Trigonometri" title="Trigonometri">Trigonometri</a>, <a href="/w/index.php?title=Transformasi_integral&action=edit&redlink=1" class="new" title="Transformasi integral (halaman belum tersedia)">Transformasi integral</a>, dll.</td></tr><tr><th colspan="2" class="infobox-header" style="background:#e0e0e0;padding-bottom:0.2em;">Domain dan Citra</th></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;"><a href="/wiki/Domain" class="mw-redirect" title="Domain">Domain dari fungsi</a></th><td class="infobox-data">(−<span style="position: relative; top: 0.2em;"><span style="font-size:130%;">∞</span></span>, +<span style="position: relative; top: 0.2em;"><span style="font-size:130%;">∞</span></span>) <sup><small>a</small></sup></td></tr><tr><th colspan="2" class="infobox-header" style="background:#e0e0e0;padding-bottom:0.2em;">Sifat umum</th></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;"><a href="/wiki/Paritas_(matematika)" title="Paritas (matematika)">Paritas fungsi</a></th><td class="infobox-data">sinus: ganjil; kosinus: genap</td></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;"><a href="/w/index.php?title=Periode_(matematika)&action=edit&redlink=1" class="new" title="Periode (matematika) (halaman belum tersedia)">Periode</a></th><td class="infobox-data">2<span class="texhtml">π</span></td></tr><tr><th colspan="2" class="infobox-header" style="background:#e0e0e0;padding-bottom:0.2em;">Nilai-nilai spesifik</th></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;">Nilai di 0</th><td class="infobox-data">sinus: 0; kosinus: 1</td></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;">Nilai maksimum</th><td class="infobox-data">sinus: (2<i>k</i><span class="texhtml">π</span> + <span class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;"><span style="display:block; line-height:1em; margin:0 0.1em;"><span class="texhtml">π</span></span><span style="display:block; line-height:1em; margin:0 0.1em; border-top:1px solid;">2</span></span>, 1)<sup><small>b</small></sup>; kosinus: (2<i>k</i><span class="texhtml">π</span>, 1)</td></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;">Nilai minimum</th><td class="infobox-data">sinus: (2<i>k</i><span class="texhtml">π</span> − <span class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;"><span style="display:block; line-height:1em; margin:0 0.1em;"><span class="texhtml">π</span></span><span style="display:block; line-height:1em; margin:0 0.1em; border-top:1px solid;">2</span></span>, −1); kosinus: (2<i>k</i><span class="texhtml">π</span> + <span class="texhtml">π</span>, -1)</td></tr><tr><th colspan="2" class="infobox-header" style="background:#e0e0e0;padding-bottom:0.2em;">Sifat khusus</th></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;"><a href="/wiki/Akar_fungsi" title="Akar fungsi">Akar</a></th><td class="infobox-data">sinus: <i>k</i><span class="texhtml">π</span>; kosinus: <i>k</i><span class="texhtml">π</span> + <span class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;"><span style="display:block; line-height:1em; margin:0 0.1em;"><span class="texhtml">π</span></span><span style="display:block; line-height:1em; margin:0 0.1em; border-top:1px solid;">2</span></span></td></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;"><a href="/wiki/Titik_kritis_(matematika)" title="Titik kritis (matematika)">Titik kritis</a></th><td class="infobox-data">sinus: <i>k</i><span class="texhtml">π</span> + <span class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;"><span style="display:block; line-height:1em; margin:0 0.1em;"><span class="texhtml">π</span></span><span style="display:block; line-height:1em; margin:0 0.1em; border-top:1px solid;">2</span></span>; kosinus: <i>k</i><span class="texhtml">π</span></td></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;"><a href="/wiki/Titik_belok" title="Titik belok"><span style="white-space: normal;">Titik belok</span></a></th><td class="infobox-data">sinus: <i>k</i><span class="texhtml">π</span>; kosinus: <i>k</i><span class="texhtml">π</span> + <span class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;"><span style="display:block; line-height:1em; margin:0 0.1em;"><span class="texhtml">π</span></span><span style="display:block; line-height:1em; margin:0 0.1em; border-top:1px solid;">2</span></span></td></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;"><a href="/wiki/Titik_tetap_(matematika)" title="Titik tetap (matematika)"><span style="white-space: normal;">Titik tetap</span></a></th><td class="infobox-data">sinus: 0; kosinus: <a href="/w/index.php?title=Bilangan_Dottie&action=edit&redlink=1" class="new" title="Bilangan Dottie (halaman belum tersedia)">Bilangan Dottie</a></td></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;"><a href="/wiki/Invers_perkalian" title="Invers perkalian">Kebalikan</a></th><td class="infobox-data">sinus: <a href="/wiki/Kosekan" title="Kosekan">kosekan</a>; cosine: <a href="/wiki/Sekan" title="Sekan">sekan</a></td></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;"><a href="/wiki/Fungsi_invers" title="Fungsi invers">Invers</a></th><td class="infobox-data">sinus: <a href="/w/index.php?title=Arcsinus&action=edit&redlink=1" class="new" title="Arcsinus (halaman belum tersedia)">arcsinus</a>; cosine: <a href="/w/index.php?title=Arccosinus&action=edit&redlink=1" class="new" title="Arccosinus (halaman belum tersedia)">arccosinus</a></td></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;"><a href="/wiki/Turunan" title="Turunan">Turunan</a></th><td class="infobox-data"><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}}\sin(x)&=\cos(x)\\[8pt]{\frac {d}{dx}}\cos(x)&=-\sin(x)\\[8pt]\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="1.1em 1.1em" 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>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {d}{dx}}\sin(x)&=\cos(x)\\[8pt]{\frac {d}{dx}}\cos(x)&=-\sin(x)\\[8pt]\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d0ab4c1ad1432ebe16e39e8f89b82da63258f15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:22.059ex; height:12.843ex;" alt="{\displaystyle {\begin{aligned}{\frac {d}{dx}}\sin(x)&=\cos(x)\\[8pt]{\frac {d}{dx}}\cos(x)&=-\sin(x)\\[8pt]\end{aligned}}}"></span></td></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;"><a href="/wiki/Integral_tak_tentu" title="Integral tak tentu">Antiturunan</a></th><td class="infobox-data"><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}\int \sin(x)\,dx&=-\cos(x)+C\\[8pt]\int \cos(x)\,dx&=\sin(x)+C\\[8pt]\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="1.1em 1.1em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo>∫</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>C</mi> </mtd> </mtr> <mtr> <mtd> <mo>∫</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>C</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\int \sin(x)\,dx&=-\cos(x)+C\\[8pt]\int \cos(x)\,dx&=\sin(x)+C\\[8pt]\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69f750edd0eb2079924aa919719829f4dc0d03c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.171ex; width:28.666ex; height:13.509ex;" alt="{\displaystyle {\begin{aligned}\int \sin(x)\,dx&=-\cos(x)+C\\[8pt]\int \cos(x)\,dx&=\sin(x)+C\\[8pt]\end{aligned}}}"></span></td></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;">Fungsi yang relevan</th><td class="infobox-data">tan, csc, sec, cot</td></tr><tr><th colspan="2" class="infobox-header" style="background:#e0e0e0;padding-bottom:0.2em;">Definisi deret</th></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;"><a href="/wiki/Deret_Taylor" title="Deret Taylor">Deret Taylor</a></th><td class="infobox-data"><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}\sin(x)&=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots \\[8pt]&={}\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}x^{2n+1}\\[8pt]\cos(x)&=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots \\[8pt]&={}\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}x^{2n}\\[8pt]\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="1.1em 1.1em 1.1em 1.1em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>x</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow> <mn>3</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mrow> <mn>5</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mrow> <mn>7</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> </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> <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> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mrow> <mn>4</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mrow> <mn>6</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> </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> <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> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sin(x)&=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots \\[8pt]&={}\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}x^{2n+1}\\[8pt]\cos(x)&=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots \\[8pt]&={}\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}x^{2n}\\[8pt]\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00027b8d48b1d8e8373bbb3e4b529f7779ac315a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -15.171ex; width:35.175ex; height:31.509ex;" alt="{\displaystyle {\begin{aligned}\sin(x)&=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots \\[8pt]&={}\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}x^{2n+1}\\[8pt]\cos(x)&=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots \\[8pt]&={}\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}x^{2n}\\[8pt]\end{aligned}}}"></span></td></tr><tr><td colspan="2" class="infobox-full-data"><hr /><style data-mw-deduplicate="TemplateStyles:r23782729">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style><div class="plainlist"><ul><li><sup>a</sup> Untuk <a href="/wiki/Bilangan_real" class="mw-redirect" title="Bilangan real">bilangan real</a>.</li><li><sup>b</sup> Variabel <i>k</i> adalah suatu <a href="/wiki/Bilangan_bulat" title="Bilangan bulat">bilangan bulat</a>.</li></ul></div></td></tr></tbody></table> <p>Dalam <a href="/wiki/Matematika" title="Matematika">matematika</a>, <b>sinus</b> dan <b>kosinus</b> adalah <a href="/wiki/Fungsi_trigonometri" title="Fungsi trigonometri">fungsi trigonometri</a> untuk <a href="/wiki/Sudut_(geometri)" title="Sudut (geometri)">sudut</a>. Sinus dan kosinus dari suatu sudut lancip didefinisikan dalam konteks <a href="/wiki/Segitiga_siku-siku" title="Segitiga siku-siku">segitiga siku-siku</a>: nilai sinus adalah rasio dari panjang sisi segitiga yang menghadap sudut tersebut (sisi tegak) terhadap panjang sisi terpanjang segitiga (hipotenusa), sedangkan nilai kosinus adalah rasio panjang sisi segitiga yang lain (sisi alas) terhadap hipotenusa. Untuk suatu sudut <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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>, fungsi sinus dan kosinus dituliskan sebagai <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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09b4b55580d6a821a07ad9fe35be88976917b10b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.572ex; height:2.176ex;" alt="{\displaystyle \sin x}"></span> and <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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/184ba70c3a71df25a25c09f34cd7f8175a9b5280" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.828ex; height:1.676ex;" alt="{\displaystyle \cos x}"></span>.<sup id="cite_ref-:1_1-0" class="reference"><a href="#cite_note-:1-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Lebih umum lagi, definisi sinus dan kosinus dapat diperluas ke sembarang nilai <a href="/wiki/Bilangan_real" class="mw-redirect" title="Bilangan real">real</a>, dalam konteks panjang suatu segmen garis pada suatu <a href="/wiki/Lingkaran_satuan" title="Lingkaran satuan">lingkaran satuan</a>. Definisi yang lebih modern menyatakan sinus dan kosinus dalam bentuk <a href="/wiki/Deret_(matematika)" title="Deret (matematika)">deret tak hingga</a>, atau solusi dari suatu <a href="/wiki/Persamaan_diferensial" title="Persamaan diferensial">persamaan diferensial</a>, yang memungkinkan memperluasnya ke bilangan negatif dan bahkan ke <a href="/wiki/Bilangan_kompleks" title="Bilangan kompleks">bilangan kompleks</a>. </p><p>Fungsi sinus dan kosinus umum digunakan untuk memodelkan fenomena <a href="/w/index.php?title=Fungsi_periodik&action=edit&redlink=1" class="new" title="Fungsi periodik (halaman belum tersedia)">periodik</a> seperti <a href="/wiki/Bunyi" title="Bunyi">bunyi</a> dan gelombang suara, posisi dan kecepatan dari osilator harmonik, intensitas cahay matahari dan panjang hari, maupun variasi temperatur sepanjang tahun. </p><p>Fungsi sinus dan kosinus dapat dilacak kembali ke fungsi <a href="/w/index.php?title=Jy%C4%81,_koti-jy%C4%81_dan_utkrama-jy%C4%81&action=edit&redlink=1" class="new" title="Jyā, koti-jyā dan utkrama-jyā (halaman belum tersedia)"><i>jyā</i> dan <i>koṭi-jyā </i></a>yang digunakan pada <a href="/wiki/Astronomi_India" title="Astronomi India">astronomi India</a> pada <a href="/wiki/Kemaharajaan_Gupta" title="Kemaharajaan Gupta">periode Gupta</a> (<i><a href="/w/index.php?title=Aryabhatiya&action=edit&redlink=1" class="new" title="Aryabhatiya (halaman belum tersedia)">Aryabhatiya</a></i>, <i><a href="/wiki/Surya_Siddhanta" title="Surya Siddhanta">Surya Siddhanta</a></i>), yang mengalami penerjemahan dari bahasa Sanskerta ke bahasa Arab, kemudian dari bahasa Arab ke bahasa Latin.<sup id="cite_ref-Boyer,_Carl_B._1991_p._210_2-0" class="reference"><a href="#cite_note-Boyer,_Carl_B._1991_p._210-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Kata "sinus" berasal dari penerjemahan <a href="/wiki/Bahasa_Latin" title="Bahasa Latin">bahasa Latin</a> yang salah oleh <a href="/w/index.php?title=Robert_of_Chester&action=edit&redlink=1" class="new" title="Robert of Chester (halaman belum tersedia)">Robert of Chester</a> untuk kata Arab <i>jiba</i>, yang selanjutnya merupakan <a href="/wiki/Alih_aksara" title="Alih aksara">transliterasi</a> dari kata Sanskerta untuk setengah busur, <i>jya-ardha</i>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Kata "cosinus" (Indonesia: "kosinus") berasal dari singkatan Latin "complementi sinus" pada abad pertengahan.<sup id="cite_ref-cosine_4-0" class="reference"><a href="#cite_note-cosine-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definisi_dengan_segitiga_siku-siku">Definisi dengan segitiga siku-siku</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sinus_dan_kosinus&veaction=edit&section=1" title="Sunting bagian: Definisi dengan segitiga siku-siku" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sinus_dan_kosinus&action=edit&section=1" title="Sunting kode sumber bagian: Definisi dengan segitiga siku-siku"><span>sunting sumber</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/Berkas:Trigono_a10.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Trigono_a10.svg/langid-220px-Trigono_a10.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Trigono_a10.svg/langid-330px-Trigono_a10.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Trigono_a10.svg/langid-440px-Trigono_a10.svg.png 2x" data-file-width="600" data-file-height="600" /></a><figcaption> Untuk suatu sudut <i>α</i>, fungsi sinus memberikan rasio panjang sisi tegak dengan panjang hipotenusa</figcaption></figure> <p>Untuk mendefinisikan sinus dan kosinus dari suatu sudut lancip <i>α</i>, mulai dengan membentuk <a href="/wiki/Segitiga_siku-siku" title="Segitiga siku-siku">segitiga siku-siku</a> yang mengandung sudut <i>α</i>; pada gambar berikut, sudut <i>α</i> pada segitiga <i>ABC</i> adalah sudut yang ingin dihitung. Ketiga sisi pada segitiga diberi nama sebagai berikut: </p> <ul><li><i>Sisi tegak</i> atau <i>sisi berlawanan</i> adalah sisi yang menghadap sudut <i>α</i>; dalam kasus ini adalah sisi <i>a</i>.</li> <li><i>Hipotenusa</i> adalah sisi yang menghadap sudut siku-siku, dalam kasus ini adalah sisi <i>h</i>. Hipotenusa selalu merupakan sisi terpanjang dari sembarang segitiga siku-siku.</li> <li><i>Sisi samping</i> adalah sisi segitiga yang tersisa, dalam kasus ini adalah sisi <i>b</i>. Sisi ini diapit oleh sudut <i>α</i> dan sudut siku-siku.</li></ul> <p>Setelah segitiga tersebut dibentuk, sinus dari sudut adalah panjang dari sisi tegak dibagi dengan panjang hipotenusa:<sup id="cite_ref-:2_5-0" class="reference"><a href="#cite_note-:2-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </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 \sin(\alpha )={\frac {\text{sisi tegak}}{\textrm {hipotenusa}}}\qquad \cos(\alpha )={\frac {\text{sisi alas}}{\textrm {hipotenusa}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mtext>sisi tegak</mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>hipotenusa</mtext> </mrow> </mfrac> </mrow> <mspace width="2em" /> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mtext>sisi alas</mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>hipotenusa</mtext> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(\alpha )={\frac {\text{sisi tegak}}{\textrm {hipotenusa}}}\qquad \cos(\alpha )={\frac {\text{sisi alas}}{\textrm {hipotenusa}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b84f75dad598d7117befbab667954c63fb76fb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:47.452ex; height:5.843ex;" alt="{\displaystyle \sin(\alpha )={\frac {\text{sisi tegak}}{\textrm {hipotenusa}}}\qquad \cos(\alpha )={\frac {\text{sisi alas}}{\textrm {hipotenusa}}}}"></span></dd></dl> <p>Fungsi-fungsi trigometri yang lain dapat didefinisikan dengan cara yang mirip; sebagai contoh, <a href="/wiki/Tangen" title="Tangen">tangen</a> dari suatu sudut adalah rasio antara sisi tegak dengan sisi alas.<sup id="cite_ref-:2_5-1" class="reference"><a href="#cite_note-:2-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>Dari definisi, nilai dari <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(\alpha )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(\alpha )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95a2a215bb78a456fe5662229c73775521b95299" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.153ex; height:2.843ex;" alt="{\displaystyle \sin(\alpha )}"></span> dan <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(\alpha )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(\alpha )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8323c66f99d1f3b7e0858fb92b0644fb0b8fba8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.408ex; height:2.843ex;" alt="{\displaystyle \cos(\alpha )}"></span> terlihat bergantung pada pemilihan segitiga siku-siku yang mengandung sudut <i>α</i>. Namun, hal ini tidak benar, karena semua segitiga siku-siku yang mengandung sudut <i>α</i> akan <a href="/w/index.php?title=Serupa_(geometri)&action=edit&redlink=1" class="new" title="Serupa (geometri) (halaman belum tersedia)">serupa</a>, sehingga rasio yang didapatkan dari semua segitiga tersebut akan sama. </p> <div class="mw-heading mw-heading2"><h2 id="Definisi_dengan_lingkaran_satuan">Definisi dengan lingkaran satuan</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sinus_dan_kosinus&veaction=edit&section=2" title="Sunting bagian: Definisi dengan lingkaran satuan" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sinus_dan_kosinus&action=edit&section=2" title="Sunting kode sumber bagian: Definisi dengan lingkaran satuan"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Berkas:Circle_cos_sin.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Circle_cos_sin.gif/220px-Circle_cos_sin.gif" decoding="async" width="220" height="132" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Circle_cos_sin.gif/330px-Circle_cos_sin.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Circle_cos_sin.gif/440px-Circle_cos_sin.gif 2x" data-file-width="650" data-file-height="390" /></a><figcaption>Animasi yang menunjukkan fungsi sinus (warna merah) digambarkan dari koordinat-<i>y</i> suatu titik pada <a href="/wiki/Lingkaran_satuan" title="Lingkaran satuan">lingkaran satuan</a> (warna hijau), pada suatu sudut <i>θ</i>. Sedangkan, koordinat-<i>x</i> dari titik tersebut menghasilkan grafik kosinus (warna biru).