Sine and cosine - Wikipedia

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<li id="toc-Unit_circle_definition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Unit_circle_definition"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Unit circle definition</span> </div> </a> <ul id="toc-Unit_circle_definition-sublist" class="vector-toc-list"> <li id="toc-Graph_of_a_function_and_its_elementary_properties" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Graph_of_a_function_and_its_elementary_properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.1</span> <span>Graph of a function and its elementary properties</span> </div> </a> <ul id="toc-Graph_of_a_function_and_its_elementary_properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Continuity_and_differentiation" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Continuity_and_differentiation"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.2</span> <span>Continuity and differentiation</span> </div> </a> <ul id="toc-Continuity_and_differentiation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Integral_and_the_usage_in_mensuration" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Integral_and_the_usage_in_mensuration"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.3</span> <span>Integral and the usage in mensuration</span> </div> </a> <ul id="toc-Integral_and_the_usage_in_mensuration-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Inverse_functions" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Inverse_functions"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.4</span> <span>Inverse functions</span> </div> </a> <ul id="toc-Inverse_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_identities" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Other_identities"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.5</span> <span>Other identities</span> </div> </a> <ul id="toc-Other_identities-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Series_and_polynomials" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Series_and_polynomials"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Series and polynomials</span> </div> </a> <ul id="toc-Series_and_polynomials-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Complex_numbers_relationship" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Complex_numbers_relationship"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Complex numbers relationship</span> </div> </a> <button aria-controls="toc-Complex_numbers_relationship-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>Toggle Complex numbers relationship subsection</span> </button> <ul id="toc-Complex_numbers_relationship-sublist" class="vector-toc-list"> <li id="toc-Complex_exponential_function_definitions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Complex_exponential_function_definitions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Complex exponential function definitions</span> </div> </a> <ul id="toc-Complex_exponential_function_definitions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Polar_coordinates" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Polar_coordinates"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Polar coordinates</span> </div> </a> <ul id="toc-Polar_coordinates-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Complex_arguments" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Complex_arguments"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Complex arguments</span> </div> </a> <ul id="toc-Complex_arguments-sublist" class="vector-toc-list"> <li id="toc-Partial_fraction_and_product_expansions_of_complex_sine" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Partial_fraction_and_product_expansions_of_complex_sine"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3.1</span> <span>Partial fraction and product expansions of complex sine</span> </div> </a> <ul id="toc-Partial_fraction_and_product_expansions_of_complex_sine-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Usage_of_complex_sine" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Usage_of_complex_sine"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3.2</span> <span>Usage of complex sine</span> </div> </a> <ul id="toc-Usage_of_complex_sine-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Complex_graphs" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Complex_graphs"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Complex graphs</span> </div> </a> <ul id="toc-Complex_graphs-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Background" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Background"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Background</span> </div> </a> <button aria-controls="toc-Background-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>Toggle Background subsection</span> </button> <ul id="toc-Background-sublist" class="vector-toc-list"> <li id="toc-Etymology" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Etymology"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Etymology</span> </div> </a> <ul id="toc-Etymology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Software_implementations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Software_implementations"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Software implementations</span> </div> </a> <button aria-controls="toc-Software_implementations-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>Toggle Software implementations subsection</span> </button> <ul id="toc-Software_implementations-sublist" class="vector-toc-list"> <li id="toc-Turns_based_implementations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Turns_based_implementations"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Turns based implementations</span> </div> </a> <ul id="toc-Turns_based_implementations-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>References</span> </div> </a> <button aria-controls="toc-References-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>Toggle References