</figcaption></figure> <p>Dalam <a href="/wiki/Trigonometri" title="Trigonometri">trigonometri</a>, <a href="/wiki/Lingkaran_satuan" title="Lingkaran satuan">lingkaran satuan</a> adalah lingkaran dengan jari-jari sebesar 1 dan berpusat di titik asal (0, 0) pada <a href="/wiki/Sistem_koordinat_Kartesius" class="mw-redirect" title="Sistem koordinat Kartesius">sistem koordinat Kartesius</a>. Misalkan suatu segmen garis melalui titik asal, membentuk sudut </p><p><i>θ</i> terhadap sisi positif dari sumbu-<i>x</i>, dan memotong lingkaran satuan pada suatu titik. Nilai koordinat<i>-x</i> dan <i>-y</i> dari titik tersebut sama dengan <span class="texhtml" style="white-space: nowrap;">cos(<i>θ</i>)</span> dan <span class="texhtml" style="white-space: nowrap;">sin(<i>θ</i>)</span>, secara berurutan. </p><p>Definisi ini konsisten dengan definisi dengan menggunakan segitiga siku-siku ketika 0 < <i>θ</i> < π/2. Karena panjang hipotenusa segitiga siku-siku di lingkaran satuan selalu bernilai 1, <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="{\textstyle \sin(\theta )={\frac {{\textrm {sisi}}\,\,{\textrm {tegak}}}{\textrm {hipotenusa}}}={\frac {{\textrm {sisi}}\,\,{\textrm {tegak}}}{1}}={\textrm {sisi}}\,\,{\textrm {tegak}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>sisi</mtext> </mrow> </mrow> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>tegak</mtext> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>hipotenusa</mtext> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>sisi</mtext> </mrow> </mrow> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>tegak</mtext> </mrow> </mrow> </mrow> <mn>1</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>sisi</mtext> </mrow> </mrow> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>tegak</mtext> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sin(\theta )={\frac {{\textrm {sisi}}\,\,{\textrm {tegak}}}{\textrm {hipotenusa}}}={\frac {{\textrm {sisi}}\,\,{\textrm {tegak}}}{1}}={\textrm {sisi}}\,\,{\textrm {tegak}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd2a2c88d4f34732f4fda43b7f7f27769f4ea127" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:40.754ex; height:4.509ex;" alt="{\textstyle \sin(\theta )={\frac {{\textrm {sisi}}\,\,{\textrm {tegak}}}{\textrm {hipotenusa}}}={\frac {{\textrm {sisi}}\,\,{\textrm {tegak}}}{1}}={\textrm {sisi}}\,\,{\textrm {tegak}}}"></span>. Lebih lanjut, panjang sisi tegak dari segitiga ini adalah koordinat-<i>y</i> dari titik. Argumen yang sama dapat dibuat untuk menunjukkan <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="{\textstyle \cos(\theta )={\frac {\text{sisi alas}}{\text{hipotenusa}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mtext>sisi alas</mtext> <mtext>hipotenusa</mtext> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \cos(\theta )={\frac {\text{sisi alas}}{\text{hipotenusa}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62a8b41c8f16f3af4afb2dfd24a39e6f7efb6415" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:17.72ex; height:4.176ex;" alt="{\textstyle \cos(\theta )={\frac {\text{sisi alas}}{\text{hipotenusa}}}}"></span> ketika 0 < <i>θ</i> < π/2, bahkan ketika menggunakan definisi dengan lingkaran satuan. nilai tangen <span class="texhtml" style="white-space: nowrap;">tan(<i>θ</i>)</span> didefinisikan sebagai <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="{\textstyle {\frac {\sin(\theta )}{\cos(\theta )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {\sin(\theta )}{\cos(\theta )}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76f858a3441aa6d03463ad723d2c9f80eba4371d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:5.086ex; height:4.843ex;" alt="{\textstyle {\frac {\sin(\theta )}{\cos(\theta )}}}"></span>, atau secara ekuivalen, sebagai kemiringan dari segmen garis. </p><p>Definisi menggunakan lingkaran satuan memiliki keuntungan bahwa nilai sudut dapat diperluas menjadi sembarang <a href="/wiki/Bilangan_real" class="mw-redirect" title="Bilangan real">bilangan real</a>. Hal ini juga dapat dicapai dengan menggunakan beberapa simetri, dan mengganggap sinus (dan kosinus) sebagai <a href="/w/index.php?title=Fungsi_periodik&action=edit&redlink=1" class="new" title="Fungsi periodik (halaman belum tersedia)">fungsi periodik</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Identitas">Identitas</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sinus_dan_kosinus&veaction=edit&section=3" title="Sunting bagian: Identitas" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sinus_dan_kosinus&action=edit&section=3" title="Sunting kode sumber bagian: Identitas"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r18844875"><div role="note" class="hatnote navigation-not-searchable">Artikel utama: <a href="/wiki/Daftar_identitas_trigonometri" title="Daftar identitas trigonometri">Daftar identitas trigonometri</a></div> <p>Fungsi sinus dan kosinus terhubung secara tepat (<i>exact</i>) oleh identitas berikut, yang berlaku untuk sembarang nilai <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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span> (dalam <a href="/wiki/Radian" title="Radian">radian</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 \sin(\theta )=\cos \left({\frac {\pi }{2}}-\theta \right)=\cos \left(\theta -{\frac {\pi }{2}}\right)}"> <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>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>θ</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(\theta )=\cos \left({\frac {\pi }{2}}-\theta \right)=\cos \left(\theta -{\frac {\pi }{2}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1951b6ceec0f33e38234a26e6de2af9145af5730" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:35.923ex; height:4.843ex;" alt="{\displaystyle \sin(\theta )=\cos \left({\frac {\pi }{2}}-\theta \right)=\cos \left(\theta -{\frac {\pi }{2}}\right)}"></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 \cos(\theta )=\sin \left({\frac {\pi }{2}}-\theta \right)=\sin \left(\theta +{\frac {\pi }{2}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>θ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(\theta )=\sin \left({\frac {\pi }{2}}-\theta \right)=\sin \left(\theta +{\frac {\pi }{2}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e1e7a6b41fedaa5a32aea071db500dbd0fad211" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:35.668ex; height:4.843ex;" alt="{\displaystyle \cos(\theta )=\sin \left({\frac {\pi }{2}}-\theta \right)=\sin \left(\theta +{\frac {\pi }{2}}\right)}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Lawan_perkalian">Lawan perkalian</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sinus_dan_kosinus&veaction=edit&section=4" title="Sunting bagian: Lawan perkalian" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sinus_dan_kosinus&action=edit&section=4" title="Sunting kode sumber bagian: Lawan perkalian"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Invers_perkalian" title="Invers perkalian">Invers perkalian</a> dari sinus adalah kosekan, yang ditulis sebagai <span class="texhtml" style="white-space: nowrap;">csc</span>, atau <span class="texhtml" style="white-space: nowrap;">cosec</span>. Kosekan memberikan rasio panjang hipotenusa terhadap panjang sisi tegak. Serupa dengan itu, invers perkalian dari kosinus adalah sekan, yang memberikan rasio panjang hipotenusa terhadap panjang sisi samping. </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 \csc(A)={\frac {1}{\sin(A)}}={\frac {\text{hipotenusa}}{\text{sisi tegak}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>csc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mtext>hipotenusa</mtext> <mtext>sisi tegak</mtext> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \csc(A)={\frac {1}{\sin(A)}}={\frac {\text{hipotenusa}}{\text{sisi tegak}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/622eb28f451a27c10a2ae4a6a94708fe6836d9ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:31.806ex; height:6.176ex;" alt="{\displaystyle \csc(A)={\frac {1}{\sin(A)}}={\frac {\text{hipotenusa}}{\text{sisi tegak}}}}"></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 \sec(A)={\frac {1}{\cos(A)}}={\frac {\text{hipotenusa}}{\text{sisi alas}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mtext>hipotenusa</mtext> <mtext>sisi alas</mtext> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sec(A)={\frac {1}{\cos(A)}}={\frac {\text{hipotenusa}}{\text{sisi alas}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8405afd97fd3d2cd9738633d7946bdc6194daa86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:32.061ex; height:6.176ex;" alt="{\displaystyle \sec(A)={\frac {1}{\cos(A)}}={\frac {\text{hipotenusa}}{\text{sisi alas}}}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Fungsi_invers">Fungsi invers</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sinus_dan_kosinus&veaction=edit&section=5" title="Sunting bagian: Fungsi invers" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sinus_dan_kosinus&action=edit&section=5" title="Sunting kode sumber bagian: Fungsi invers"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Berkas:Arcsine_Arccosine.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Arcsine_Arccosine.svg/220px-Arcsine_Arccosine.svg.png" decoding="async" width="220" height="403" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Arcsine_Arccosine.svg/330px-Arcsine_Arccosine.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Arcsine_Arccosine.svg/440px-Arcsine_Arccosine.svg.png 2x" data-file-width="240" data-file-height="440" /></a><figcaption>The usual principal values of the <span class="texhtml" style="white-space: nowrap;">arcsin(<i>x</i>)</span> and <span class="texhtml" style="white-space: nowrap;">arccos(<i>x</i>)</span> functions graphed on the Cartesian plane</figcaption></figure> <p><a href="/wiki/Fungsi_invers" title="Fungsi invers">Fungsi invers</a> dari sinus adalah arcsinus (ditulis sebagai arcsin atau asin), atau invers sinus (<span class="texhtml" style="white-space: nowrap;">sin<sup>−1</sup></span>). Fungsi invers dari kosinus adalah arccosinus, yang ditulis sebagai arccos, acos, atau <span class="texhtml" style="white-space: nowrap;">cos<sup>−1</sup></span>. (simbol -1 pada bentuk <span class="texhtml" style="white-space: nowrap;">sin<sup>−1</sup></span> dan <span class="texhtml" style="white-space: nowrap;">cos<sup>−1</sup></span> melambangkan invers dari suatu fungsi, bukan <a href="/wiki/Eksponensiasi" title="Eksponensiasi">perpangkatan</a>.) Karena sinus dan kosinus bukan <a href="/wiki/Fungsi_injektif" title="Fungsi injektif">fungsi injektif</a>, invers mereka hanya fungsi invers secara parsial. Sebagai contoh, <span class="texhtml" style="white-space: nowrap;">sin(0) = 0</span>, tapi begitu pula dengan <span class="texhtml" style="white-space: nowrap;">sin(<i>π</i>) = 0</span>, <span class="texhtml" style="white-space: nowrap;">sin(2<i>π</i>) = 0</span> dst. Hal ini menyebabkan arcsinus bernilai banyak: <span class="texhtml" style="white-space: nowrap;">arcsin(0) = 0</span>, tapi juga <span class="texhtml" style="white-space: nowrap;">arcsin(0) = <i>π</i></span>, <span class="texhtml" style="white-space: nowrap;">arcsin(0) = 2<i>π</i></span>, dst. Ketika hanya satu nilai yang diperlukan, nilai fungsi dapat dibatasi hanya ke <a href="/w/index.php?title=Cabang_utama&action=edit&redlink=1" class="new" title="Cabang utama (halaman belum tersedia)">cabang utama</a>-nya saja. Dengan pembatasan ini, untuk setiap nilai <i>x</i> di <a href="/wiki/Ranah_fungsi" title="Ranah fungsi">domain</a>, ekspresi <span class="texhtml" style="white-space: nowrap;">arcsin(<i>x</i>)</span> akan menghasilkan satu nilai tunggal, yang disebut <a href="/w/index.php?title=Nilai_utama&action=edit&redlink=1" class="new" title="Nilai utama (halaman belum tersedia)">nilai utama</a>. <a href="/wiki/Selang_(matematika)" title="Selang (matematika)">Selang</a> nilai (standar) untuk nilai utama berkisar dari <span class="texhtml" style="white-space: nowrap;">-<i>π</i>/2</span> sampai <span class="texhtml" style="white-space: nowrap;"><i>π</i>/2</span>, dan selang nilai (standar) untuk arccos berkisar dari 0 sampai <span class="texhtml" style="white-space: nowrap;"><i>π</i></span>. </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 \left({\frac {\text{sisi tegak}}{\text{hipotenusa}}}\right)=\arccos \left({\frac {\text{sisi alas}}{\text{hipotenusa}}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mi>arcsin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mtext>sisi tegak</mtext> <mtext>hipotenusa</mtext> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>arccos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mtext>sisi alas</mtext> <mtext>hipotenusa</mtext> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =\arcsin \left({\frac {\text{sisi tegak}}{\text{hipotenusa}}}\right)=\arccos \left({\frac {\text{sisi alas}}{\text{hipotenusa}}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4fb7af9c8d7893fb22b21d6756dc73967c7f5e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:50.619ex; height:6.176ex;" alt="{\displaystyle \theta =\arcsin \left({\frac {\text{sisi tegak}}{\text{hipotenusa}}}\right)=\arccos \left({\frac {\text{sisi alas}}{\text{hipotenusa}}}\right).}"></span></dd></dl> <p>dengan (untuk suatu bilangan bulat <i>k</i>): </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}\sin(y)=x\iff &y=\arcsin(x)+2\pi k,{\text{ atau }}\\&y=\pi -\arcsin(x)+2\pi k\\\cos(y)=x\iff &y=\arccos(x)+2\pi k,{\text{ atau }}\\&y=-\arccos(x)+2\pi k\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> </mtd> <mtd> <mi>y</mi> <mo>=</mo> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mi>k</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> atau </mtext> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi>y</mi> <mo>=</mo> <mi>π</mi> <mo>−</mo> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mi>k</mi> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> </mtd> <mtd> <mi>y</mi> <mo>=</mo> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mi>k</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> atau </mtext> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi>y</mi> <mo>=</mo> <mo>−</mo> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mi>k</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sin(y)=x\iff &y=\arcsin(x)+2\pi k,{\text{ atau }}\\&y=\pi -\arcsin(x)+2\pi k\\\cos(y)=x\iff &y=\arccos(x)+2\pi k,{\text{ atau }}\\&y=-\arccos(x)+2\pi k\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/175f2700fe16a795fdd1a610fd0473ad2028491e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:44.381ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}\sin(y)=x\iff &y=\arcsin(x)+2\pi k,{\text{ atau }}\\&y=\pi -\arcsin(x)+2\pi k\\\cos(y)=x\iff &y=\arccos(x)+2\pi k,{\text{ atau }}\\&y=-\arccos(x)+2\pi k\end{aligned}}}"></span></dd></dl> <p>Berdasarkan definisi, arcsin dan arccos memenuhi persamaan-persamaan berikut: </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 \sin(\arcsin(x))=x\qquad \cos(\arccos(x))=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mspace width="2em" /> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(\arcsin(x))=x\qquad \cos(\arccos(x))=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25b4ef466eeb61838eb9b7b8692bebf636a9423f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.932ex; height:2.843ex;" alt="{\displaystyle \sin(\arcsin(x))=x\qquad \cos(\arccos(x))=x}"></span></dd></dl> <p>dan </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(\sin(\theta ))=\theta \quad &{\text{untuk}}\quad -{\frac {\pi }{2}}\leq \theta \leq {\frac {\pi }{2}}\\\arccos(\cos(\theta ))=\theta \quad &{\text{untuk}}\quad 0\leq \theta \leq \pi \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> <mo stretchy="false">(</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>θ</mi> <mspace width="1em" /> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>untuk</mtext> </mrow> <mspace width="1em" /> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>≤</mo> <mi>θ</mi> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>θ</mi> <mspace width="1em" /> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>untuk</mtext> </mrow> <mspace width="1em" /> <mn>0</mn> <mo>≤</mo> <mi>θ</mi> <mo>≤</mo> <mi>π</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\arcsin(\sin(\theta ))=\theta \quad &{\text{untuk}}\quad -{\frac {\pi }{2}}\leq \theta \leq {\frac {\pi }{2}}\\\arccos(\cos(\theta ))=\theta \quad &{\text{untuk}}\quad 0\leq \theta \leq \pi \end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cce794ef712bed86920b232b0fb8aa647bc89ba6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.074ex; margin-bottom: -0.264ex; width:44.097ex; height:7.843ex;" alt="{\displaystyle {\begin{aligned}\arcsin(\sin(\theta ))=\theta \quad &{\text{untuk}}\quad -{\frac {\pi }{2}}\leq \theta \leq {\frac {\pi }{2}}\\\arccos(\cos(\theta ))=\theta \quad &{\text{untuk}}\quad 0\leq \theta \leq \pi \end{aligned}}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Kalkulus">Kalkulus</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sinus_dan_kosinus&veaction=edit&section=6" title="Sunting bagian: Kalkulus" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sinus_dan_kosinus&action=edit&section=6" title="Sunting kode sumber bagian: Kalkulus"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r18844875"><div role="note" class="hatnote navigation-not-searchable">Lihat pula: <a href="/wiki/Daftar_integral_dari_fungsi_trigonometri" title="Daftar integral dari fungsi trigonometri">Daftar integral dari fungsi trigonometri</a> dan <a href="/wiki/Diferensiasi_fungsi_trigonometri" title="Diferensiasi fungsi trigonometri">Diferensiasi fungsi trigonometri</a></div> <p><a href="/wiki/Turunan" title="Turunan">Turunan</a> dari sinus dan kosinus adalah: </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}{dx}}\sin(x)=\cos(x)\qquad {\frac {d}{dx}}\cos(x)=-\sin(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dx}}\sin(x)=\cos(x)\qquad {\frac {d}{dx}}\cos(x)=-\sin(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2506ae422c8e8bb716f6f63d7f2ea0999a82ae4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:45.064ex; height:5.509ex;" alt="{\displaystyle {\frac {d}{dx}}\sin(x)=\cos(x)\qquad {\frac {d}{dx}}\cos(x)=-\sin(x)}"></span></dd></dl> <p>Sedangkan <a href="/wiki/Integral_tak_tentu" title="Integral tak tentu">antiturunan</a> mereka adalah: </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 \int \sin(x)\,dx=-\cos(x)+C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \sin(x)\,dx=-\cos(x)+C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27a8d0ab37dfd2b61f8e6e39fc3c6bf5ad6b9244" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.659ex; height:5.676ex;" alt="{\displaystyle \int \sin(x)\,dx=-\cos(x)+C}"></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 \int \cos(x)\,dx=\sin(x)+C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \cos(x)\,dx=\sin(x)+C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1047a00ec9a318928fc7657bbe5f464ba9364ca8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.463ex; height:5.676ex;" alt="{\displaystyle \int \cos(x)\,dx=\sin(x)+C}"></span></dd></dl> <p>dengan <i>C</i> menyatakan <a href="/wiki/Konstanta_integrasi" title="Konstanta integrasi">konstanta integrasi</a>.<sup id="cite_ref-:1_1-1" class="reference"><a href="#cite_note-:1-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Sinus dan kosinus muncul sebagai solusi untuk <a href="/wiki/Persamaan_diferensial" title="Persamaan diferensial">persamaan diferensial</a> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f''(x)=-kf(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>k</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f''(x)=-kf(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56ecd4145d4e824d9dadffc85a47327dec649ba4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.132ex; height:3.009ex;" alt="{\displaystyle f''(x)=-kf(x)}"></span> </p><p>Persamaan tersebut muncul pada banyak sistem fisik, seperti pada <a href="/wiki/Bandul" title="Bandul">pendulum</a> maupun pada beban yang terikat pada suatu pegas. Solusi persamaan ini adalah: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=C_{1}\sin(x{\sqrt {k}})+C_{2}\cos(x{\sqrt {k}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>k</mi> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>k</mi> </msqrt> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=C_{1}\sin(x{\sqrt {k}})+C_{2}\cos(x{\sqrt {k}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15e1fee0267b699f94f884aed0a3dfe456fe512" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.102ex; height:3.176ex;" alt="{\displaystyle f(x)=C_{1}\sin(x{\sqrt {k}})+C_{2}\cos(x{\sqrt {k}})}"></span> </p><p>Ketika <i>k</i> = 1, solusi unik dengan f(0) = 0 dan f'(0) = 1 adalah fungsi sinus, dan solusi unik dengan f(0) = 1 dan f'(0) = 0 adalah fungsi kosinus. </p> <div class="mw-heading mw-heading3"><h3 id="Identitas_trigonometri_Pythagoras">Identitas trigonometri Pythagoras</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sinus_dan_kosinus&veaction=edit&section=7" title="Sunting bagian: Identitas trigonometri Pythagoras" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sinus_dan_kosinus&action=edit&section=7" title="Sunting kode sumber bagian: Identitas trigonometri Pythagoras"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Hubungan dasar antara fungsi sinus dan kosinus termuat dalam <a href="/w/index.php?title=Identitas_trigonometri_Pythagoras&action=edit&redlink=1" class="new" title="Identitas trigonometri Pythagoras (halaman belum tersedia)">identitas trigonometri Pythagoras</a>:<sup id="cite_ref-:1_1-2" class="reference"><a href="#cite_note-:1-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </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 \cos ^{2}(\theta )+\sin ^{2}(\theta )=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos ^{2}(\theta )+\sin ^{2}(\theta )=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43794398352606bc4711133afa5eb2d6be348f50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.976ex; height:3.176ex;" alt="{\displaystyle \cos ^{2}(\theta )+\sin ^{2}(\theta )=1}"></span></dd></dl> <p>dengan sin<sup>2</sup>(<i>x</i>) menyatakan (sin(<i>x</i>))<sup>2</sup>. </p> <div class="mw-heading mw-heading3"><h3 id="Rumus_sudut_ganda">Rumus sudut ganda</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sinus_dan_kosinus&veaction=edit&section=8" title="Sunting bagian: Rumus sudut ganda" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sinus_dan_kosinus&action=edit&section=8" title="Sunting kode sumber bagian: Rumus sudut ganda"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Berkas:SinSquared.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/SinSquared.png/220px-SinSquared.png" decoding="async" width="220" height="136" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/SinSquared.png/330px-SinSquared.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c9/SinSquared.png/440px-SinSquared.png 2x" data-file-width="850" data-file-height="525" /></a><figcaption>Fungsi sinus diwarnai biru dan fungsi kuadrat sinus (sin<sup>2</sup>(x)) diwarnai merah. Sumbu-<i>x</i> dinyatakan dalam satuan radian.</figcaption></figure> <p>Sinus dan kosinus memenuhi persamaan-persamaan sudut ganda berikut: </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 \sin(2\theta )=2\sin(\theta )\cos(\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(2\theta )=2\sin(\theta )\cos(\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49b9120d4e69a660935b978d66f352fd2e645199" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.719ex; height:2.843ex;" alt="{\displaystyle \sin(2\theta )=2\sin(\theta )\cos(\theta )}"></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 \cos(2\theta )=\cos ^{2}(\theta )-\sin ^{2}(\theta )=2\cos ^{2}(\theta )-1=1-2\sin ^{2}(\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(2\theta )=\cos ^{2}(\theta )-\sin ^{2}(\theta )=2\cos ^{2}(\theta )-1=1-2\sin ^{2}(\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b899dd7bcb5d07da13e95c16b104361f3ba261ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:58.163ex; height:3.176ex;" alt="{\displaystyle \cos(2\theta )=\cos ^{2}(\theta )-\sin ^{2}(\theta )=2\cos ^{2}(\theta )-1=1-2\sin ^{2}(\theta )}"></span></dd></dl> <p>Rumus sudut ganda kosinus menyimpulkan bahwa fungsi sin<sup>2</sup> dan cos<sup>2</sup> adalah gelombang sinus yang mengalami penggeseran dan skalarisasi. Secara spesifik,<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </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 \sin ^{2}(\theta )={\frac {1-\cos(2\theta )}{2}}\qquad \cos ^{2}(\theta )={\frac {1+\cos(2\theta )}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> <mspace width="2em" /> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin ^{2}(\theta )={\frac {1-\cos(2\theta )}{2}}\qquad \cos ^{2}(\theta )={\frac {1+\cos(2\theta )}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4dc6bb37905a99fd21458b0011b90ebfcf6e9092" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:49.129ex; height:5.676ex;" alt="{\displaystyle \sin ^{2}(\theta )={\frac {1-\cos(2\theta )}{2}}\qquad \cos ^{2}(\theta )={\frac {1+\cos(2\theta )}{2}}}"></span></dd></dl> <p>Gambar berikut menyajikan fungsi sinus dan fungsi kuadrat sinus, masing-masing diwarnai dengan warna biru dan warna merah. Kedua grafik fungsi memiliki bentuk yang sama, namun memiliki jangkauan fungsi dan periode yang berbeda. Fungsi kuadrat sinus bernilai non-negatif, namun memiliki periode dua kali lebih cepat. </p> <div class="mw-heading mw-heading2"><h2 id="Sifat_yang_berhubungan_dengan_kuadran">Sifat yang berhubungan dengan kuadran</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sinus_dan_kosinus&veaction=edit&section=9" title="Sunting bagian: Sifat yang berhubungan dengan kuadran" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sinus_dan_kosinus&action=edit&section=9" title="Sunting kode sumber bagian: Sifat yang berhubungan dengan kuadran"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/Berkas:Quadrants_01_Pengo.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c7/Quadrants_01_Pengo.svg/175px-Quadrants_01_Pengo.svg.png" decoding="async" width="175" height="175" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c7/Quadrants_01_Pengo.svg/263px-Quadrants_01_Pengo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c7/Quadrants_01_Pengo.svg/350px-Quadrants_01_Pengo.svg.png 2x" data-file-width="724" data-file-height="724" /></a><figcaption>Keempat kuadran pada sistem koordinat Kartesius.</figcaption></figure> <p>Tabel berikut menyajikan banyak sifat penting dari fungsi sinus (tanda fungsi, kemonotonan, dan kecekungan), disusun berdasarkan kuadran dari nilai argumen (sudut). Untuk argumen di luar tabel, informasi yang bersesuaian dapat diperoleh dengan menggunakan sifat periodik <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(\alpha +360^{\circ })=\sin(\alpha )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>+</mo> <msup> <mn>360</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(\alpha +360^{\circ })=\sin(\alpha )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d585c1d1d9db6ada94b1034f502aaf323766e6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.786ex; height:2.843ex;" alt="{\displaystyle \sin(\alpha +360^{\circ })=\sin(\alpha )}"></span> dari fungsi sinus. </p> <table class="wikitable" style="text-align:center;"> <tbody><tr> <th rowspan="2"><a href="/wiki/Sistem_koordinat_Kartesius#Kuadran_dan_oktan" class="mw-redirect" title="Sistem koordinat Kartesius">Kuadran</a> </th> <th colspan="2">Sudut </th> <th colspan="3">Sinus </th> <th colspan="3">Kosinus </th></tr> <tr> <th><a href="/wiki/Derajat_(satuan_sudut)" title="Derajat (satuan sudut)">Derajat</a> </th> <th><a href="/wiki/Radian" title="Radian">Radian</a> </th> <th><a href="/wiki/Tanda_(matematika)" title="Tanda (matematika)">Tanda</a> </th> <th><a href="/wiki/Fungsi_monotonik" class="mw-redirect" title="Fungsi monotonik">Kemonotonan</a> </th> <th><a href="/wiki/Fungsi_konveks" class="mw-redirect" title="Fungsi konveks">Kecekungan</a> </th> <th><a href="/wiki/Tanda_(matematika)" title="Tanda (matematika)">Tanda</a> </th> <th><a href="/w/index.php?title=Monotonic_function&action=edit&redlink=1" class="new" title="Monotonic function (halaman belum tersedia)">Kemonotonan</a> </th> <th><a href="/w/index.php?title=Convex_function&action=edit&redlink=1" class="new" title="Convex function (halaman belum tersedia)">Kecekungan</a> </th></tr> <tr> <td style="text-align:left;">Kuadran I </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 0^{\circ }<x<90^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo><</mo> <mi>x</mi> <mo><</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0^{\circ }<x<90^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cd94c157a7ab3ac44ece2b4a7e2a805c5e3774b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.122ex; height:2.343ex;" alt="{\displaystyle 0^{\circ }<x<90^{\circ }}"></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 0<x<{\frac {\pi }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>x</mi> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<x<{\frac {\pi }{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27cc01a58038dcde40cba7a9801f6ae56972a8ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.857ex; height:4.676ex;" alt="{\displaystyle 0<x<{\frac {\pi }{2}}}"></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 +}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}"></span> </td> <td>naik </td> <td>konkaf </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 +}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}"></span> </td> <td>turun </td> <td>konkaf </td></tr> <tr> <td style="text-align:left;">Kuadran II </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 90^{\circ }<x<180^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo><</mo> <mi>x</mi> <mo><</mo> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 90^{\circ }<x<180^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7369b464d2a5d0f9d988d98d356c17552dd4973" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.447ex; height:2.343ex;" alt="{\displaystyle 90^{\circ }<x<180^{\circ }}"></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 {\frac {\pi }{2}}<x<\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo><</mo> <mi>x</mi> <mo><</mo> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\pi }{2}}<x<\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db6042419fb74dd6d63a74b866cc672d7f9a8e1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.027ex; height:4.676ex;" alt="{\displaystyle {\frac {\pi }{2}}<x<\pi }"></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 +}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}"></span> </td> <td>turun </td> <td>konkaf </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 -}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04bd52ce670743d3b61bec928a7ec9f47309eb36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle -}"></span> </td> <td>turun </td> <td>konveks </td></tr> <tr> <td style="text-align:left;">Kuadran III </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 180^{\circ }<x<270^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo><</mo> <mi>x</mi> <mo><</mo> <msup> <mn>270</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 180^{\circ }<x<270^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1c54b3e507d6bcc171945407efe3f8010e8b93f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.61ex; height:2.343ex;" alt="{\displaystyle 180^{\circ }<x<270^{\circ }}"></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 \pi <x<{\frac {3\pi }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo><</mo> <mi>x</mi> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi <x<{\frac {3\pi }{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c10cf0babe2aceaf56a8c04d17d61f183ea84da8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.189ex; height:5.176ex;" alt="{\displaystyle \pi <x<{\frac {3\pi }{2}}}"></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 -}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04bd52ce670743d3b61bec928a7ec9f47309eb36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle -}"></span> </td> <td>turun </td> <td>konveks </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 -}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04bd52ce670743d3b61bec928a7ec9f47309eb36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle -}"></span> </td> <td>naik </td> <td>konveks </td></tr> <tr> <td style="text-align:left;">Kuadran IV </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 270^{\circ }<x<360^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>270</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo><</mo> <mi>x</mi> <mo><</mo> <msup> <mn>360</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 270^{\circ }<x<360^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/637855480e837d2b6e96aecfef878b4bce1d24f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.61ex; height:2.343ex;" alt="{\displaystyle 270^{\circ }<x<360^{\circ }}"></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 {\frac {3\pi }{2}}<x<2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo><</mo> <mi>x</mi> <mo><</mo> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {3\pi }{2}}<x<2\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5d15bf5b904f030224fb1f461d2725ccef32c5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.352ex; height:5.176ex;" alt="{\displaystyle {\frac {3\pi }{2}}<x<2\pi }"></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 -}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04bd52ce670743d3b61bec928a7ec9f47309eb36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle -}"></span> </td> <td>naik </td> <td>konveks </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 +}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}"></span> </td> <td>naik </td> <td>konkaf </td></tr></tbody></table> <figure typeof="mw:File/Thumb"><a href="/wiki/Berkas:Sine_quads_01_Pengo.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Sine_quads_01_Pengo.svg/250px-Sine_quads_01_Pengo.svg.png" decoding="async" width="250" height="108" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Sine_quads_01_Pengo.svg/375px-Sine_quads_01_Pengo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Sine_quads_01_Pengo.svg/500px-Sine_quads_01_Pengo.svg.png 2x" data-file-width="1065" data-file-height="459" /></a><figcaption>Kuadran-kuadran pada lingkaran satuan dan fungsi sin(<i>x</i>), menggunakan <a href="/wiki/Sistem_koordinat_Kartesius" class="mw-redirect" title="Sistem koordinat Kartesius">sistem koordinat Kartesius</a>. </figcaption></figure> <p>Tabel berikut menyajikan informasi dasar terkait nilai sinus dan kosinus pada batas selang kuadran. </p> <table class="wikitable" style="text-align: center"> <tbody><tr> <th rowspan="2"><a href="/wiki/Derajat_(satuan_sudut)" title="Derajat (satuan sudut)">Derajat</a> </th> <th rowspan="2"><a href="/wiki/Radian" title="Radian">Radian</a> </th> <th colspan="2"><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(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a990a5545cac26c1c6821dca95d898bc80fe3a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.995ex; height:2.843ex;" alt="{\displaystyle \sin(x)}"></span> </th> <th colspan="2"><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(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb9af7ed6f44822021b74bb82b431022c7fd66b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.25ex; height:2.843ex;" alt="{\displaystyle \cos(x)}"></span> </th></tr> <tr> <th>Nilai </th> <th>Jenis titik </th> <th>Nilai </th> <th>Jenis titik </th></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 0^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd0e1e92cf5770c2bfbb1de8b4b7bf904c9deef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.217ex; height:2.343ex;" alt="{\displaystyle 0^{\circ }}"></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 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></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 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span> </td> <td align="left"><a href="/wiki/Akar_fungsi" title="Akar fungsi">Akar</a>, <a href="/wiki/Titik_belok" title="Titik belok">belok</a> </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 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span> </td> <td align="left"><a href="/wiki/Maksimum_dan_minimum" title="Maksimum dan minimum">Maksimum</a> </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 90^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 90^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c326d317eddef3ad3e6625e018a708e290a039f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.343ex;" alt="{\displaystyle 90^{\circ }}"></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 {\frac {\pi }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\pi }{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98f98bef5d4981ff6e2aa827d4699e347fb30db2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.168ex; height:4.676ex;" alt="{\displaystyle {\frac {\pi }{2}}}"></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 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span> </td> <td align="left"><a href="/wiki/Maksimum_dan_minimum" title="Maksimum dan minimum">Maksimum</a> </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 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span> </td> <td align="left"><a href="/wiki/Akar_fungsi" title="Akar fungsi">Akar</a>, <a href="/wiki/Titik_belok" title="Titik belok">belok</a> </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 180^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 180^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5d0431ce231935522dc0cb52df7f2b406cdadc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.542ex; height:2.343ex;" alt="{\displaystyle 180^{\circ }}"></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 \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"></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 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span> </td> <td align="left"><a href="/wiki/Akar_fungsi" title="Akar fungsi">Akar</a>, <a href="/wiki/Titik_belok" title="Titik belok">belok</a> </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 -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/704fb0427140d054dd267925495e78164fee9aac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}"></span> </td> <td align="left"><a href="/wiki/Maksimum_dan_minimum" title="Maksimum dan minimum">Minimum</a> </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 270^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>270</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 270^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d46a7b7ced26e658ef8ae2a8feaf5fe3113193f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.542ex; height:2.343ex;" alt="{\displaystyle 270^{\circ }}"></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 {\frac {3\pi }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {3\pi }{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9da88627f444ec5ae4ed69cfb5ff5d87763a857d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.331ex; height:5.176ex;" alt="{\displaystyle {\frac {3\pi }{2}}}"></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 -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/704fb0427140d054dd267925495e78164fee9aac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}"></span> </td> <td align="left"><a href="/wiki/Maksimum_dan_minimum" title="Maksimum dan minimum">Minimum</a> </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 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span> </td> <td align="left"><a href="/wiki/Akar_fungsi" title="Akar fungsi">Akar</a>, <a href="/wiki/Titik_belok" title="Titik belok">belok</a> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Definisi_dengan_deret">Definisi dengan deret</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sinus_dan_kosinus&veaction=edit&section=10" title="Sunting bagian: Definisi dengan deret" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sinus_dan_kosinus&action=edit&section=10" title="Sunting kode sumber bagian: Definisi dengan deret"><span>sunting sumber</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/Berkas:Taylorsine.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Taylorsine.svg/220px-Taylorsine.svg.png" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Taylorsine.svg/330px-Taylorsine.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Taylorsine.svg/440px-Taylorsine.svg.png 2x" data-file-width="1600" data-file-height="1200" /></a><figcaption>Fungsi sinus (biru) dapat dihampiri dengan baik dengan <a href="/wiki/Polinomial_Taylor" class="mw-redirect" title="Polinomial Taylor">polinomial Taylor</a>-nya yang berderajat 7 (merah muda) untuk nilai sudut di antara -<span class="texhtml">π</span> sampai <span class="texhtml">π</span>.</figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Berkas:Sine.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Sine.gif/220px-Sine.gif" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Sine.gif/330px-Sine.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Sine.gif/440px-Sine.gif 2x" data-file-width="800" data-file-height="600" /></a><figcaption>Animasi berikut menunjukkan efek menambahkan lebih banyak suku pada deret Taylor menyebabkannya menjadi lebih mirip dengan fungsi sinus.</figcaption></figure> <p>Turunan secara terus-menerus dari fungsi sinus di 0 dapat digunakan untuk menentukan bentuk <a href="/wiki/Deret_Taylor" title="Deret Taylor">deret Taylor</a>-nya. Dengan hanya menggunakan geometri dan sifat dari <a href="/wiki/Limit" class="mw-disambig" title="Limit">limit</a>, dapat ditunjukkan bahwa <a href="/wiki/Turunan" title="Turunan">turunan</a> dari sinus adalah kosinus, dan turunan kosinus adalah negatif dari sinus. Hal ini mengartikan secara turunan dari sin(x) secara berurutan adalah cos(x), -sin(x), -cos(x), sin(x), dan berulang lagi dari awal. Turunan ke-(4<i>n</i>+<i>k</i>), dievaluasi di titik 0 adalah: </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 \sin ^{(4n+k)}(0)={\begin{cases}0&{\text{ketika }}k=0\\1&{\text{ketika }}k=1\\0&{\text{ketika }}k=2\\-1&{\text{ketika }}k=3\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>4</mn> <mi>n</mi> <mo>+</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>ketika </mtext> </mrow> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>ketika </mtext> </mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>ketika </mtext> </mrow> <mi>k</mi> <mo>=</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>ketika </mtext> </mrow> <mi>k</mi> <mo>=</mo> <mn>3</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin ^{(4n+k)}(0)={\begin{cases}0&{\text{ketika }}k=0\\1&{\text{ketika }}k=1\\0&{\text{ketika }}k=2\\-1&{\text{ketika }}k=3\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4259e380551029bbf117a3a3394b9011712d67e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:34.745ex; height:11.176ex;" alt="{\displaystyle \sin ^{(4n+k)}(0)={\begin{cases}0&{\text{ketika }}k=0\\1&{\text{ketika }}k=1\\0&{\text{ketika }}k=2\\-1&{\text{ketika }}k=3\end{cases}}}"></span></dd></dl> <p>Dengan simbol pada tika atas (<i>superscript</i>) menyatakan turunan secara berulang. Hal ini dapat digunakan untuk menghitung ekspansi deret Taylor di titik x = 0 (dalam radian):<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </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}\sin(x)&=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots \\[8pt]&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}x^{2n+1}\\[8pt]\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="1.1em 1.1em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>x</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow> <mn>3</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mrow> <mn>5</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mrow> <mn>7</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></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> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sin(x)&=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots \\[8pt]&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}x^{2n+1}\\[8pt]\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/def345e147219a7892eb8140dfeb1c77b29dce38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:34.919ex; height:14.843ex;" alt="{\displaystyle {\begin{aligned}\sin(x)&=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots \\[8pt]&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}x^{2n+1}\\[8pt]\end{aligned}}}"></span></dd></dl> <p>Menghitung turunan untuk setiap suku menghasilkan deret Taylor untuk kosinus: </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}\cos(x)&=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots \\[8pt]&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}x^{2n}\\[8pt]\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="1.1em 1.1em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mrow> <mn>4</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mrow> <mn>6</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></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> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\cos(x)&=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots \\[8pt]&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}x^{2n}\\[8pt]\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72f8792e2fd4203f00339519200068cdd1652b08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:35.007ex; height:14.843ex;" alt="{\displaystyle {\begin{aligned}\cos(x)&=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots \\[8pt]&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}x^{2n}\\[8pt]\end{aligned}}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Bentuk_pecahan_berulang">Bentuk pecahan berulang</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sinus_dan_kosinus&veaction=edit&section=11" title="Sunting bagian: Bentuk pecahan berulang" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sinus_dan_kosinus&action=edit&section=11" title="Sunting kode sumber bagian: Bentuk pecahan berulang"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Fungsi sinus dan kosinus dapat direpresentasikan dalam bentuk <a href="/w/index.php?title=Pecahan_berulang&action=edit&redlink=1" class="new" title="Pecahan berulang (halaman belum tersedia)">pecahan berulang yang diperumum</a>, yang dapat dihasilkan dengan menggunakan <a href="/w/index.php?title=Rumus_pecahan_berulang_Euler&action=edit&redlink=1" class="new" title="Rumus pecahan berulang Euler (halaman belum tersedia)">rumus pecahan berulang Euler</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 \sin(x)={\cfrac {x}{1+{\cfrac {x^{2}}{2\cdot 3-x^{2}+{\cfrac {2\cdot 3x^{2}}{4\cdot 5-x^{2}+{\cfrac {4\cdot 5x^{2}}{6\cdot 7-x^{2}+\ddots }}}}}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>⋅</mo> <mn>3</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>⋅</mo> <mn>3</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mo>⋅</mo> <mn>5</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mo>⋅</mo> <mn>5</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> <mo>⋅</mo> <mn>7</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>⋱</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(x)={\cfrac {x}{1+{\cfrac {x^{2}}{2\cdot 3-x^{2}+{\cfrac {2\cdot 3x^{2}}{4\cdot 5-x^{2}+{\cfrac {4\cdot 5x^{2}}{6\cdot 7-x^{2}+\ddots }}}}}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1520b2ba509fd410d7968f178fa3fce414eb8b6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -15.338ex; width:56.272ex; height:19.509ex;" alt="{\displaystyle \sin(x)={\cfrac {x}{1+{\cfrac {x^{2}}{2\cdot 3-x^{2}+{\cfrac {2\cdot 3x^{2}}{4\cdot 5-x^{2}+{\cfrac {4\cdot 5x^{2}}{6\cdot 7-x^{2}+\ddots }}}}}}}}.