subsection</span> </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-Footnotes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Footnotes"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Footnotes</span> </div> </a> <ul id="toc-Footnotes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Citations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Citations"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>Citations</span> </div> </a> <ul id="toc-Citations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Works_cited" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Works_cited"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.3</span> <span>Works cited</span> </div> </a> <ul id="toc-Works_cited-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>External links</span> </div> </a> <ul id="toc-External_links-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="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" 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Available in 9 languages" > <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-9" 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">9 languages</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 – Bangla" lang="bn" hreflang="bn" data-title="সাইন ও কোসাইন" data-language-autonym="বাংলা" data-language-local-name="Bangla" 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 – German" lang="de" hreflang="de" data-title="Sinus und Kosinus" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Sinus_dan_kosinus" title="Sinus dan kosinus – Indonesian" lang="id" hreflang="id" data-title="Sinus dan kosinus" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</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 – Georgian" lang="ka" hreflang="ka" data-title="სინუსი და კოსინუსი" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Sini,_kosini_na_tanjenti" title="Sini, kosini na tanjenti – Swahili" lang="sw" hreflang="sw" data-title="Sini, kosini na tanjenti" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target"><span>Kiswahili</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 – Dutch" lang="nl" hreflang="nl" data-title="Sinus en cosinus" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Trigonometriske_funksjoner#Sinus,_cosinus_og_tangens" title="Trigonometriske funksjoner – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Trigonometriske funksjoner" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</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 – Ukrainian" lang="uk" hreflang="uk" data-title="Синус та косинус" data-language-autonym="Українська" data-language-local-name="Ukrainian" 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 – Chinese" lang="zh" 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id="mw-content-subtitle"><span class="mw-redirectedfrom">(Redirected from <a href="/w/index.php?title=Sine&redirect=no" class="mw-redirect" title="Sine">Sine</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Fundamental trigonometric functions</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.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}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">"Sine" and "Cosine" redirect here. For other uses, see <a href="/wiki/Sine_(disambiguation)" class="mw-disambig" title="Sine (disambiguation)">Sine (disambiguation)</a> and <a href="/wiki/Cosine_(disambiguation)" class="mw-disambig" title="Cosine (disambiguation)">Cosine (disambiguation)</a>. "Sine" is not to be confused with <a href="/wiki/Sign" title="Sign">Sign</a>, <a href="/wiki/Sign_(mathematics)" title="Sign (mathematics)">Sign (mathematics)</a> or the <a href="/wiki/Sign_function" title="Sign function">sign function</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1257001546">.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%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</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;">Sine and cosine</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/File:Sine_cosine_one_period.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Sine_cosine_one_period.svg/250px-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/500px-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;">General information</th></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;">General definition</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(\theta )={\frac {\textrm {opposite}}{\textrm {hypotenuse}}}\\[8pt]&\cos(\theta )={\frac {\textrm {adjacent}}{\textrm {hypotenuse}}}\\[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> <mtd> <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 class="MJX-TeXAtom-ORD"> <mtext>opposite</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>hypotenuse</mtext> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mtext>adjacent</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>hypotenuse</mtext> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&\sin(\theta )={\frac {\textrm {opposite}}{\textrm {hypotenuse}}}\\[8pt]&\cos(\theta )={\frac {\textrm {adjacent}}{\textrm {hypotenuse}}}\\[8pt]\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52374fd3474dfab1331993d6c170e9cac82f4a4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:22.143ex; height:13.843ex;" alt="{\displaystyle {\begin{aligned}&\sin(\theta )={\frac {\textrm {opposite}}{\textrm {hypotenuse}}}\\[8pt]&\cos(\theta )={\frac {\textrm {adjacent}}{\textrm {hypotenuse}}}\\[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;">Fields of application</th><td class="infobox-data"><a href="/wiki/Trigonometry" title="Trigonometry">Trigonometry</a>, <a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a>, etc.</td></tr></tbody></table> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, <b>sine</b> and <b>cosine</b> are <a href="/wiki/Trigonometric_functions" title="Trigonometric functions">trigonometric functions</a> of an <a href="/wiki/Angle" title="Angle">angle</a>. The sine and cosine of an acute <a href="/wiki/Angle" title="Angle">angle</a> are defined in the context of a <a href="/wiki/Right_triangle" title="Right triangle">right triangle</a>: for the specified angle, its sine is the ratio of the length of the side opposite that angle to the length of the longest side of the <a href="/wiki/Triangle" title="Triangle">triangle</a> (the <a href="/wiki/Hypotenuse" title="Hypotenuse">hypotenuse</a>), and the cosine is the <a href="/wiki/Ratio" title="Ratio">ratio</a> of the length of the adjacent leg to that of the <a href="/wiki/Hypotenuse" title="Hypotenuse">hypotenuse</a>. For an angle <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>, the sine and cosine functions are denoted as <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> 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(\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>. </p><p>The definitions of sine and cosine have been extended to any <a href="/wiki/Real_number" title="Real number">real</a> value in terms of the lengths of certain line segments in a <a href="/wiki/Unit_circle" title="Unit circle">unit circle</a>. More modern definitions express the sine and cosine as <a href="/wiki/Series_(mathematics)" title="Series (mathematics)">infinite series</a>, or as the solutions of certain <a href="/wiki/Differential_equation" title="Differential equation">differential equations</a>, allowing their extension to arbitrary positive and negative values and even to <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>. </p><p>The sine and cosine functions are commonly used to model <a href="/wiki/Periodic_function" title="Periodic function">periodic</a> phenomena such as <a href="/wiki/Sound" title="Sound">sound</a> and <a href="/wiki/Light_waves" class="mw-redirect" title="Light waves">light waves</a>, the position and velocity of harmonic oscillators, sunlight intensity and day length, and average temperature variations throughout the year. They can be traced to the <a href="/wiki/Jy%C4%81,_koti-jy%C4%81_and_utkrama-jy%C4%81" title="Jyā, koti-jyā and utkrama-jyā"><i>jyā</i> and <i>koṭi-jyā</i></a> functions used in <a href="/wiki/Indian_astronomy" title="Indian astronomy">Indian astronomy</a> during the <a href="/wiki/Gupta_period" class="mw-redirect" title="Gupta period">Gupta period</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Elementary_descriptions">Elementary descriptions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine_and_cosine&action=edit&section=1" title="Edit section: Elementary descriptions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Right-angled_triangle_definition">Right-angled triangle definition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine_and_cosine&action=edit&section=2" title="Edit section: Right-angled triangle definition"><span>edit</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/File:Trigono_sine_en2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Trigono_sine_en2.svg/250px-Trigono_sine_en2.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Trigono_sine_en2.svg/330px-Trigono_sine_en2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Trigono_sine_en2.svg/500px-Trigono_sine_en2.svg.png 2x" data-file-width="600" data-file-height="600" /></a><figcaption>For the angle <span class="texhtml"><i>α</i></span>, the sine function gives the ratio of the length of the opposite side to the length of the hypotenuse.</figcaption></figure> <p>To define the sine and cosine of an acute angle <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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" /></span>, start with a <a href="/wiki/Right_triangle" title="Right triangle">right triangle</a> that contains an angle of measure <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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" /></span>; in the accompanying figure, angle <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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" /></span> in a right triangle <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 ABC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ABC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e55b44cfd965fbdc7a328d5db8a35a619db0971" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.273ex; height:2.176ex;" alt="{\displaystyle ABC}" /></span> is the angle of interest. The three sides of the triangle are named as follows:<sup id="cite_ref-FOOTNOTEYoung2017[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA27_27]_1-0" class="reference"><a href="#cite_note-FOOTNOTEYoung2017[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA27_27]-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <ul><li>The <i>opposite side</i> is the side opposite to the angle of interest; in this case, it is <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>.</li> <li>The <i>hypotenuse</i> is the side opposite the right angle; in this case, it is <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 h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" /></span>. The hypotenuse is always the longest side of a right-angled triangle.</li> <li>The <i>adjacent side</i> is the remaining side; in this case, it is <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>. It forms a side of (and is adjacent to) both the angle of interest and the right angle.</li></ul> <p>Once such a triangle is chosen, the sine of the angle is equal to the length of the opposite side divided by the length of the hypotenuse, and the cosine of the angle is equal to the length of the adjacent side divided by the length of the hypotenuse:<sup id="cite_ref-FOOTNOTEYoung2017[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA27_27]_1-1" class="reference"><a href="#cite_note-FOOTNOTEYoung2017[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA27_27]-1"><span class="cite-bracket">[</span>1<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 \sin(\alpha )={\frac {\text{opposite}}{\text{hypotenuse}}},\qquad \cos(\alpha )={\frac {\text{adjacent}}{\text{hypotenuse}}}.}"> <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>opposite</mtext> <mtext>hypotenuse</mtext> </mfrac> </mrow> <mo>,</mo> <mspace width="2em"></mspace> <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>adjacent</mtext> <mtext>hypotenuse</mtext> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(\alpha )={\frac {\text{opposite}}{\text{hypotenuse}}},\qquad \cos(\alpha )={\frac {\text{adjacent}}{\text{hypotenuse}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0cbaaff1eba0ec735281921a03965ab3a36763b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:49.647ex; height:5.843ex;" alt="{\displaystyle \sin(\alpha )={\frac {\text{opposite}}{\text{hypotenuse}}},\qquad \cos(\alpha )={\frac {\text{adjacent}}{\text{hypotenuse}}}.