}"></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 \cos(x)={\cfrac {1}{1+{\cfrac {x^{2}}{1\cdot 2-x^{2}+{\cfrac {1\cdot 2x^{2}}{3\cdot 4-x^{2}+{\cfrac {3\cdot 4x^{2}}{5\cdot 6-x^{2}+\ddots }}}}}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>⋅</mo> <mn>2</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>⋅</mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>⋅</mo> <mn>4</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>⋅</mo> <mn>4</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> <mo>⋅</mo> <mn>6</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>⋱</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(x)={\cfrac {1}{1+{\cfrac {x^{2}}{1\cdot 2-x^{2}+{\cfrac {1\cdot 2x^{2}}{3\cdot 4-x^{2}+{\cfrac {3\cdot 4x^{2}}{5\cdot 6-x^{2}+\ddots }}}}}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83243e1d7a1909b0203c7af3f0710741b5f86fb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -15.338ex; width:56.528ex; height:19.509ex;" alt="{\displaystyle \cos(x)={\cfrac {1}{1+{\cfrac {x^{2}}{1\cdot 2-x^{2}+{\cfrac {1\cdot 2x^{2}}{3\cdot 4-x^{2}+{\cfrac {3\cdot 4x^{2}}{5\cdot 6-x^{2}+\ddots }}}}}}}}.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Hukum_sinus">Hukum sinus</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sinus_dan_kosinus&veaction=edit&section=12" title="Sunting bagian: Hukum sinus" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sinus_dan_kosinus&action=edit&section=12" title="Sunting kode sumber bagian: Hukum sinus"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r18844875"><div role="note" class="hatnote navigation-not-searchable">Artikel utama: <a href="/wiki/Hukum_sinus" class="mw-redirect" title="Hukum sinus">Hukum sinus</a></div> <p><a href="/wiki/Hukum_sinus" class="mw-redirect" title="Hukum sinus">Hukum sinus</a> menyatakan bahwa untuk sembarang segitiga dengan sisi-sisi <i>a</i>, <i>b</i>, dan <i>c</i>, dan sudut-sudut yang menghadap sisi-sisi tersebut secara berurutan adalah <i>A</i>, <i>B</i>, dan <i>C</i>, berlaku:<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\sin A}{a}}={\frac {\sin B}{b}}={\frac {\sin C}{c}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>A</mi> </mrow> <mi>a</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>B</mi> </mrow> <mi>b</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>C</mi> </mrow> <mi>c</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\sin A}{a}}={\frac {\sin B}{b}}={\frac {\sin C}{c}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04be582cf7d80eed2b241465484e2e5f0f10b691" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:24.354ex; height:5.509ex;" alt="{\displaystyle {\frac {\sin A}{a}}={\frac {\sin B}{b}}={\frac {\sin C}{c}}.}"></span>Pernyataan di atas ekuivalen dengan persamaan berikut: </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 {a}{\sin A}}={\frac {b}{\sin B}}={\frac {c}{\sin C}}=2R,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>A</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>B</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>C</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> <mi>R</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a}{\sin A}}={\frac {b}{\sin B}}={\frac {c}{\sin C}}=2R,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1419ccbac53bfe833611e68a722eef65eee1245d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:30.379ex; height:5.509ex;" alt="{\displaystyle {\frac {a}{\sin A}}={\frac {b}{\sin B}}={\frac {c}{\sin C}}=2R,}"></span></dd></dl> <p>dengan <i>R</i> menyatakan radius dari <a href="/wiki/Lingkaran_luar" title="Lingkaran luar">lingkaran luar</a> segitiga. Hal tersebut dapat ditunjukkan dengan membagi segitiga menjadi dua segitiga siku-siku dan menggunakan definisi sinus. </p><p>Hukum sinus berguna untuk menghitung panjang sisi segitiga yang tidak diketahui, jika nilai dua sudut dan satu sisi diketahui. Hal ini umum terjadi ketika melakukan <a href="/wiki/Triangulasi" title="Triangulasi">triangulasi</a>, suatu teknik untuk menentukan panjang yang tidak diketahui dengan mengukur dua nilai sudut yang terpisah dalam jarak yang ditentukan sebelumnya. </p> <div class="mw-heading mw-heading2"><h2 id="Hukum_kosinus">Hukum kosinus</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sinus_dan_kosinus&veaction=edit&section=13" title="Sunting bagian: Hukum kosinus" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sinus_dan_kosinus&action=edit&section=13" title="Sunting kode sumber bagian: Hukum kosinus"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r18844875"><div role="note" class="hatnote navigation-not-searchable">Artikel utama: <a href="/wiki/Hukum_kosinus" class="mw-redirect" title="Hukum kosinus">hukum kosinus</a></div> <p><a href="/wiki/Hukum_kosinus" class="mw-redirect" title="Hukum kosinus">Hukum kosinus</a> menyatakan bahwa untuk sembarang segitiga dengan sisi-sisi <i>a</i>, <i>b</i>, dan <i>c</i>, dan sudut-sudut yang menghadap sisi-sisi tersebut secara berurutan adalah <i>A</i>, <i>B</i>, dan <i>C</i>, berlaku:<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{2}+b^{2}-2ab\cos(C)=c^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>a</mi> <mi>b</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{2}+b^{2}-2ab\cos(C)=c^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcb76d51ffdeaa7b6a12ed382b57650997d0769c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.64ex; height:3.176ex;" alt="{\displaystyle a^{2}+b^{2}-2ab\cos(C)=c^{2}}"></span>Untuk kasus dengan <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 C=\pi /2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C=\pi /2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2d9a0497ccff2a204b58a64af42266b67f8c1c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.522ex; height:2.843ex;" alt="{\displaystyle C=\pi /2}"></span>, nilai <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(C)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(C)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26f6ccfb305c2971b178289ff727ccb9060e4cb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.948ex; height:2.843ex;" alt="{\displaystyle \cos(C)=0}"></span> dan hukum ini berubah menjadi <a href="/wiki/Teorema_Pythagoras" title="Teorema Pythagoras">teorema Pythagoras</a>: Untuk segitiga siku-siku, berlaku hubungan <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 a^{2}+b^{2}=c^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{2}+b^{2}=c^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92333b53991e3ea02f5d6384bac4911ae3060a1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.983ex; height:3.009ex;" alt="{\displaystyle a^{2}+b^{2}=c^{2},}"></span> dengan <i>c</i> menyatakan panjang hipotenusa. </p> <div class="mw-heading mw-heading2"><h2 id="Nilai-nilai_spesial">Nilai-nilai spesial</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sinus_dan_kosinus&veaction=edit&section=14" title="Sunting bagian: Nilai-nilai spesial" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sinus_dan_kosinus&action=edit&section=14" title="Sunting kode sumber bagian: Nilai-nilai spesial"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r18844875"><div role="note" class="hatnote navigation-not-searchable">Lihat pula: <a href="/wiki/Fungsi_trigonometrik#Nilai_aljabar_sederhana" class="mw-redirect" title="Fungsi trigonometrik">Fungsi trigonometrik § Nilai aljabar sederhana</a></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Berkas:Unit_circle_angles.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Unit_circle_angles.svg/350px-Unit_circle_angles.svg.png" decoding="async" width="350" height="350" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Unit_circle_angles.svg/525px-Unit_circle_angles.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Unit_circle_angles.svg/700px-Unit_circle_angles.svg.png 2x" data-file-width="720" data-file-height="720" /></a><figcaption>Beberapa sudut yang umum (<i>θ</i>) tergambarkan pada <a href="/wiki/Lingkaran_satuan" title="Lingkaran satuan">lingkaran satuan</a>. Sudut-sudut dinyatakan dalam satuan derajat dan radian, beserta titik-titik perpotongan di lingkaran satuan dengan koordinat (cos(<i>θ</i>), sin(<i>θ</i>)).</figcaption></figure> <p>Untuk beberapa sudut <i>x</i> dalam satuan derajat, nilai dari fungsi sin(<i>x</i>) dan cos(<i>x</i>) dapat dinyatakan dengan sederhana. Berikut adalah tabel beberapa nilai sudut dan fungsi tersebut: </p> <table class="wikitable"> <tbody><tr> <th colspan="3">Sudut, <i>x</i> </th> <th colspan="2">sin(<i>x</i>) </th> <th colspan="2">cos(<i>x</i>) </th></tr> <tr> <th><a href="/wiki/Derajat_(satuan_sudut)" title="Derajat (satuan sudut)">Derajat</a> </th> <th><a href="/wiki/Radian" title="Radian">Radians</a> </th> <th><a href="/wiki/Putaran_(sudut)" title="Putaran (sudut)">Putaran</a> </th> <th>Eksak </th> <th>Desimal </th> <th>Eksak </th> <th>Desimal </th></tr> <tr> <td>0° </td> <td>0 </td> <td>0 </td> <td>0 </td> <td>0 </td> <td>1 </td> <td>1 </td></tr> <tr> <td>15° </td> <td><span class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;"><span style="display:block; line-height:1em; margin:0 0.1em;">1</span><span style="display:block; line-height:1em; margin:0 0.1em; border-top:1px solid;">12</span></span>π </td> <td><span class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;"><span style="display:block; line-height:1em; margin:0 0.1em;">1</span><span style="display:block; line-height:1em; margin:0 0.1em; border-top:1px solid;">24</span></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 {\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> <mn>4</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78c627f0ebe511a980e4f73e69210a111a36bda0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.873ex; height:5.843ex;" alt="{\displaystyle {\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}}"></span> </td> <td>0.2588 </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 {\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> <mn>4</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3b490d0765455c4d35af5b84445852edea3f2a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.873ex; height:5.843ex;" alt="{\displaystyle {\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}}"></span> </td> <td>0.9659 </td></tr> <tr> <td>30° </td> <td><span class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;"><span style="display:block; line-height:1em; margin:0 0.1em;">1</span><span style="display:block; line-height:1em; margin:0 0.1em; border-top:1px solid;">6</span></span>π </td> <td><span class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;"><span style="display:block; line-height:1em; margin:0 0.1em;">1</span><span style="display:block; line-height:1em; margin:0 0.1em; border-top:1px solid;">12</span></span> </td> <td><span class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;"><span style="display:block; line-height:1em; margin:0 0.1em;">1</span><span style="display:block; line-height:1em; margin:0 0.1em; border-top:1px solid;">2</span></span> </td> <td>0.5 </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 {\frac {\sqrt {3}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>3</mn> </msqrt> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\sqrt {3}}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4864a0c173339d1d88e89ca3c943f016744c879a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.934ex; height:5.843ex;" alt="{\displaystyle {\frac {\sqrt {3}}{2}}}"></span> </td> <td>0.8660 </td></tr> <tr> <td>45° </td> <td><span class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;"><span style="display:block; line-height:1em; margin:0 0.1em;">1</span><span style="display:block; line-height:1em; margin:0 0.1em; border-top:1px solid;">4</span></span>π </td> <td><span class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;"><span style="display:block; line-height:1em; margin:0 0.1em;">1</span><span style="display:block; line-height:1em; margin:0 0.1em; border-top:1px solid;">8</span></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 {\frac {\sqrt {2}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>2</mn> </msqrt> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\sqrt {2}}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fb9b5960bf5eae3065db9c23495e465f5fef61e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.934ex; height:5.843ex;" alt="{\displaystyle {\frac {\sqrt {2}}{2}}}"></span> </td> <td>0.7071 </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 {\frac {\sqrt {2}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>2</mn> </msqrt> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\sqrt {2}}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fb9b5960bf5eae3065db9c23495e465f5fef61e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.934ex; height:5.843ex;" alt="{\displaystyle {\frac {\sqrt {2}}{2}}}"></span> </td> <td>0.7071 </td></tr> <tr> <td>60° </td> <td><span class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;"><span style="display:block; line-height:1em; margin:0 0.1em;">1</span><span style="display:block; line-height:1em; margin:0 0.1em; border-top:1px solid;">3</span></span>π </td> <td><span class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;"><span style="display:block; line-height:1em; margin:0 0.1em;">1</span><span style="display:block; line-height:1em; margin:0 0.1em; border-top:1px solid;">6</span></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 {\frac {\sqrt {3}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>3</mn> </msqrt> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\sqrt {3}}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4864a0c173339d1d88e89ca3c943f016744c879a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.934ex; height:5.843ex;" alt="{\displaystyle {\frac {\sqrt {3}}{2}}}"></span> </td> <td>0.8660 </td> <td><span class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;"><span style="display:block; line-height:1em; margin:0 0.1em;">1</span><span style="display:block; line-height:1em; margin:0 0.1em; border-top:1px solid;">2</span></span> </td> <td>0.5 </td></tr> <tr> <td>75° </td> <td><span class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;"><span style="display:block; line-height:1em; margin:0 0.1em;">5</span><span style="display:block; line-height:1em; margin:0 0.1em; border-top:1px solid;">12</span></span>π </td> <td><span class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;"><span style="display:block; line-height:1em; margin:0 0.1em;">5</span><span style="display:block; line-height:1em; margin:0 0.1em; border-top:1px solid;">24</span></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 {\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> <mn>4</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3b490d0765455c4d35af5b84445852edea3f2a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.873ex; height:5.843ex;" alt="{\displaystyle {\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}}"></span> </td> <td>0.9659 </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 {\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> <mn>4</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78c627f0ebe511a980e4f73e69210a111a36bda0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.873ex; height:5.843ex;" alt="{\displaystyle {\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}}"></span> </td> <td>0.2588 </td></tr> <tr> <td>90° </td> <td><span class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;"><span style="display:block; line-height:1em; margin:0 0.1em;">1</span><span style="display:block; line-height:1em; margin:0 0.1em; border-top:1px solid;">2</span></span>π </td> <td><span class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;"><span style="display:block; line-height:1em; margin:0 0.1em;">1</span><span style="display:block; line-height:1em; margin:0 0.1em; border-top:1px solid;">4</span></span> </td> <td>1 </td> <td>1 </td> <td>0 </td> <td>0 </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Titik_tetap">Titik tetap</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sinus_dan_kosinus&veaction=edit&section=15" title="Sunting bagian: Titik tetap" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sinus_dan_kosinus&action=edit&section=15" title="Sunting kode sumber bagian: Titik tetap"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r18844875"><div role="note" class="hatnote navigation-not-searchable">Artikel utama: <a href="/w/index.php?title=Bilangan_Dottie&action=edit&redlink=1" class="new" title="Bilangan Dottie (halaman belum tersedia)">Bilangan Dottie</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Berkas:Cosine_fixed_point.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Cosine_fixed_point.svg/220px-Cosine_fixed_point.svg.png" decoding="async" width="220" height="176" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Cosine_fixed_point.svg/330px-Cosine_fixed_point.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Cosine_fixed_point.svg/440px-Cosine_fixed_point.svg.png 2x" data-file-width="600" data-file-height="480" /></a><figcaption>Iterasi titik tetap <i>x<sub>n</sub></i><sub>+1</sub> = cos(<i>x<sub>n</sub></i>) dengan nilai awal <i>x</i><sub>0</sub> = -1 akan konvergen ke bilangan Dottie.</figcaption></figure> <p>Nol adalah satu-satunya <a href="/w/index.php?title=Titik_tetap&action=edit&redlink=1" class="new" title="Titik tetap (halaman belum tersedia)">titik tetap</a> real yang dimiliki fungsi sinus; dengan kata lain, perpotongan antara fungsi sinus dan fungsi identitas <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 f(x)=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f690285952308aa49e3c6aac892df31cad6d1b06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.846ex; height:2.843ex;" alt="{\displaystyle f(x)=x}"></span> hanya terjadi di titik <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 x=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=0}"></span>. Satu-satunya titik tetap (real) dari fungsi kosinus adalah <a href="/w/index.php?title=Bilangan_Dottie&action=edit&redlink=1" class="new" title="Bilangan Dottie (halaman belum tersedia)">bilangan Dottie</a>. Dengan kata lain, bilangan Dottie adalah akar real unik dari persamaan <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(x)=x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(x)=x.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a70f9a16d7002842bae6c8f9475c9a2460ca61e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.325ex; height:2.843ex;" alt="{\displaystyle \cos(x)=x.}"></span> Ekspansi desimal dari bilangan Dottie adalah <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 0.739085...}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0.739085...</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0.739085...}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3670b51bd58cc0cf501926062e7b8adab043a7db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.725ex; height:2.176ex;" alt="{\displaystyle 0.739085...}"></span>.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Panjang_busur">Panjang busur</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sinus_dan_kosinus&veaction=edit&section=16" title="Sunting bagian: Panjang busur" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sinus_dan_kosinus&action=edit&section=16" title="Sunting kode sumber bagian: Panjang busur"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Panjang busur dari kurva sinus diantara <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 a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> dan <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 b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> adalah <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="{\textstyle \int _{a}^{b}\!{\sqrt {1+\cos ^{2}(x)}}\,dx={\sqrt {2}}(\operatorname {E} (b,1/{\sqrt {2}})-\operatorname {E} (a,1/{\sqrt {2}}))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>,</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \int _{a}^{b}\!{\sqrt {1+\cos ^{2}(x)}}\,dx={\sqrt {2}}(\operatorname {E} (b,1/{\sqrt {2}})-\operatorname {E} (a,1/{\sqrt {2}}))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00ee9825d510c7763117fe7f5d581957b8300333" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:51.788ex; height:3.676ex;" alt="{\textstyle \int _{a}^{b}\!{\sqrt {1+\cos ^{2}(x)}}\,dx={\sqrt {2}}(\operatorname {E} (b,1/{\sqrt {2}})-\operatorname {E} (a,1/{\sqrt {2}}))}"></span>, dengan <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 {E} (\varphi ,k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} (\varphi ,k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1520125a116a29984e7bca67df1f10ef35292b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.157ex; height:2.843ex;" alt="{\displaystyle \operatorname {E} (\varphi ,k)}"></span> menyatakan <a href="/w/index.php?title=Integral_eliptik&action=edit&redlink=1" class="new" title="Integral eliptik (halaman belum tersedia)">integral eliptik tak lengkap jenis kedua</a> (<i>incomplete elliptic integral of the second kind</i>) dengan modulus <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span>. </p><p>Panjang busur untuk satu periode penuh adalah <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="{\textstyle L=4{\sqrt {2\pi ^{3}}}/\Gamma (1/4)^{2}+\Gamma (1/4)^{2}/{\sqrt {2\pi }}=7.640395578\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mrow> <mo>=</mo> <mn>7.640395578</mn> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle L=4{\sqrt {2\pi ^{3}}}/\Gamma (1/4)^{2}+\Gamma (1/4)^{2}/{\sqrt {2\pi }}=7.640395578\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7e6e11ac10e88162284e1116b54485f53efdb63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:55.014ex; height:3.343ex;" alt="{\textstyle L=4{\sqrt {2\pi ^{3}}}/\Gamma (1/4)^{2}+\Gamma (1/4)^{2}/{\sqrt {2\pi }}=7.640395578\ldots }"></span> dengan <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 \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }"></span> menyatakan <a href="/wiki/Fungsi_gamma" title="Fungsi gamma">fungsi gamma</a>. Solusi persamaan ini dapat dihitung dengan cepat menggunakan <a href="/w/index.php?title=Rataan_arimetik-geometrik&action=edit&redlink=1" class="new" title="Rataan arimetik-geometrik (halaman belum tersedia)">rataan arimetik-geometrik</a>: <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="{\textstyle L=2\operatorname {M} (1,{\sqrt {2}})+2\pi /\operatorname {M} (1,{\sqrt {2}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mn>2</mn> <mi mathvariant="normal">M</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">M</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle L=2\operatorname {M} (1,{\sqrt {2}})+2\pi /\operatorname {M} (1,{\sqrt {2}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11c6cdc6799afbad0006ad01492c3d37c965e60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.585ex; height:3.176ex;" alt="{\textstyle L=2\operatorname {M} (1,{\sqrt {2}})+2\pi /\operatorname {M} (1,{\sqrt {2}})}"></span>.<sup id="cite_ref-ams_9-0" class="reference"><a href="#cite_note-ams-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Lebih lanjut, <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 L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> sebenarnya adalah keliling dari elips dengan panjang sumbu semi-major sebesar <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 {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"></span> dan panjang sumbu semi-minor sebesar <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 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span>.<sup id="cite_ref-ams_9-1" class="reference"><a href="#cite_note-ams-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>Panjang busur dari kurva sinus dari <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 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span> sampai <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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> adalah <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 Lx/(2\pi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Lx/(2\pi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4155019040aff7240ebdbf18272142080ff7022b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.379ex; height:2.843ex;" alt="{\displaystyle Lx/(2\pi )}"></span>, ditambah suatu faktor koreksi yang bernilai periodik dalam <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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> dan memiliki periode <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 \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"></span>. <a href="/wiki/Deret_Fourier" title="Deret Fourier">Deret Fourier</a> dari faktor koreksi ini dapat dituliskan dalam bentuk tertutup dengan menggunakan sutu fungsi spesial. Panjang busur kurva sinus dari <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 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span> sampai <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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> adalah<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {\operatorname {M} (1,{\sqrt {2}})}{\pi }}+{\frac {1}{\operatorname {M} (1,{\sqrt {2}})}}\right)x+{\sqrt {2}}\sum _{n=1}^{\infty }{\frac {(-1)^{n}{\tbinom {n-3/2}{n}}}{2^{3n}n}}{}_{2}F_{1}\left(n-{\frac {1}{2}},n+{\frac {1}{2}};2n+1;{\frac {1}{2}}\right)\sin(2nx),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">M</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> </mrow> <mi>π</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi mathvariant="normal">M</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow 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> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>−</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> </mrow> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mi>n</mi> </mrow> </msup> <mi>n</mi> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mi>n</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>;</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>;</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {\operatorname {M} (1,{\sqrt {2}})}{\pi }}+{\frac {1}{\operatorname {M} (1,{\sqrt {2}})}}\right)x+{\sqrt {2}}\sum _{n=1}^{\infty }{\frac {(-1)^{n}{\tbinom {n-3/2}{n}}}{2^{3n}n}}{}_{2}F_{1}\left(n-{\frac {1}{2}},n+{\frac {1}{2}};2n+1;{\frac {1}{2}}\right)\sin(2nx),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02d1fdc6357baec5cc32bf018a7713d33d475261" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:93.269ex; height:7.676ex;" alt="{\displaystyle \left({\frac {\operatorname {M} (1,{\sqrt {2}})}{\pi }}+{\frac {1}{\operatorname {M} (1,{\sqrt {2}})}}\right)x+{\sqrt {2}}\sum _{n=1}^{\infty }{\frac {(-1)^{n}{\tbinom {n-3/2}{n}}}{2^{3n}n}}{}_{2}F_{1}\left(n-{\frac {1}{2}},n+{\frac {1}{2}};2n+1;{\frac {1}{2}}\right)\sin(2nx),}"></span>dengan <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 {}_{2}F_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}_{2}F_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7506f01afc6742d7168ab0cb201337ca2a216ea5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.603ex; height:2.509ex;" alt="{\displaystyle {}_{2}F_{1}}"></span> menyatakan <a href="/wiki/Fungsi_hipergeometris" title="Fungsi hipergeometris">fungsi hipergeometrik</a>. Suku-suku pada ekspresi di atas dapat diaproksimasi dengan<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1.21600672x+0.10317093\sin(2x)-0.00220445\sin(4x)+0.00012584\sin(6x)-0.00001011\sin(8x)+\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1.21600672</mn> <mi>x</mi> <mo>+</mo> <mn>0.10317093</mn> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>0.00220445</mn> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>4</mn> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>0.00012584</mn> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>6</mn> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>0.00001011</mn> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>8</mn> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1.21600672x+0.10317093\sin(2x)-0.00220445\sin(4x)+0.00012584\sin(6x)-0.00001011\sin(8x)+\cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbd704803b904454bb716555b11093348e938295" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:103.976ex; height:2.843ex;" alt="{\displaystyle 1.21600672x+0.10317093\sin(2x)-0.00220445\sin(4x)+0.00012584\sin(6x)-0.00001011\sin(8x)+\cdots }"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Hubungan_dengan_bilangan_kompleks">Hubungan dengan bilangan kompleks</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sinus_dan_kosinus&veaction=edit&section=17" title="Sunting bagian: Hubungan dengan bilangan kompleks" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sinus_dan_kosinus&action=edit&section=17" title="Sunting kode sumber bagian: Hubungan dengan bilangan kompleks"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r18844875"><div role="note" class="hatnote navigation-not-searchable">Artikel utama: <a href="/wiki/Fungsi_trigonometrik#Hubungan_dengan_fungsi_eksponensial_(Rumus_Euler)" class="mw-redirect" title="Fungsi trigonometrik">Fungsi trigonometrik § Hubungan dengan fungsi eksponensial (Rumus Euler)</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Berkas:Sinus_und_Kosinus_am_Einheitskreis_3.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Sinus_und_Kosinus_am_Einheitskreis_3.svg/220px-Sinus_und_Kosinus_am_Einheitskreis_3.svg.png" decoding="async" width="220" height="215" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Sinus_und_Kosinus_am_Einheitskreis_3.svg/330px-Sinus_und_Kosinus_am_Einheitskreis_3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Sinus_und_Kosinus_am_Einheitskreis_3.svg/440px-Sinus_und_Kosinus_am_Einheitskreis_3.svg.png 2x" data-file-width="418" data-file-height="409" /></a><figcaption><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(\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aaac7b75cda6d5570780075aa2622d27b21117cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.011ex; height:2.843ex;" alt="{\displaystyle \cos(\theta )}"></span> dan <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 )}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acafc444aea85d63a40dabf84f035a6b4955a948" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.755ex; height:2.843ex;" alt="{\displaystyle \sin(\theta )}"></span> secara berurutan adalah nilai dari bagian real dan bagian imajiner dari <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 }}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{i\theta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b4b8e67ee479d68e0e5040aaf87eff99214c90f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.654ex; height:2.676ex;" alt="{\displaystyle e^{i\theta }}"></span>.</figcaption></figure> <p>Sinus dan kosinus digunakan untuk menggabungkan bagian real dan bagian imajiner dari suatu <a href="/wiki/Bilangan_kompleks" title="Bilangan kompleks">bilangan kompleks</a> dengan bentuk <a href="/w/index.php?title=Koordinat_polar&action=edit&redlink=1" class="new" title="Koordinat polar (halaman belum tersedia)">koordinat polar</a> (<i>r</i>, <i>φ</i>):<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=r(\cos(\varphi )+i\sin(\varphi ))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>r</mi> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=r(\cos(\varphi )+i\sin(\varphi ))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14700b82a5aaccf19b79afba82f911a196c85546" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.7ex; height:2.843ex;" alt="{\displaystyle z=r(\cos(\varphi )+i\sin(\varphi ))}"></span>Bagian real dan bagian imajiner dari bilangan kompleks tersebut adalah:<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Re} (z)=r\cos(\varphi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>r</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Re} (z)=r\cos(\varphi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49186c9491824a31134efbb28ce8c39678255751" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.615ex; height:2.843ex;" alt="{\displaystyle \operatorname {Re} (z)=r\cos(\varphi )}"></span>dengan <i>r</i> dan <i>φ</i> menyatakan magnitudo dan sudut dari bilangan kompleks <i>z</i>. </p><p>Untuk sembarang <a href="/wiki/Bilangan_real" class="mw-redirect" title="Bilangan real">bilangan real</a> <i>θ</i>, <a href="/wiki/Rumus_Euler" title="Rumus Euler">rumus Euler</a> menyatakan bahwa:<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{i\theta }=\cos(\theta )+i\sin(\theta )}"> <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>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{i\theta }=\cos(\theta )+i\sin(\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73c8c9fd86d0c91988ada4114e91df800f9dbbcc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.549ex; height:3.176ex;" alt="{\displaystyle e^{i\theta }=\cos(\theta )+i\sin(\theta )}"></span>Dengan demikian, jika koordinat polar dari <i>z</i> adalah (<i>r</i>, <i>φ</i>), maka <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 z=re^{i\varphi }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>r</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>φ</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=re^{i\varphi }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a087c772212e7375cb321d83fc1fcc715cd0ed2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.84ex; height:2.676ex;" alt="{\displaystyle z=re^{i\varphi }.}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Argumen_kompleks">Argumen kompleks</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sinus_dan_kosinus&veaction=edit&section=18" title="Sunting bagian: Argumen kompleks" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sinus_dan_kosinus&action=edit&section=18" title="Sunting kode sumber bagian: Argumen kompleks"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Berkas:Complex_sin.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/45/Complex_sin.jpg/220px-Complex_sin.jpg" decoding="async" width="220" height="251" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/45/Complex_sin.jpg/330px-Complex_sin.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/45/Complex_sin.jpg/440px-Complex_sin.jpg 2x" data-file-width="944" data-file-height="1079" /></a><figcaption><a href="/w/index.php?title=Pewarnaan_domain&action=edit&redlink=1" class="new" title="Pewarnaan domain (halaman belum tersedia)">Pewarnaan domain</a> dari fungsi sin(<i>z</i>) pada bidang kompleks. Tingkat kecerahan menyatakan magnitudo, sedangkan hue menyatakan argumen kompleks.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Berkas:Sin_z_vector_field_02_Pengo.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Sin_z_vector_field_02_Pengo.svg/220px-Sin_z_vector_field_02_Pengo.svg.png" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Sin_z_vector_field_02_Pengo.svg/330px-Sin_z_vector_field_02_Pengo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/96/Sin_z_vector_field_02_Pengo.svg/440px-Sin_z_vector_field_02_Pengo.svg.png 2x" data-file-width="720" data-file-height="540" /></a><figcaption>sin(<i>z</i>) sebagai suatu medan vektor.</figcaption></figure> <p>Dengan menerapkan definisi deret dari fungsi sinus dan kosinus ke suatu argumen kompleks <i>z</i>, didapatkan hubungan: </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}\sin(z)&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}z^{2n+1}\\&={\frac {e^{iz}-e^{-iz}}{2i}}\\&={\frac {\sinh \left(iz\right)}{i}}\\&=-i\sinh \left(iz\right)\\\cos(z)&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}z^{2n}\\&={\frac {e^{iz}+e^{-iz}}{2}}\\&=\cosh(iz)\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></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> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>z</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>z</mi> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>i</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sinh</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mi>z</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mi>i</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>i</mi> <mi>sinh</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mi>z</mi> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></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> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>z</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>z</mi> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>cosh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>i</mi> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sin(z)&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}z^{2n+1}\\&={\frac {e^{iz}-e^{-iz}}{2i}}\\&={\frac {\sinh \left(iz\right)}{i}}\\&=-i\sinh \left(iz\right)\\\cos(z)&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}z^{2n}\\&={\frac {e^{iz}+e^{-iz}}{2}}\\&=\cosh(iz)\\\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77ff8035628f5a90a7aced4bb8fb5b4a2b0c3a52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -18.072ex; margin-bottom: -0.266ex; width:28.684ex; height:37.843ex;" alt="{\displaystyle {\begin{aligned}\sin(z)&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}z^{2n+1}\\&={\frac {e^{iz}-e^{-iz}}{2i}}\\&={\frac {\sinh \left(iz\right)}{i}}\\&=-i\sinh \left(iz\right)\\\cos(z)&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}z^{2n}\\&={\frac {e^{iz}+e^{-iz}}{2}}\\&=\cosh(iz)\\\end{aligned}}}"></span></dd></dl> <p>dengan sinh dand cosh adalah <a href="/w/index.php?title=Sinus_dan_cosinus_hiperbolik&action=edit&redlink=1" class="new" title="Sinus dan cosinus hiperbolik (halaman belum tersedia)">sinus dan cosinus hiperbolik</a>. Kedua fungsi ini merupakan <a href="/w/index.php?title=Fungsi_entire&action=edit&redlink=1" class="new" title="Fungsi entire (halaman belum tersedia)">fungsi entire</a>. </p><p>Terkadang fungsi sinus dan kosinus kompleks lebih cocok dinyatakan dalam bentuk bagian real dan bagian imajiner. Pada kasus ini, persamaan-persamaan berikut dapat digunakan: </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}\sin(x+iy)&=\sin(x)\cos(iy)+\cos(x)\sin(iy)\\&=\sin(x)\cosh(y)+i\cos(x)\sinh(y)\\\cos(x+iy)&=\cos(x)\cos(iy)-\sin(x)\sin(iy)\\&=\cos(x)\cosh(y)-i\sin(x)\sinh(y)\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>i</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>i</mi> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>cosh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>i</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>sinh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>i</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>i</mi> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>cosh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>sinh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sin(x+iy)&=\sin(x)\cos(iy)+\cos(x)\sin(iy)\\&=\sin(x)\cosh(y)+i\cos(x)\sinh(y)\\\cos(x+iy)&=\cos(x)\cos(iy)-\sin(x)\sin(iy)\\&=\cos(x)\cosh(y)-i\sin(x)\sinh(y)\\\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22162f3c99776ef5203c9c7fc8d418162bdcfd99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:46.429ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}\sin(x+iy)&=\sin(x)\cos(iy)+\cos(x)\sin(iy)\\&=\sin(x)\cosh(y)+i\cos(x)\sinh(y)\\\cos(x+iy)&=\cos(x)\cos(iy)-\sin(x)\sin(iy)\\&=\cos(x)\cosh(y)-i\sin(x)\sinh(y)\\\end{aligned}}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Grafik_kompleks">Grafik kompleks</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sinus_dan_kosinus&veaction=edit&section=19" title="Sunting bagian: Grafik kompleks" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sinus_dan_kosinus&action=edit&section=19" title="Sunting kode sumber bagian: Grafik kompleks"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <table> <caption><b>Fungsi sinus pada bidang kompleks</b> </caption> <tbody><tr> <td><figure class="mw-halign-none" typeof="mw:File"><a href="/wiki/Berkas:Complex_sin_real_01_Pengo.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Complex_sin_real_01_Pengo.svg/173px-Complex_sin_real_01_Pengo.svg.png" decoding="async" width="173" height="130" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Complex_sin_real_01_Pengo.svg/260px-Complex_sin_real_01_Pengo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Complex_sin_real_01_Pengo.svg/346px-Complex_sin_real_01_Pengo.svg.png 2x" data-file-width="720" data-file-height="540" /></a><figcaption></figcaption></figure> </td> <td><figure class="mw-halign-none" typeof="mw:File"><a href="/wiki/Berkas:Complex_sin_imag_01_Pengo.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Complex_sin_imag_01_Pengo.svg/173px-Complex_sin_imag_01_Pengo.svg.png" decoding="async" width="173" height="130" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Complex_sin_imag_01_Pengo.svg/260px-Complex_sin_imag_01_Pengo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Complex_sin_imag_01_Pengo.svg/346px-Complex_sin_imag_01_Pengo.svg.png 2x" data-file-width="720" data-file-height="540" /></a><figcaption></figcaption></figure> </td> <td><figure class="mw-halign-none" typeof="mw:File"><a href="/wiki/Berkas:Complex_sin_abs_01_Pengo.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Complex_sin_abs_01_Pengo.svg/173px-Complex_sin_abs_01_Pengo.svg.png" decoding="async" width="173" height="130" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Complex_sin_abs_01_Pengo.svg/260px-Complex_sin_abs_01_Pengo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Complex_sin_abs_01_Pengo.svg/346px-Complex_sin_abs_01_Pengo.svg.png 2x" data-file-width="720" data-file-height="540" /></a><figcaption></figcaption></figure> </td></tr> <tr> <td>komponen real </td> <td>komponen imajiner </td> <td>magnitudo </td></tr></tbody></table> <table> <caption><b>Fungsi arcsinus pada bidang kompleks</b> </caption> <tbody><tr> <td><figure class="mw-halign-none" typeof="mw:File"><a href="/wiki/Berkas:Complex_arcsin_real_01_Pengo.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Complex_arcsin_real_01_Pengo.svg/173px-Complex_arcsin_real_01_Pengo.svg.png" decoding="async" width="173" height="130" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Complex_arcsin_real_01_Pengo.svg/260px-Complex_arcsin_real_01_Pengo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Complex_arcsin_real_01_Pengo.svg/346px-Complex_arcsin_real_01_Pengo.svg.png 2x" data-file-width="720" data-file-height="540" /></a><figcaption></figcaption></figure> </td> <td><figure class="mw-halign-none" typeof="mw:File"><a href="/wiki/Berkas:Complex_arcsin_imag_01_Pengo.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Complex_arcsin_imag_01_Pengo.svg/173px-Complex_arcsin_imag_01_Pengo.svg.png" decoding="async" width="173" height="130" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Complex_arcsin_imag_01_Pengo.svg/260px-Complex_arcsin_imag_01_Pengo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Complex_arcsin_imag_01_Pengo.svg/346px-Complex_arcsin_imag_01_Pengo.svg.png 2x" data-file-width="720" data-file-height="540" /></a><figcaption></figcaption></figure> </td> <td><figure class="mw-halign-none" typeof="mw:File"><a href="/wiki/Berkas:Complex_arcsin_abs_01_Pengo.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Complex_arcsin_abs_01_Pengo.svg/173px-Complex_arcsin_abs_01_Pengo.svg.png" decoding="async" width="173" height="130" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Complex_arcsin_abs_01_Pengo.svg/260px-Complex_arcsin_abs_01_Pengo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Complex_arcsin_abs_01_Pengo.svg/346px-Complex_arcsin_abs_01_Pengo.svg.png 2x" data-file-width="720" data-file-height="540" /></a><figcaption></figcaption></figure> </td></tr> <tr> <td>komponen real </td> <td>komponen imajiner </td> <td>magnitudo </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Sejarah">Sejarah</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sinus_dan_kosinus&veaction=edit&section=20" title="Sunting bagian: Sejarah" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sinus_dan_kosinus&action=edit&section=20" title="Sunting kode sumber bagian: Sejarah"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r18844875"><div role="note" class="hatnote navigation-not-searchable">Artikel utama: <a href="/w/index.php?title=Trigonometric_functions&action=edit&redlink=1" class="new" title="Trigonometric functions (halaman belum tersedia)">Trigonometric functions § History</a>, dan <a href="/w/index.php?title=History_of_trigonometry&action=edit&redlink=1" class="new" title="History of trigonometry (halaman belum tersedia)">History of trigonometry</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Berkas:Khalili_Collection_Islamic_Art_sci_0040.1_CROP.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Khalili_Collection_Islamic_Art_sci_0040.1_CROP.jpg/220px-Khalili_Collection_Islamic_Art_sci_0040.1_CROP.jpg" decoding="async" width="220" height="166" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Khalili_Collection_Islamic_Art_sci_0040.1_CROP.jpg/330px-Khalili_Collection_Islamic_Art_sci_0040.1_CROP.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Khalili_Collection_Islamic_Art_sci_0040.1_CROP.jpg/440px-Khalili_Collection_Islamic_Art_sci_0040.1_CROP.jpg 2x" data-file-width="2415" data-file-height="1821" /></a><figcaption>Quadrant from 1840s <a href="/wiki/Ottoman_Empire" class="mw-redirect" title="Ottoman Empire">Ottoman Turkey</a> with axes for looking up the sine and <a href="/w/index.php?title=Versine&action=edit&redlink=1" class="new" title="Versine (halaman belum tersedia)">versine</a> of angles</figcaption></figure> <p>Walaupun ilmu trigonometri dapat dilacak jauh ke masa lalu, penggunaan <a href="/wiki/Fungsi_trigonometri" title="Fungsi trigonometri">fungsi trigonometri</a> seperti yang digunakan saat ini dikembangkan pada zaman pertengahan. Fungsi <a href="/w/index.php?title=Busur_(geometri)&action=edit&redlink=1" class="new" title="Busur (geometri) (halaman belum tersedia)">busur</a> ditemukan oleh <a href="/wiki/Hipparchus" class="mw-redirect" title="Hipparchus">Hipparchus</a> dari <a href="/wiki/%C4%B0znik" class="mw-redirect" title="İznik">Nicaea</a> (180–125 SM) dan <a href="/wiki/Ptolemy" class="mw-redirect" title="Ptolemy">Ptolemy</a> dari <a href="/wiki/Mesir_Romawi" class="mw-redirect" title="Mesir Romawi">Mesir Romawi</a> (90–165 M). Secara spesifik, lihat <a href="/w/index.php?title=Tabel_busur_Ptolemy&action=edit&redlink=1" class="new" title="Tabel busur Ptolemy (halaman belum tersedia)">tabel busur Ptolemy</a>. </p><p>Fungsi sinus dan versine (1 − kosinus) dapat dilacak ke fungsi <a href="/w/index.php?title=Jy%C4%81,_koti-jy%C4%81_dan_utkrama-jy%C4%81&action=edit&redlink=1" class="new" title="Jyā, koti-jyā dan utkrama-jyā (halaman belum tersedia)"><i>jyā</i> dan <i>koṭi-jyā</i></a> yang digunakan pada bidang <a href="/wiki/Astronomi_India" title="Astronomi India">astronomi India</a> (<i><a href="/w/index.php?title=Aryabhatiya&action=edit&redlink=1" class="new" title="Aryabhatiya (halaman belum tersedia)">Aryabhatiya</a></i>, <i><a href="/wiki/Surya_Siddhanta" title="Surya Siddhanta">Surya Siddhanta</a></i>) di masa <a href="/wiki/Kemaharajaan_Gupta" title="Kemaharajaan Gupta">kemaharajaan Gupta</a> (320-550 M). Fungsi ini selanjutnya mengalami penerjemahan dari bahasa Sanskerta ke bahasa Arab, dilanjutkan dari bahasa Arab ke bahasa Latin.<sup id="cite_ref-Boyer,_Carl_B._1991_p._210_2-1" class="reference"><a href="#cite_note-Boyer,_Carl_B._1991_p._210-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Keenam fungsi trigonometrik yang digunakan saat ini sudah dikenal di masa <a href="/wiki/Matematika_Islam" title="Matematika Islam">matematika Islam</a> pada abas ke-9, termasuk <a href="/wiki/Hukum_sinus" class="mw-redirect" title="Hukum sinus">hukum sinus</a> yang digunakan untuk menyelesaikan permasalahan yang berkaitan dengan segitiga.<sup id="cite_ref-Gingerich_1986_11-0" class="reference"><a href="#cite_note-Gingerich_1986-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> Dengan pengecualian sinus, kelima fungsi trigonometrik lain ditemukan oleh matematikawan Arab; yakni fungsi kosinus, tangen, kotangen, sekan, dan kosekan.<sup id="cite_ref-Gingerich_1986_11-1" class="reference"><a href="#cite_note-Gingerich_1986-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Mu%E1%B8%A5ammad_bin_M%C5%ABs%C4%81_al-Khaw%C4%81rizm%C4%AB" class="mw-redirect" title="Muḥammad bin Mūsā al-Khawārizmī">Al-Khwārizmī</a> (sekitar 780–850) membentuk tabel nilai sinus, kosinus, dan tangen.<sup id="cite_ref-Sesiano_12-0" class="reference"><a href="#cite_note-Sesiano-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Britannica_13-0" class="reference"><a href="#cite_note-Britannica-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Al-Battani" title="Al-Battani">Muhammad ibn Jābir al-Harrānī al-Battānī</a> (853–929) menemukan fungsi invers dari sekan dan kosekan, dan membentuk tabel kosekan pertama kalinya untuk setiap sudut dari 1° sampai 90°.<sup id="cite_ref-Britannica_13-1" class="reference"><a href="#cite_note-Britannica-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p><p>Publikasi pertama yang menggunakan singkatan 'sin', 'cos', dan 'tan' adalah oleh matematikawan Prancis <a href="/w/index.php?title=Albert_Girard&action=edit&redlink=1" class="new" title="Albert Girard (halaman belum tersedia)">Albert Girard</a>; singkatan ini selanjutnya disebarluaskan oleh Euler (lihat bagian dibawah). Tulisan <i>Opus palatinum de triangulis</i> oleh <a href="/w/index.php?title=Georg_Joachim_Rheticus&action=edit&redlink=1" class="new" title="Georg Joachim Rheticus (halaman belum tersedia)">Georg Joachim Rheticus</a>, seorang siswa <a href="/wiki/Copernicus" class="mw-redirect" title="Copernicus">Copernicus</a>, mungkin adalah yang pertama di Eropa, yang mendefinisikan fungsi-fungsi trigonometri langsung dari segitiga siku-siku ketimbang menggunakan lingkaran. Tulisan ini juga mengikutkan tabel nilai untuk keenam fungsi trigonometrik; tulisan ini diselesaikan oleh seorang siswanya, Valentin Otho, pada tahun 1596. </p><p>Dalam suatu makalah yang diterbitkan pada tahun 1682, <a href="/wiki/Gottfried_Leibniz" title="Gottfried Leibniz">Leibniz</a> membuktikan bahwa sin <i>x</i> bukanlah suatu <a href="/wiki/Fungsi_aljabar" title="Fungsi aljabar">fungsi aljabar</a> dari <i>x</i>.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Roger_Cotes" title="Roger Cotes">Roger Cotes</a> menghitung turunan dari sinus dalam karyanya <i>Harmonia Mensurarum</i> (1722).<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> Karya <i>Introductio in analysin infinitorum</i> (1748) oleh <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> adalah yang paling signifikan di Eropa, dalam memberikan dasar analitik mengenai fungsi-fungsi trigonometrik. Dalam karya ini, fungsi-fungsi dinyatakan sebagai deret tak hingga, menyajikan "<a href="/wiki/Rumus_Euler" title="Rumus Euler">rumus Euler</a>", dan penggunaan singkatan <i>sin., cos., tang., cot., sec.,</i> dan <i>cosec</i> yang mirip dengan penggunaan saat ini<i>.</i><sup id="cite_ref-boyer_16-0" class="reference"><a href="#cite_note-boyer-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Etimologi">Etimologi</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sinus_dan_kosinus&veaction=edit&section=21" title="Sunting bagian: Etimologi" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sinus_dan_kosinus&action=edit&section=21" title="Sunting kode sumber bagian: Etimologi"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Secara <a href="/wiki/Etimologi" title="Etimologi">etimologi</a>, kata <i>sinus</i> berasal dari kata <a href="/wiki/Sanskerta" class="mw-redirect" title="Sanskerta">Sanskerta</a> untuk busur, <i>jiva</i>* (<i>jya</i> adalah sinonim yang lebih populer). Kata ini mengalami <a href="/wiki/Alih_aksara" title="Alih aksara">transliterasi</a> ke <a href="/wiki/Bahasa_Arab" title="Bahasa Arab">bahasa Arab</a> sebagai <i>jiba</i> جيب, yang sebenarnya tidak memiliki arti dari bahasa Arab, dan disingkat sebagai <i>jb</i> جب . Karena bahasa Arab ditulis tanpa menggunakan vokal pendek, "jb" secara keliru diintepretasikan sebagai kata <i>jaib</i> جيب, yang berarti "dada". Ketika teks Arab diterjemahkan ke bahasa Latin oleh <a href="/w/index.php?title=Gerard_of_Cremona&action=edit&redlink=1" class="new" title="Gerard of Cremona (halaman belum tersedia)">Gerard of Cremona</a> pada abad ke-12, ia menggunakan kata yang setara untuk "dada", <i><a href="https://id.wiktionary.org/wiki/sinus" class="extiw" title="wikt:sinus">sinus</a></i> (yang berarti "dada" atau "teluk" atau "lipatan").<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> Gerard mungkin bukan yang pertama kali menggunakan terjemahan ini; Robert dari Chester sepertinya terlebih dahulu melakukannya dan terdapat bukti penggunaan pada masa yang lebih lawas.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> Bentuk bahasa Inggris <i>sine</i> diperkenalkan pada tahun 1590-an. Kata "cosine" berasal dari singkatan bahasa Latin masa pertengahan untuk frasa "complementi sinus".<sup id="cite_ref-cosine_4-1" class="reference"><a href="#cite_note-cosine-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Implementasi_perangkat_lunak">Implementasi perangkat lunak</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sinus_dan_kosinus&veaction=edit&section=22" title="Sunting bagian: Implementasi perangkat lunak" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sinus_dan_kosinus&action=edit&section=22" title="Sunting kode sumber bagian: Implementasi perangkat lunak"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Tidak ada standardisasi algoritma untuk menghitung fungsi sinus dan kosinus. <a href="/w/index.php?title=IEEE_754-2008&action=edit&redlink=1" class="new" title="IEEE 754-2008 (halaman belum tersedia)">IEEE 754-2008</a>, standar paling umum digunakan untuk komputasi <a href="/wiki/Titik_kambang" class="mw-redirect" title="Titik kambang">titik-kambang</a> (<i>floating-point</i>), tidak membahas cara menghitung fungsi trigonometri seperti sinus.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> Algoritma yang dikembangkan untuk menghitung sinus dapat disesuaikan dengan menimbang aspek kecepatan, akurasi, portabilitas, maupun jangkauan input yang dapat diproses. Hal ini menyebabkan munculnya banyak algoritma berbeda, khususnya untuk menyelesaikan kasus khusus seperti menghitung input yang sangat besar, seperti <code>sin(10<sup>22</sup>)</code>. </p><p>Optimisasi pemrograman yang umum, digunakan khususnya dalam grafik 3D, adalah dengan menghitung tabel nilai sinus terlebih dahulu, misalnya satu nilai untuk setiap derajat, lalu melakukan <a href="/wiki/Interpolasi_(matematika)" title="Interpolasi (matematika)">interpolasi linear</a> dengan menggunakan dua nilai derajat yang paling dekat dengan nilai input. Hal ini memungkinkan hasil ditentukan dari pencarian tabel ketimbang melakukan perhitungan secara <i>real-time</i>. Namun pada arsitektur <a href="/wiki/Unit_Pemroses_Sentral" title="Unit Pemroses Sentral">CPU</a> modern, metode ini mungkin tidak memberikan keuntungan yang berarti.<sup class="noprint Inline-Template"><span title="Kalimat yang diikuti tag ini membutuhkan rujukan. since October 2012" style="white-space: nowrap;">[<i><a href="/wiki/Wikipedia:Kutip_sumber_tulisan" title="Wikipedia:Kutip sumber tulisan">butuh rujukan</a></i>]</span></sup> Beberapa arsitektur CPU memiliki instruksi dasar untuk menghitung sinus, contohnya Intel x87 FPUs since the 80387. </p><p>Fungsi sinus dan kosinus, beserta fungsi-fungsi trigonometrik lainnya, umum tersedia pada berbagai bahasa pemrograman dan platform. Dalam komputasi, mereka umumnya dikenal sebagai <code>sin</code> dan <code>cos</code>. Pada bahasa pemrograman, <code>sin</code> dan <code>cos</code> umumnya merupakan fungsi dasar (<i>built-in</i>) atau dapat dijumpai di <a href="/wiki/Pustaka_(perangkat_lunak)" title="Pustaka (perangkat lunak)">pustaka</a> matematika standar. Sebagai contoh, <a href="/w/index.php?title=Pustaka_standar_C&action=edit&redlink=1" class="new" title="Pustaka standar C (halaman belum tersedia)">pustaka standar C</a> mendefinisikan fungsi sinus di dalam pustaka <code>math.h</code> sebagai <code>sin(<a href="/w/index.php?title=Double-precision_floating-point_format&action=edit&redlink=1" class="new" title="Double-precision floating-point format (halaman belum tersedia)">double</a>)</code>, <code>sinf(<a href="/w/index.php?title=Single-precision_floating-point_format&action=edit&redlink=1" class="new" title="Single-precision floating-point format (halaman belum tersedia)">float</a>)</code>, dan <code>sinl(<a href="/w/index.php?title=Long_double&action=edit&redlink=1" class="new" title="Long double (halaman belum tersedia)">long double</a>)</code>. Parameter untuk setiap fungsi ini adalah nilai <a href="/wiki/Titik_kambang" class="mw-redirect" title="Titik kambang">titik-kambang</a>, dalam satuan radian. Setiap fungsi akan menghasilkan <a href="/wiki/Tipe_data" title="Tipe data">tipe data</a> yang sama dengan inputnya. Banyak fungsi trigonometrik lainnya juga didefinisikan di <code>math.h</code>, seperti kosinus, arcsinus, dan sinus hiperbolik (sinh). Mirip dengan itu, <a href="/wiki/Python_(programming_language)" class="mw-redirect" title="Python (programming language)">Python</a> mendefinisikan <code>math.sin(x)</code> dan <code>math.cos(x)</code> yang termuat dalam modul <code>math</code>. Fungsi sinus dan kosinus kompleks tersedia dalam modul <code>cmath</code>, contohnya <code>cmath.sin(z)</code> . </p> <div class="mw-heading mw-heading3"><h3 id="Implementasi_berdasarkan_satuan_putaran">Implementasi berdasarkan satuan putaran</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sinus_dan_kosinus&veaction=edit&section=23" title="Sunting bagian: Implementasi berdasarkan satuan putaran" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sinus_dan_kosinus&action=edit&section=23" title="Sunting kode sumber bagian: Implementasi berdasarkan satuan putaran"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Beberapa pustaka perangkat lunak memiliki implementasi sinus dan kosinus menggunakan sudut yang dinyatakan dalam setengah-<a href="/wiki/Putaran_(sudut)" title="Putaran (sudut)">putaran</a>; nilai dari setengah-putaran adalah sudut sebesar 180 derajat atau <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 \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"></span> radian. Menyatakan sudut dalam satuan putaran atau setengah-putaran memiliki keuntungan akurasi dan efisiensi pada beberapa kasus.<sup id="cite_ref-matlab_21-0" class="reference"><a href="#cite_note-matlab-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-r_22-0" class="reference"><a href="#cite_note-r-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> Pada <a href="/wiki/MATLAB" title="MATLAB">MATLAB</a>, <a href="/w/index.php?title=OpenCL&action=edit&redlink=1" class="new" title="OpenCL (halaman belum tersedia)">OpenCL</a>, <a href="/wiki/R_(bahasa_pemrograman)" title="R (bahasa pemrograman)">R</a>, <a href="/w/index.php?title=Julia_(bahasa_pemrograman)&action=edit&redlink=1" class="new" title="Julia (bahasa pemrograman) (halaman belum tersedia)">Julia</a>, <a href="/w/index.php?title=CUDA&action=edit&redlink=1" class="new" title="CUDA (halaman belum tersedia)">CUDA</a>, dan <a href="/wiki/ARM" class="mw-redirect" title="ARM">ARM</a>, fungsi-fungsi ini disebut dengan <code>sinpi</code> dan <code>cospi</code>.<sup id="cite_ref-matlab_21-1" class="reference"><a href="#cite_note-matlab-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-r_22-1" class="reference"><a href="#cite_note-r-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> Sebagai contoh, <code>sinpi(x)</code> akan mengevaluasi nilai to <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(\pi x),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(\pi x),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4f4681ee034a3d894ce28068c5f6bacfa86e8c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.973ex; height:2.843ex;" alt="{\displaystyle \sin(\pi x),}"></span> dengan <i>x</i> dinyatakan dalam satuan radian. </p><p>Keuntungan akurasi timbul dari kemampuan merepresentasikan sudut-sudut penting, seperti satu putaran penuh, setengah putaran, dan seperempat putaran, dengan sempurna (<i>losslessly</i>) dalam bentuk titik-kambang (<i>floating-point</i>) biner atau titik-tetap (<i>fixed-point</i>). Sebagai kontras, menyatakan <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 2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73efd1f6493490b058097060a572606d2c550a06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }"></span>, <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 \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"></span>, dan <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="{\textstyle {\frac {\pi }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {\pi }{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/313947d4107765408f902d3eaee6a719e7f01dae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.778ex; height:3.176ex;" alt="{\textstyle {\frac {\pi }{2}}}"></span> dalam bentuk titik-kambang biner akan selalu menyebabkan hilangnya akurasi akibat galat pemotongan. Satuan putaran juga memiliki keuntungan akurasi dan efisiensi ketika menghitung modulo satu periode. Komputasi modulo 1 putaran atau modulo 2 setengah-putaran dapat dinyatakan dengan sempurna dan dihitung dengan efisien, baik dalam bentuk titik-kambang maupun titik-tetap. Sebagai contoh, komputasi modulo 1 atau modulo 2 untuk nilai biner dalam bentuk titik-tetap hanya memerlukan operasi penggeseran bit (<i>bit shift</i>) dan operator bit-demi-bit (<i>bitwise</i>) AND. Sebagai kontras, komputasi modulo <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="{\textstyle {\frac {\pi }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {\pi }{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/313947d4107765408f902d3eaee6a719e7f01dae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.778ex; height:3.176ex;" alt="{\textstyle {\frac {\pi }{2}}}"></span> menghasilkan galat karena ketidakakuratan dalam merepresentasikan <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="{\textstyle {\frac {\pi }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {\pi }{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/313947d4107765408f902d3eaee6a719e7f01dae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.778ex; height:3.176ex;" alt="{\textstyle {\frac {\pi }{2}}}"></span>. </p><p>Untuk aplikasi yang melibatkan sensor sudut, sensor umumnya menyajikan hasil pengukuran yang kompatibel dengan satuan putaran atau setengah-putaran. Sebagai contoh, suatu sensor sudut mungkin memberikan nilai 0 sampai 4096 ketika melakukan satu putaran penuh.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> Jika setengah-putaran digunakan sebagai satuan pengukuran sudut, maka nilai yang diberikan sensor dapat dipetakan dengan sempurna (<i>lossless</i>) ke tipe data titik-tetap dengan 11 bit dibelakang koma. Sebagai kontras, jika radian yang digunakan untuk menghitung sudut, ketidakakuratan sebesar <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="{\textstyle {\frac {\pi }{2048}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2048</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {\pi }{2048}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af6a95707656082f4af5a498fb84cd8a03827286" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:4.124ex; height:3.343ex;" alt="{\textstyle {\frac {\pi }{2048}}}"></span> terjadi ketika nilai dari sensor diubah menjadi satuan radian. </p> <div class="mw-heading mw-heading2"><h2 id="Bacaan_lebih_lanjut">Bacaan lebih lanjut</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sinus_dan_kosinus&veaction=edit&section=24" title="Sunting bagian: Bacaan lebih lanjut" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sinus_dan_kosinus&action=edit&section=24" title="Sunting kode sumber bagian: Bacaan lebih lanjut"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite class="citation book">Kurnianingsih, Sri (2007). <i>Matematika SMA dan MA 2A Untuk Kelas XI Semester 1 Program IPA</i>. Jakarta: Esis/Erlangga. <a href="/wiki/Istimewa:Sumber_buku/9797345025" class="internal mw-magiclink-isbn">ISBN 979-734-502-5</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Matematika+SMA+dan+MA+2A+Untuk+Kelas+XI+Semester+1+Program+IPA&rft.place=Jakarta&rft.pub=Esis%2FErlangga&rft.date=2007&rft.aulast=Kurnianingsih&rft.aufirst=Sri&rfr_id=info%3Asid%2Fid.wikipedia.org%3ASinus+dan+kosinus" class="Z3988"><span style="display:none;"> </span></span> <span style="display:none;font-size:100%" class="error citation-comment">Parameter <code style="color:inherit; border:inherit; padding:inherit;">|coauthors=</code> yang tidak diketahui mengabaikan (<code style="color:inherit; border:inherit; padding:inherit;">|author=</code> yang disarankan) (<a href="/wiki/Bantuan:Galat_CS1#parameter_ignored_suggest" title="Bantuan:Galat CS1">bantuan</a>)</span></li> <li><cite id="CITEREFTraupman,_Ph.D.1966" class="citation">Traupman, Ph.D., John C. (1966), <span class="plainlinks"><a rel="nofollow" class="external text" href="https://archive.org/details/boysgirlsbookabo00gard_0"><i>The New College Latin & English Dictionary</i><span style="padding-left:0.15em"><span typeof="mw:File"><span title="Perlu mendaftar (gratis)"><img alt="Perlu mendaftar (gratis)" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Lock-blue-alt-2.svg/9px-Lock-blue-alt-2.svg.png" decoding="async" width="9" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Lock-blue-alt-2.svg/14px-Lock-blue-alt-2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Lock-blue-alt-2.svg/18px-Lock-blue-alt-2.svg.png 2x" data-file-width="512" data-file-height="813" /></span></span></span></a></span>, Toronto: Bantam, <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/0-553-27619-0" title="Istimewa:Sumber buku/0-553-27619-0">0-553-27619-0</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+New+College+Latin+%26+English+Dictionary&rft.place=Toronto&rft.pub=Bantam&rft.date=1966&rft.isbn=0-553-27619-0&rft.aulast=Traupman%2C+Ph.D.&rft.aufirst=John+C.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fboysgirlsbookabo00gard_0&rfr_id=info%3Asid%2Fid.wikipedia.org%3ASinus+dan+kosinus" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite id="CITEREFWebster1969" class="citation"><i>Webster's Seventh New Collegiate Dictionary</i>, Springfield: G. & C. Merriam Company, 1969</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Webster%27s+Seventh+New+Collegiate+Dictionary&rft.place=Springfield&rft.pub=G.+%26+C.+Merriam+Company&rft.date=1969&rfr_id=info%3Asid%2Fid.wikipedia.org%3ASinus+dan+kosinus" class="Z3988"><span style="display:none;"> </span></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Referensi">Referensi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sinus_dan_kosinus&veaction=edit&section=25" title="Sunting bagian: Referensi" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sinus_dan_kosinus&action=edit&section=25" title="Sunting kode sumber bagian: Referensi"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-:1-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-:1_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:1_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:1_1-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><cite class="citation web">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Sine.html">"Sine"</a>. <i>mathworld.wolfram.com</i> (dalam bahasa Inggris)<span class="reference-accessdate">. Diakses tanggal <span class="nowrap">2020-08-29</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=mathworld.wolfram.com&rft.atitle=Sine&rft.aulast=Weisstein&rft.aufirst=Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FSine.html&rfr_id=info%3Asid%2Fid.wikipedia.org%3ASinus+dan+kosinus" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-Boyer,_Carl_B._1991_p._210-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Boyer,_Carl_B._1991_p._210_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Boyer,_Carl_B._1991_p._210_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="/w/index.php?title=Uta_Merzbach&action=edit&redlink=1" class="new" title="Uta Merzbach (halaman belum tersedia)">Uta C. Merzbach</a>, <a href="/w/index.php?title=Carl_B._Boyer&action=edit&redlink=1" class="new" title="Carl B. Boyer (halaman belum tersedia)">Carl B. Boyer</a> (2011), A History of Mathematics, Hoboken, N.J.: John Wiley & Sons, 3rd ed., p. 189.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Victor J. Katz (2008), <i>A History of Mathematics</i>, Boston: Addison-Wesley, 3rd. ed., p. 253, sidebar 8.1. <cite class="citation web"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20150414065531/http://deti-bilingual.com/wp-content/uploads/2014/06/3rd-Edition-Victor-J.-Katz-A-History-of-Mathematics-Pearson-2008.pdf">"Archived copy"</a> <span style="font-size:85%;">(PDF)</span>. Diarsipkan dari <a rel="nofollow" class="external text" href="http://deti-bilingual.com/wp-content/uploads/2014/06/3rd-Edition-Victor-J.-Katz-A-History-of-Mathematics-Pearson-2008.pdf">versi asli</a> <span style="font-size:85%;">(PDF)</span> tanggal 2015-04-14<span class="reference-accessdate">. Diakses tanggal <span class="nowrap">2015-04-09</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Archived+copy&rft_id=http%3A%2F%2Fdeti-bilingual.com%2Fwp-content%2Fuploads%2F2014%2F06%2F3rd-Edition-Victor-J.-Katz-A-History-of-Mathematics-Pearson-2008.pdf&rfr_id=info%3Asid%2Fid.wikipedia.org%3ASinus+dan+kosinus" class="Z3988"><span style="display:none;"> </span></span> <span style="display:none;font-size:100%" class="error citation-comment">Parameter <code style="color:inherit; border:inherit; padding:inherit;">|url-status=</code> yang tidak diketahui akan diabaikan (<a href="/wiki/Bantuan:Galat_CS1#parameter_ignored" title="Bantuan:Galat CS1">bantuan</a>)</span></span> </li> <li id="cite_note-cosine-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-cosine_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-cosine_4-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite class="citation web"><a rel="nofollow" class="external text" href="https://www.etymonline.com/word/cosine">"cosine"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=cosine&rft_id=https%3A%2F%2Fwww.etymonline.com%2Fword%2Fcosine&rfr_id=info%3Asid%2Fid.wikipedia.org%3ASinus+dan+kosinus" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-:2-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-:2_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:2_5-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite class="citation web"><a rel="nofollow" class="external text" href="https://www.mathsisfun.com/sine-cosine-tangent.html">"Sine, Cosine, Tangent"</a>. <i>www.mathsisfun.com</i><span class="reference-accessdate">. Diakses tanggal <span class="nowrap">2020-08-29</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.mathsisfun.com&rft.atitle=Sine%2C+Cosine%2C+Tangent&rft_id=https%3A%2F%2Fwww.mathsisfun.com%2Fsine-cosine-tangent.html&rfr_id=info%3Asid%2Fid.wikipedia.org%3ASinus+dan+kosinus" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><cite class="citation web"><a rel="nofollow" class="external text" href="https://calculus.subwiki.org/wiki/Sine-squared_function#Identities">"Sine-squared function"</a><span class="reference-accessdate">. Diakses tanggal <span class="nowrap">August 9,</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Sine-squared+function&rft_id=https%3A%2F%2Fcalculus.subwiki.org%2Fwiki%2FSine-squared_function%23Identities&rfr_id=info%3Asid%2Fid.wikipedia.org%3ASinus+dan+kosinus" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">See Ahlfors, pages 43–44.</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><cite class="citation web"><a rel="nofollow" class="external text" href="https://oeis.org/A003957">"OEIS A003957"</a>. <i>oeis.org</i><span class="reference-accessdate">. Diakses tanggal <span class="nowrap">2019-05-26</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=oeis.org&rft.atitle=OEIS+A003957&rft_id=https%3A%2F%2Foeis.org%2FA003957&rfr_id=info%3Asid%2Fid.wikipedia.org%3ASinus+dan+kosinus" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-ams-9"><span class="mw-cite-backlink">^ <a href="#cite_ref-ams_9-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-ams_9-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite class="citation web">Adlaj, Semjon (2012). <a rel="nofollow" class="external text" href="https://www.ams.org/notices/201208/rtx120801094p.pdf">"An Eloquent Formula for the Perimeter of an Ellipse"</a> <span style="font-size:85%;">(PDF)</span>. <i>American Mathematical Society</i>. hlm. 1097.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=American+Mathematical+Society&rft.atitle=An+Eloquent+Formula+for+the+Perimeter+of+an+Ellipse&rft.pages=1097&rft.date=2012&rft.aulast=Adlaj&rft.aufirst=Semjon&rft_id=https%3A%2F%2Fwww.ams.org%2Fnotices%2F201208%2Frtx120801094p.pdf&rfr_id=info%3Asid%2Fid.wikipedia.org%3ASinus+dan+kosinus" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><cite class="citation web"><a rel="nofollow" class="external text" href="https://functions.wolfram.com/EllipticIntegrals/EllipticE2/06/01/10/0009/">"Incomplete elliptic integral of the second kind: Series representations (Formula 08.04.06.0003)"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Incomplete+elliptic+integral+of+the+second+kind%3A+Series+representations+%28Formula+08.04.06.0003%29&rft_id=https%3A%2F%2Ffunctions.wolfram.com%2FEllipticIntegrals%2FEllipticE2%2F06%2F01%2F10%2F0009%2F&rfr_id=info%3Asid%2Fid.wikipedia.org%3ASinus+dan+kosinus" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-Gingerich_1986-11"><span class="mw-cite-backlink">^ <a href="#cite_ref-Gingerich_1986_11-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Gingerich_1986_11-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite class="citation magazine">Gingerich, Owen (1986). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20131019140821/http://faculty.kfupm.edu.sa/PHYS/alshukri/PHYS215/Islamic_astronomy.htm">"Islamic Astronomy"</a>. <i><a href="/wiki/Scientific_American" title="Scientific American">Scientific American</a></i>. Vol. 254. hlm. 74. Diarsipkan dari <a rel="nofollow" class="external text" href="http://faculty.kfupm.edu.sa/PHYS/alshukri/PHYS215/Islamic_astronomy.htm">versi asli</a> tanggal 2013-10-19<span class="reference-accessdate">. Diakses tanggal <span class="nowrap">2010-07-13</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Scientific+American&rft.atitle=Islamic+Astronomy&rft.volume=254&rft.pages=74&rft.date=1986&rft.aulast=Gingerich&rft.aufirst=Owen&rft_id=http%3A%2F%2Ffaculty.kfupm.edu.sa%2FPHYS%2Falshukri%2FPHYS215%2FIslamic_astronomy.htm&rfr_id=info%3Asid%2Fid.wikipedia.org%3ASinus+dan+kosinus" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-Sesiano-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-Sesiano_12-0">^</a></b></span> <span class="reference-text">Jacques Sesiano, "Islamic mathematics", p. 157, in <cite class="citation book"><a href="/w/index.php?title=Helaine_Selin&action=edit&redlink=1" class="new" title="Helaine Selin (halaman belum tersedia)">Selin, Helaine</a>; <a href="/w/index.php?title=Ubiratan_D%27Ambrosio&action=edit&redlink=1" class="new" title="Ubiratan D'Ambrosio (halaman belum tersedia)">D'Ambrosio, Ubiratan</a>, ed. (2000). <i>Mathematics Across Cultures: The History of Non-western Mathematics</i>. <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer Science+Business Media</a>. <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/978-1-4020-0260-1" title="Istimewa:Sumber buku/978-1-4020-0260-1">978-1-4020-0260-1</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+Across+Cultures%3A+The+History+of+Non-western+Mathematics&rft.pub=Springer+Science%2BBusiness+Media&rft.date=2000&rft.isbn=978-1-4020-0260-1&rfr_id=info%3Asid%2Fid.wikipedia.org%3ASinus+dan+kosinus" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-Britannica-13"><span class="mw-cite-backlink">^ <a href="#cite_ref-Britannica_13-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Britannica_13-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite class="citation web"><a rel="nofollow" class="external text" href="http://www.britannica.com/EBchecked/topic/605281/trigonometry">"trigonometry"</a>. Encyclopedia Britannica.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=trigonometry&rft.pub=Encyclopedia+Britannica&rft_id=http%3A%2F%2Fwww.britannica.com%2FEBchecked%2Ftopic%2F605281%2Ftrigonometry&rfr_id=info%3Asid%2Fid.wikipedia.org%3ASinus+dan+kosinus" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><cite class="citation book">Nicolás Bourbaki (1994). <span class="plainlinks"><a rel="nofollow" class="external text" href="https://archive.org/details/elementsofhistor0000bour"><i>Elements of the History of Mathematics</i><span style="padding-left:0.15em"><span typeof="mw:File"><span title="Perlu mendaftar (gratis)"><img alt="Perlu mendaftar (gratis)" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Lock-blue-alt-2.svg/9px-Lock-blue-alt-2.svg.png" decoding="async" width="9" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Lock-blue-alt-2.svg/14px-Lock-blue-alt-2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Lock-blue-alt-2.svg/18px-Lock-blue-alt-2.svg.png 2x" data-file-width="512" data-file-height="813" /></span></span></span></a></span>. Springer. <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/9783540647676" title="Istimewa:Sumber buku/9783540647676">9783540647676</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elements+of+the+History+of+Mathematics&rft.pub=Springer&rft.date=1994&rft.isbn=9783540647676&rft.au=Nicol%C3%A1s+Bourbaki&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Felementsofhistor0000bour&rfr_id=info%3Asid%2Fid.wikipedia.org%3ASinus+dan+kosinus" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text">"<a rel="nofollow" class="external text" href="http://www.math.usma.edu/people/rickey/hm/CalcNotes/Sine-Deriv.pdf">Why the sine has a simple derivative</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110720102700/http://www.math.usma.edu/people/rickey/hm/CalcNotes/Sine-Deriv.pdf">Diarsipkan</a> 2011-07-20 di <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>.", in <i><a rel="nofollow" class="external text" href="http://www.math.usma.edu/people/rickey/hm/CalcNotes/default.htm">Historical Notes for Calculus Teachers</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110720102613/http://www.math.usma.edu/people/rickey/hm/CalcNotes/default.htm">Diarsipkan</a> 2011-07-20 di <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>.</i> by <a rel="nofollow" class="external text" href="http://www.math.usma.edu/people/rickey/">V. Frederick Rickey</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110720102654/http://www.math.usma.edu/people/rickey/">Diarsipkan</a> 2011-07-20 di <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>.</span> </li> <li id="cite_note-boyer-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-boyer_16-0">^</a></b></span> <span class="reference-text">See Merzbach, Boyer (2011).</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text">Eli Maor (1998), <i>Trigonometric Delights</i>, Princeton: Princeton University Press, p. 35-36.</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text">Victor J. Katz (2008), <i>A History of Mathematics</i>, Boston: Addison-Wesley, 3rd. ed., p. 253, sidebar 8.1. <cite class="citation web"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20150414065531/http://deti-bilingual.com/wp-content/uploads/2014/06/3rd-Edition-Victor-J.-Katz-A-History-of-Mathematics-Pearson-2008.pdf">"Archived copy"</a> <span style="font-size:85%;">(PDF)</span>. Diarsipkan dari <a rel="nofollow" class="external text" href="http://deti-bilingual.com/wp-content/uploads/2014/06/3rd-Edition-Victor-J.-Katz-A-History-of-Mathematics-Pearson-2008.pdf">versi asli</a> <span style="font-size:85%;">(PDF)</span> tanggal 2015-04-14<span class="reference-accessdate">. Diakses tanggal <span class="nowrap">2015-04-09</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Archived+copy&rft_id=http%3A%2F%2Fdeti-bilingual.com%2Fwp-content%2Fuploads%2F2014%2F06%2F3rd-Edition-Victor-J.-Katz-A-History-of-Mathematics-Pearson-2008.pdf&rfr_id=info%3Asid%2Fid.wikipedia.org%3ASinus+dan+kosinus" class="Z3988"><span style="display:none;"> </span></span> <span style="display:none;font-size:100%" class="error citation-comment">Parameter <code style="color:inherit; border:inherit; padding:inherit;">|url-status=</code> yang tidak diketahui akan diabaikan (<a href="/wiki/Bantuan:Galat_CS1#parameter_ignored" title="Bantuan:Galat CS1">bantuan</a>)</span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><cite id="CITEREFSmith1958" class="citation">Smith, D.E. (1958) [1925], <i>History of Mathematics</i>, <b>I</b>, Dover, hlm. 202, <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/0-486-20429-4" title="Istimewa:Sumber buku/0-486-20429-4">0-486-20429-4</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=History+of+Mathematics&rft.pages=202&rft.pub=Dover&rft.date=1958&rft.isbn=0-486-20429-4&rft.aulast=Smith&rft.aufirst=D.E.&rfr_id=info%3Asid%2Fid.wikipedia.org%3ASinus+dan+kosinus" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text">Grand Challenges of Informatics, Paul Zimmermann. September 20, 2006 – p. 14/31 <cite class="citation web"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20110716205108/http://www.jaist.ac.jp/~bjorner/ae-is-budapest/talks/Sept20pm2_Zimmermann.pdf">"Archived copy"</a> <span style="font-size:85%;">(PDF)</span>. Diarsipkan dari <a rel="nofollow" class="external text" href="http://www.jaist.ac.jp/~bjorner/ae-is-budapest/talks/Sept20pm2_Zimmermann.pdf">versi asli</a> <span style="font-size:85%;">(PDF)</span> tanggal 2011-07-16<span class="reference-accessdate">. Diakses tanggal <span class="nowrap">2010-09-11</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Archived+copy&rft_id=http%3A%2F%2Fwww.jaist.ac.jp%2F~bjorner%2Fae-is-budapest%2Ftalks%2FSept20pm2_Zimmermann.pdf&rfr_id=info%3Asid%2Fid.wikipedia.org%3ASinus+dan+kosinus" class="Z3988"><span style="display:none;"> </span></span> <span style="display:none;font-size:100%" class="error citation-comment">Parameter <code style="color:inherit; border:inherit; padding:inherit;">|url-status=</code> yang tidak diketahui akan diabaikan (<a href="/wiki/Bantuan:Galat_CS1#parameter_ignored" title="Bantuan:Galat CS1">bantuan</a>)</span></span> </li> <li id="cite_note-matlab-21"><span class="mw-cite-backlink">^ <a href="#cite_ref-matlab_21-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-matlab_21-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">"<a rel="nofollow" class="external text" href="https://www.mathworks.com/help/matlab/ref/double.sinpi.html">MATLAB Documentation sinpi</a></span> </li> <li id="cite_note-r-22"><span class="mw-cite-backlink">^ <a href="#cite_ref-r_22-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-r_22-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">"<a rel="nofollow" class="external text" href="https://www.rdocumentation.org/packages/base/versions/3.5.3/topics/Trig">R Documentation sinpi</a></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text">"<a rel="nofollow" class="external text" href="https://www.khronos.org/registry/OpenCL/sdk/1.0/docs/man/xhtml/sin.html">OpenCL Documentation sinpi</a></span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text">"<a rel="nofollow" class="external text" href="http://www.jlhub.com/julia/manual/en/function/sinpi">Julia Documentation sinpi</a></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text">"<a rel="nofollow" class="external text" href="https://docs.nvidia.com/cuda/cuda-math-api/group__CUDA__MATH__DOUBLE.html">CUDA Documentation sinpi</a></span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text">"<a rel="nofollow" class="external text" href="https://developer.arm.com/docs/100614/latest/b-opencl-built-in-functions/b2-math-functions">ARM Documentation sinpi</a></span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text">"<a rel="nofollow" class="external text" href="https://www.allegromicro.com/en/Products/Magnetic-Linear-And-Angular-Position-Sensor-ICs/Angular-Position-Sensor-ICs/AAS33051.aspx">ALLEGRO Angle Sensor Datasheet</a> <a rel="nofollow" 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class="navbox-group" style="width:1%">Fungsi polinomial</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Fungsi_konstan" title="Fungsi konstan">Fungsi konstan</a> (0)</li> <li><a href="/wiki/Fungsi_linear" class="mw-redirect" title="Fungsi linear">Fungsi linear</a> (1)</li> <li><a href="/wiki/Fungsi_kuadrat" title="Fungsi kuadrat">Fungsi kuadrat</a> (2)</li> <li><a href="/wiki/Fungsi_kubik" title="Fungsi kubik">Fungsi kubik</a> (3)</li> <li><a href="/wiki/Fungsi_kuartik" title="Fungsi kuartik">Fungsi kuartik</a> (4)</li> <li><a href="/wiki/Fungsi_kuintik" title="Fungsi kuintik">Fungsi kuintik</a> (5)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Fungsi aljabar</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Fungsi_rasional" title="Fungsi rasional">Fungsi rasional</a></li> <li><a href="/wiki/Fungsi_eksponensial" title="Fungsi eksponensial">Fungsi eksponensial</a> <ul><li><a href="/wiki/Fungsi_Lambert_W" title="Fungsi Lambert W">Lambert W</a></li> <li><a href="/wiki/Superakar" class="mw-redirect" title="Superakar">Superakar</a></li></ul></li> <li><a href="/wiki/Fungsi_hiperbolik" title="Fungsi hiperbolik">Fungsi hiperbolik</a></li> <li><a href="/wiki/Fungsi_logaritma" class="mw-redirect" title="Fungsi logaritma">Fungsi logaritma</a> <ul><li>Berdasarkan basis <ul><li><a href="/wiki/Logaritma_biner" title="Logaritma biner">2</a></li> <li><a href="/wiki/Logaritma_natural" class="mw-redirect" title="Logaritma natural"><span class="texhtml mvar" style="font-style:italic;">e</span></a></li> <li><a href="/wiki/Logaritma_umum" title="Logaritma umum">10</a></li></ul></li> <li><a href="/w/index.php?title=Logaritma_teriterasi&action=edit&redlink=1" class="new" title="Logaritma teriterasi (halaman belum tersedia)">teriterasi</a></li> <li><a href="/w/index.php?title=Superlogaritma&action=edit&redlink=1" class="new" title="Superlogaritma (halaman belum tersedia)">Superlogaritma</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Fungsi dalam<br />teori bilangan</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Fungsi_M%C3%B6bius" title="Fungsi Möbius">Fungsi Möbius</a></li> <li><a href="/w/index.php?title=Fungsi_partisi&action=edit&redlink=1" class="new" title="Fungsi partisi (halaman belum tersedia)">Fungsi partisi</a></li> <li><a href="/w/index.php?title=Fungsi_perhitungan_bilangan_prima&action=edit&redlink=1" class="new" title="Fungsi perhitungan bilangan prima (halaman belum tersedia)">Fungsi perhitungan bilangan prima</a></li> <li><a href="/wiki/Fungsi_phi_Euler" title="Fungsi phi Euler">Fungsi phi Euler</a></li> <li><a href="/w/index.php?title=Fungsi_sigma&action=edit&redlink=1" class="new" title="Fungsi sigma (halaman belum tersedia)">Fungsi sigma</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Fungsi trigonometri</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Sinus_(trigonometri)" class="mw-redirect" title="Sinus (trigonometri)">Sinus</a></li> <li><a href="/wiki/Kosinus" class="mw-redirect" title="Kosinus">Kosinus</a></li> <li><a href="/wiki/Tangen" title="Tangen">Tangen</a></li> <li><a href="/wiki/Sekan" title="Sekan">Sekan</a></li> <li><a href="/wiki/Kosekan" title="Kosekan">Kosekan</a></li> <li><a href="/wiki/Kotangen" title="Kotangen">Kotangen</a></li> <li><a href="/w/index.php?title=Versinus&action=edit&redlink=1" class="new" title="Versinus (halaman belum tersedia)">Versinus</a></li> <li><a href="/w/index.php?title=Koversinus&action=edit&redlink=1" class="new" title="Koversinus (halaman belum tersedia)">Koversinus</a></li> <li><a href="/w/index.php?title=Verkosinus&action=edit&redlink=1" class="new" title="Verkosinus (halaman belum tersedia)">Verkosinus</a></li> <li><a href="/w/index.php?title=Koverkosinus&action=edit&redlink=1" class="new" title="Koverkosinus (halaman belum tersedia)">Koverkosinus</a></li> <li><a href="/w/index.php?title=Ekssekan&action=edit&redlink=1" class="new" title="Ekssekan (halaman belum tersedia)">Ekssekan</a></li> <li><a href="/w/index.php?title=Ekskosekan&action=edit&redlink=1" class="new" title="Ekskosekan (halaman belum tersedia)">Ekskosekan</a></li> <li><a href="/w/index.php?title=Haversinus&action=edit&redlink=1" class="new" title="Haversinus (halaman belum tersedia)">Haversinus</a></li> <li><a href="/w/index.php?title=Hakoversinus&action=edit&redlink=1" class="new" title="Hakoversinus (halaman belum tersedia)">Hakoversinus</a></li> <li><a href="/w/index.php?title=Haverkosinus&action=edit&redlink=1" class="new" title="Haverkosinus (halaman belum tersedia)">Haverkosinus</a></li> <li><a href="/w/index.php?title=Hakoverkosinus&action=edit&redlink=1" class="new" title="Hakoverkosinus (halaman belum tersedia)">Hakoverkosinus</a></li></ul> <p><br /> </p> <ul><li><a href="/w/index.php?title=Fungsi_Gudermann&action=edit&redlink=1" class="new" title="Fungsi Gudermann (halaman belum tersedia)">Gudermann</a></li> <li><a href="/w/index.php?title=Fungsi_sinc&action=edit&redlink=1" class="new" title="Fungsi sinc (halaman belum tersedia)">sinc</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Fungsi berdasarkan<br />huruf Yunani</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/w/index.php?title=Fungsi_beta&action=edit&redlink=1" class="new" title="Fungsi beta (halaman belum tersedia)">Fungsi beta</a> <ul><li><a href="/w/index.php?title=Fungsi_beta_Dirichlet&action=edit&redlink=1" class="new" title="Fungsi beta Dirichlet (halaman belum tersedia)">Dirichlet</a></li> <li><a href="/w/index.php?title=Fungsi_beta_taklengkap&action=edit&redlink=1" class="new" title="Fungsi beta taklengkap (halaman belum tersedia)">taklengkap</a></li></ul></li> <li>Fungsi chi <ul><li><a href="/w/index.php?title=Fungsi_chi_Legendre&action=edit&redlink=1" class="new" title="Fungsi chi Legendre (halaman belum tersedia)">Legendre</a></li></ul></li> <li><a href="/w/index.php?title=Fungsi_delta&action=edit&redlink=1" class="new" title="Fungsi delta (halaman belum tersedia)">Fungsi delta</a> <ul><li><a href="/wiki/Fungsi_delta_Dirac" title="Fungsi delta Dirac">Fungsi delta Dirac</a></li> <li><a href="/wiki/Fungsi_delta_Kronecker" title="Fungsi delta Kronecker">Fungsi delta Kronecker</a></li> <li><a href="/w/index.php?title=Potensial_fungsi_delta&action=edit&redlink=1" class="new" title="Potensial fungsi delta (halaman belum tersedia)">potensial delta</a></li></ul></li> <li>Fungsi eta <ul><li><a href="/w/index.php?title=Fungsi_eta_Dirichlet&action=edit&redlink=1" class="new" title="Fungsi eta Dirichlet (halaman belum tersedia)">Dirichlet</a></li></ul></li> <li><a href="/wiki/Fungsi_gamma" title="Fungsi gamma">Fungsi gamma</a> <ul><li><a href="/w/index.php?title=Fungsi_digamma&action=edit&redlink=1" class="new" title="Fungsi digamma (halaman belum tersedia)">Fungsi digamma</a></li> <li><a href="/w/index.php?title=Fungsi-G_Barnes&action=edit&redlink=1" class="new" title="Fungsi-G Barnes (halaman belum tersedia)">Barnes</a></li> <li><a href="/w/index.php?title=Fungsi-G_Meijer&action=edit&redlink=1" class="new" title="Fungsi-G Meijer (halaman belum tersedia)">Meijer</a></li> <li><a href="/w/index.php?title=Fungsi_gamma_banyak&action=edit&redlink=1" class="new" title="Fungsi gamma banyak (halaman belum tersedia)">banyak</a></li> <li><a href="/w/index.php?title=Fungsi_gamma_eliptik&action=edit&redlink=1" class="new" title="Fungsi gamma eliptik (halaman belum tersedia)">eliptik</a></li> <li><a href="/w/index.php?title=Fungsi_gamma_Hadamard&action=edit&redlink=1" class="new" title="Fungsi gamma Hadamard (halaman belum tersedia)">Hadamard</a></li> <li><a href="/w/index.php?title=Fungsi_gamma_multivariabel&action=edit&redlink=1" class="new" title="Fungsi gamma multivariabel (halaman belum tersedia)">multivariabel</a></li> <li><a href="/w/index.php?title=Fungsi_gamma_p-adik&action=edit&redlink=1" class="new" title="Fungsi gamma p-adik (halaman belum tersedia)"><i>p</i>-adik</a></li> <li><a href="/w/index.php?title=Fungsi_gamma-q&action=edit&redlink=1" class="new" title="Fungsi gamma-q (halaman belum tersedia)"><i>q</i></a></li> <li><a href="/w/index.php?title=Fungsi_gamma_taklengkap&action=edit&redlink=1" class="new" title="Fungsi gamma taklengkap (halaman belum tersedia)">taklengkap</a></li> <li><a href="/wiki/Fungsi_poligamma" title="Fungsi poligamma">Fungsi poligamma</a></li> <li><a href="/w/index.php?title=Fungsi_trigamma&action=edit&redlink=1" class="new" title="Fungsi trigamma (halaman belum tersedia)">Fungsi trigamma</a></li></ul></li> <li>Fungsi lambda <ul><li><a href="/w/index.php?title=Fungsi_lambda_Dirchlet&action=edit&redlink=1" class="new" title="Fungsi lambda Dirchlet (halaman belum tersedia)">Dirchlet</a></li> <li><a href="/w/index.php?title=Fungsi_lambda_modular&action=edit&redlink=1" class="new" title="Fungsi lambda modular (halaman belum tersedia)">modular</a></li> <li><a href="/w/index.php?title=Fungsi_von_Mangoldt&action=edit&redlink=1" class="new" title="Fungsi von Mangoldt (halaman belum tersedia)">von Mangoldt</a></li></ul></li> <li>Fungsi mu <ul><li><a href="/w/index.php?title=Fungsi_%CE%BC_M%C3%B6bius&action=edit&redlink=1" class="new" title="Fungsi μ Möbius (halaman belum tersedia)">Möbius</a></li></ul></li> <li>Fungsi phi <ul><li><a href="/wiki/Fungsi_phi_Euler" title="Fungsi phi Euler">Euler</a></li></ul></li> <li><a href="/w/index.php?title=Fungsi_pi&action=edit&redlink=1" class="new" title="Fungsi pi (halaman belum tersedia)">Fungsi pi</a></li> <li><a href="/w/index.php?title=Fungsi_sigma&action=edit&redlink=1" class="new" title="Fungsi sigma (halaman belum tersedia)">Fungsi sigma</a> <ul><li><a href="/w/index.php?title=Fungsi_sigma_Weierstrass&action=edit&redlink=1" class="new" title="Fungsi sigma Weierstrass (halaman belum tersedia)">Weierstrass</a></li></ul></li> <li><a href="/wiki/Fungsi_theta" title="Fungsi theta">Fungsi theta</a></li> <li><a href="/w/index.php?title=Fungsi_zeta&action=edit&redlink=1" class="new" title="Fungsi zeta (halaman belum tersedia)">Fungsi zeta</a> <ul><li><a href="/w/index.php?title=Fungsi_zeta_Hurwitz&action=edit&redlink=1" class="new" title="Fungsi zeta Hurwitz (halaman belum tersedia)">Hurwitz</a></li> <li><a href="/wiki/Fungsi_zeta_Riemann" title="Fungsi zeta Riemann">Riemann</a></li> <li><a href="/w/index.php?title=Fungsi_zeta_Weierstrass&action=edit&redlink=1" class="new" title="Fungsi zeta Weierstrass (halaman belum tersedia)">Weierstrass</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Fungsi berdasarkan<br />nama matematikawan</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/w/index.php?title=Fungsi_Airy&action=edit&redlink=1" class="new" title="Fungsi Airy (halaman belum tersedia)">Airy</a></li> <li><a href="/w/index.php?title=Fungsi_Ackermann&action=edit&redlink=1" class="new" title="Fungsi Ackermann (halaman belum tersedia)">Ackermann</a></li> <li><a href="/w/index.php?title=Fungsi_Bessel&action=edit&redlink=1" class="new" title="Fungsi Bessel (halaman belum tersedia)">Bessel</a></li> <li><a href="/w/index.php?title=Fungsi_Bessel%E2%80%93Clifford&action=edit&redlink=1" class="new" title="Fungsi Bessel–Clifford (halaman belum tersedia)">Bessel–Clifford</a></li> <li><a href="/w/index.php?title=Fungsi_Bottcher&action=edit&redlink=1" class="new" title="Fungsi Bottcher (halaman belum tersedia)">Bottcher</a></li> <li><a href="/w/index.php?title=Polinomial_Chebyshev&action=edit&redlink=1" class="new" title="Polinomial Chebyshev (halaman belum tersedia)">Chebyshev</a></li> <li><a href="/w/index.php?title=Fungsi_Clausen&action=edit&redlink=1" class="new" title="Fungsi Clausen (halaman belum tersedia)">Clausen</a></li> <li><a href="/w/index.php?title=Fungsi_Dawson&action=edit&redlink=1" class="new" title="Fungsi Dawson (halaman belum tersedia)">Dawson</a></li> <li><a href="/w/index.php?title=Fungsi_Dirichlet&action=edit&redlink=1" class="new" title="Fungsi Dirichlet (halaman belum tersedia)">Dirichlet</a> <ul><li><a href="/w/index.php?title=Fungsi_beta_Dirichlet&action=edit&redlink=1" class="new" title="Fungsi beta Dirichlet (halaman belum tersedia)">beta</a></li> <li><a href="/w/index.php?title=Fungsi_eta_Dirichlet&action=edit&redlink=1" class="new" title="Fungsi eta Dirichlet (halaman belum tersedia)">eta</a></li> <li><a href="/w/index.php?title=Fungsi-L_Dirichlet&action=edit&redlink=1" class="new" title="Fungsi-L Dirichlet (halaman belum tersedia)">L</a></li> <li><a href="/w/index.php?title=Fungsi_lambda_Dirchlet&action=edit&redlink=1" class="new" title="Fungsi lambda Dirchlet (halaman belum tersedia)">lambda</a></li></ul></li> <li><a href="/w/index.php?title=Fungsi_Faddeeva&action=edit&redlink=1" class="new" title="Fungsi Faddeeva (halaman belum tersedia)">Faddeeva</a></li> <li>Fermi–Dirac <ul><li><a href="/w/index.php?title=Integral_Fermi%E2%80%93Dirac_lengkap&action=edit&redlink=1" class="new" title="Integral Fermi–Dirac lengkap (halaman belum tersedia)">lengkap</a></li> <li><a href="/w/index.php?title=Integral_Fermi%E2%80%93Dirac_taklengkap&action=edit&redlink=1" class="new" title="Integral Fermi–Dirac taklengkap (halaman belum tersedia)">taklengkap</a></li></ul></li> <li><a href="/wiki/Integral_Fresnel" title="Integral Fresnel">Fresnel</a></li> <li><a href="/wiki/Fungsi-H_Fox" title="Fungsi-H Fox">Fox</a></li> <li><a href="/w/index.php?title=Fungsi_Gudermann&action=edit&redlink=1" class="new" title="Fungsi Gudermann (halaman belum tersedia)">Gudermann</a></li> <li><a href="/wiki/Polinomial_Hermite" title="Polinomial Hermite">Hermite</a></li> <li><a href="/w/index.php?title=Fungsi_Jacob&action=edit&redlink=1" class="new" title="Fungsi Jacob (halaman belum tersedia)">Fungsi Jacob</a> <ul><li><a href="/w/index.php?title=Fungsi_eliptik_Jacobi&action=edit&redlink=1" class="new" title="Fungsi eliptik Jacobi (halaman belum tersedia)">eliptik Jacobi</a></li></ul></li> <li><a href="/w/index.php?title=Fungsi_Kelvin&action=edit&redlink=1" class="new" title="Fungsi Kelvin (halaman belum tersedia)">Kelvin</a></li> <li><a href="/w/index.php?title=Fungsi_Kummer&action=edit&redlink=1" class="new" title="Fungsi Kummer (halaman belum tersedia)">Fungsi Kummer</a></li> <li>Fungsi Lambert <ul><li><a href="/wiki/Fungsi_Lambert_W" title="Fungsi Lambert W">W</a></li></ul></li> <li><a href="/w/index.php?title=Fungsi_Lam%C3%A9&action=edit&redlink=1" class="new" title="Fungsi Lamé (halaman belum tersedia)">Lamé</a></li> <li><a href="/w/index.php?title=Polinomial_Laguerre&action=edit&redlink=1" class="new" title="Polinomial Laguerre (halaman belum tersedia)">Laguerre</a></li> <li><a href="/w/index.php?title=Fungsi_Legendre&action=edit&redlink=1" class="new" title="Fungsi Legendre (halaman belum tersedia)">Legendre</a> <ul><li><a href="/w/index.php?title=Fungsi_chi_Legendre&action=edit&redlink=1" class="new" title="Fungsi chi Legendre (halaman belum tersedia)">chi</a></li> <li><a href="/w/index.php?title=Fungsi_Legendre_iring&action=edit&redlink=1" class="new" title="Fungsi Legendre iring (halaman belum tersedia)">iring</a></li></ul></li> <li><a href="/w/index.php?title=Fungsi_Liouville&action=edit&redlink=1" class="new" title="Fungsi Liouville (halaman belum tersedia)">Liouville</a></li> <li><a href="/w/index.php?title=Fungsi_Mathieu&action=edit&redlink=1" class="new" title="Fungsi Mathieu (halaman belum tersedia)">Mathieu</a></li> <li><a href="/w/index.php?title=Fungsi-G_Meijer&action=edit&redlink=1" class="new" title="Fungsi-G Meijer (halaman belum tersedia)">Meijer</a></li> <li><a href="/w/index.php?title=Fungsi_Mittag-Leffler&action=edit&redlink=1" class="new" title="Fungsi Mittag-Leffler (halaman belum tersedia)">Mittag-Leffler</a></li> <li><a href="/w/index.php?title=Transenden_Painlev%C3%A9&action=edit&redlink=1" class="new" title="Transenden Painlevé (halaman belum tersedia)">Painlevé</a></li> <li><a href="/w/index.php?title=Fungsi_Riemann&action=edit&redlink=1" class="new" title="Fungsi Riemann (halaman belum tersedia)">Riemann</a> <ul><li><a href="/w/index.php?title=Fungsi_xi_Riemann&action=edit&redlink=1" class="new" title="Fungsi xi Riemann (halaman belum tersedia)">xi</a></li> <li><a href="/wiki/Fungsi_zeta_Riemann" title="Fungsi zeta Riemann">zeta</a></li></ul></li> <li><a href="/w/index.php?title=Fungsi_Riesz&action=edit&redlink=1" class="new" title="Fungsi Riesz (halaman belum tersedia)">Riesz</a></li> <li><a href="/w/index.php?title=Fungsi_Scorer&action=edit&redlink=1" class="new" title="Fungsi Scorer (halaman belum tersedia)">Scorer</a></li> <li><a href="/w/index.php?title=Fungsi_Spence&action=edit&redlink=1" class="new" title="Fungsi Spence (halaman belum tersedia)">Spence</a></li> <li><a href="/w/index.php?title=Fungsi_von_Mangoldt&action=edit&redlink=1" class="new" title="Fungsi von Mangoldt (halaman belum tersedia)">von Mangoldt</a></li> <li><a href="/wiki/Fungsi_Weierstrass" title="Fungsi Weierstrass">Weierstrass</a> <ul><li><a href="/w/index.php?title=Fungsi_eliptik_Weierstrass&action=edit&redlink=1" class="new" title="Fungsi eliptik Weierstrass (halaman belum tersedia)">eliptik</a></li> <li><a href="/w/index.php?title=Fungsi_eta_Weierstrass&action=edit&redlink=1" class="new" title="Fungsi eta Weierstrass (halaman belum tersedia)">eta</a></li> <li><a href="/w/index.php?title=Fungsi_sigma_Weierstrass&action=edit&redlink=1" class="new" title="Fungsi sigma Weierstrass (halaman belum tersedia)">sigma</a></li> <li><a href="/w/index.php?title=Fungsi_zeta_Weierstrass&action=edit&redlink=1" class="new" title="Fungsi zeta Weierstrass (halaman belum tersedia)">zeta</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Fungsi khusus</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Fungsi_bagian_bilangan_bulat" class="mw-redirect" title="Fungsi bagian bilangan bulat">Fungsi bagian bilangan bulat</a> <ul><li><a href="/wiki/Fungsi_bilangan_bulat_terbesar" class="mw-redirect" title="Fungsi bilangan bulat terbesar">Fungsi bilangan bulat terbesar</a></li> <li><a href="/wiki/Fungsi_bilangan_bulat_terkecil" class="mw-redirect" title="Fungsi bilangan bulat terkecil">Fungsi bilangan bulat terkecil</a></li></ul></li> <li><a href="/w/index.php?title=Fungsi_gergaji&action=edit&redlink=1" class="new" title="Fungsi gergaji (halaman belum tersedia)">Fungsi gergaji</a></li> <li><a href="/wiki/Fungsi_indikator" title="Fungsi indikator">Fungsi indikator</a></li> <li><a href="/wiki/Nilai_mutlak" class="mw-redirect" title="Nilai mutlak">Fungsi nilai mutlak</a></li> <li><a href="/w/index.php?title=Fungsi_persegi&action=edit&redlink=1" class="new" title="Fungsi persegi (halaman belum tersedia)">Fungsi persegi</a></li> <li><a href="/w/index.php?title=Fungsi_segitiga&action=edit&redlink=1" class="new" title="Fungsi segitiga (halaman belum tersedia)">Fungsi segitiga</a></li> <li><a href="/wiki/Fungsi_tanda" title="Fungsi tanda">Fungsi tanda</a></li> <li><a href="/w/index.php?title=Fungsi_tangga&action=edit&redlink=1" class="new" title="Fungsi tangga (halaman belum tersedia)">Fungsi tangga</a> <ul><li><a href="/wiki/Fungsi_tangga_Heaviside" title="Fungsi tangga Heaviside">Fungsi tangga Heaviside</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Fungsi lainnya</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/w/index.php?title=Purata_aritmetik-geometrik&action=edit&redlink=1" class="new" title="Purata aritmetik-geometrik (halaman belum tersedia)">Aritmetik-geometrik</a></li> <li><a href="/w/index.php?title=Fungsi_eliptik&action=edit&redlink=1" class="new" title="Fungsi eliptik (halaman belum tersedia)">eliptik</a></li> <li><a href="/wiki/Fungsi_hiperbolik" title="Fungsi hiperbolik">Fungsi hiperbolik</a> <ul><li><a href="/w/index.php?title=Fungsi_hiperbolik_konfluen&action=edit&redlink=1" class="new" title="Fungsi hiperbolik konfluen (halaman belum tersedia)">konfluen</a></li></ul></li> <li><a href="/w/index.php?title=Fungsi_K&action=edit&redlink=1" class="new" title="Fungsi K (halaman belum tersedia)">K</a></li> <li><a href="/w/index.php?title=Fungsi_sinkrotron&action=edit&redlink=1" class="new" title="Fungsi sinkrotron (halaman belum tersedia)">sinkrotron</a></li> <li><a href="/w/index.php?title=Fungsi_tabung_parabolik&action=edit&redlink=1" class="new" title="Fungsi tabung parabolik (halaman belum tersedia)">tabung parabolik</a></li> <li><a href="/w/index.php?title=Fungsi_tanda_tanya_Minkowski&action=edit&redlink=1" class="new" title="Fungsi tanda tanya Minkowski (halaman belum tersedia)">tanda tanya Minkowski</a></li> <li><a href="/w/index.php?title=Pentasi&action=edit&redlink=1" class="new" title="Pentasi (halaman belum tersedia)">Pentasi</a></li> <li><a href="/w/index.php?title=Distribusi-t_Student&action=edit&redlink=1" class="new" title="Distribusi-t Student (halaman belum tersedia)">Student</a></li> <li><a href="/wiki/Tetrasi" title="Tetrasi">Tetrasi</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="">Diperoleh dari "<a dir="ltr" href="https://id.wikipedia.org/w/index.php?title=Sinus_dan_kosinus&oldid=22345579">https://id.wikipedia.org/w/index.php?title=Sinus_dan_kosinus&oldid=22345579</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Istimewa:Daftar_kategori" title="Istimewa:Daftar kategori">Kategori</a>: <ul><li><a href="/wiki/Kategori:Sudut" title="Kategori:Sudut">Sudut</a></li><li><a href="/w/index.php?title=Kategori:Fungsi_trigonometri&action=edit&redlink=1" class="new" title="Kategori:Fungsi trigonometri (halaman belum tersedia)">Fungsi 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