}" /></span> </p><p>The other trigonometric functions of the angle can be defined similarly; for example, the <a href="/wiki/Trigonometric_functions#Right-angled_triangle_definitions" title="Trigonometric functions">tangent</a> is the ratio between the opposite and adjacent sides or equivalently the ratio between the sine and cosine functions. The <a href="/wiki/Multiplicative_inverse" title="Multiplicative inverse">reciprocal</a> of sine is cosecant, which gives the ratio of the hypotenuse length to the length of the opposite side. Similarly, the reciprocal of cosine is secant, which gives the ratio of the hypotenuse length to that of the adjacent side. The cotangent function is the ratio between the adjacent and opposite sides, a reciprocal of a tangent function. These functions can be formulated as:<sup id="cite_ref-FOOTNOTEYoung2017[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA27_27]_1-2" class="reference"><a href="#cite_note-FOOTNOTEYoung2017[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA27_27]-1"><span class="cite-bracket">[</span>1<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 {\begin{aligned}\tan(\theta )&={\frac {\sin(\theta )}{\cos(\theta )}}={\frac {\text{opposite}}{\text{adjacent}}},\\\cot(\theta )&={\frac {1}{\tan(\theta )}}={\frac {\text{adjacent}}{\text{opposite}}},\\\csc(\theta )&={\frac {1}{\sin(\theta )}}={\frac {\text{hypotenuse}}{\text{opposite}}},\\\sec(\theta )&={\frac {1}{\cos(\theta )}}={\frac {\textrm {hypotenuse}}{\textrm {adjacent}}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <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> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mtext>opposite</mtext> <mtext>adjacent</mtext> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>cot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mtext>adjacent</mtext> <mtext>opposite</mtext> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>csc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mtext>hypotenuse</mtext> <mtext>opposite</mtext> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>sec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mtext>hypotenuse</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>adjacent</mtext> </mrow> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\tan(\theta )&={\frac {\sin(\theta )}{\cos(\theta )}}={\frac {\text{opposite}}{\text{adjacent}}},\\\cot(\theta )&={\frac {1}{\tan(\theta )}}={\frac {\text{adjacent}}{\text{opposite}}},\\\csc(\theta )&={\frac {1}{\sin(\theta )}}={\frac {\text{hypotenuse}}{\text{opposite}}},\\\sec(\theta )&={\frac {1}{\cos(\theta )}}={\frac {\textrm {hypotenuse}}{\textrm {adjacent}}}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6a3c9e1e99b68fbda583330226dc87136a62315" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.171ex; width:32.983ex; height:25.509ex;" alt="{\displaystyle {\begin{aligned}\tan(\theta )&={\frac {\sin(\theta )}{\cos(\theta )}}={\frac {\text{opposite}}{\text{adjacent}}},\\\cot(\theta )&={\frac {1}{\tan(\theta )}}={\frac {\text{adjacent}}{\text{opposite}}},\\\csc(\theta )&={\frac {1}{\sin(\theta )}}={\frac {\text{hypotenuse}}{\text{opposite}}},\\\sec(\theta )&={\frac {1}{\cos(\theta )}}={\frac {\textrm {hypotenuse}}{\textrm {adjacent}}}.\end{aligned}}}" /></span> </p> <div class="mw-heading mw-heading3"><h3 id="Special_angle_measures">Special angle measures</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine_and_cosine&action=edit&section=3" title="Edit section: Special angle measures"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As stated, the values <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> 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(\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> appear to depend on the choice of a right triangle containing an angle of measure <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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" /></span>. However, this is not the case as all such triangles are <a href="/wiki/Similarity_(geometry)" title="Similarity (geometry)">similar</a>, and so the ratios are the same for each of them. For example, each <a href="/wiki/Catheti" class="mw-redirect" title="Catheti">leg</a> of the 45-45-90 right triangle is 1 unit, and its hypotenuse is <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>; therefore, <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 45^{\circ }=\cos 45^{\circ }={\frac {\sqrt {2}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <msup> <mn>45</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <msup> <mn>45</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>2</mn> </msqrt> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sin 45^{\circ }=\cos 45^{\circ }={\frac {\sqrt {2}}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35373a42f10dc6f660ac17f1243294010fea43fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:22.723ex; height:4.176ex;" alt="{\textstyle \sin 45^{\circ }=\cos 45^{\circ }={\frac {\sqrt {2}}{2}}}" /></span>.<sup id="cite_ref-FOOTNOTEYoung2017[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA36_36]_2-0" class="reference"><a href="#cite_note-FOOTNOTEYoung2017[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA36_36]-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The following table shows the special value of each input for both sine and cosine with the domain between <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 0<\alpha <{\frac {\pi }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>α</mi> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle 0<\alpha <{\frac {\pi }{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d878d68144536cb7db4805f440b6cf3e0b82825b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:10.625ex; height:3.176ex;" alt="{\textstyle 0<\alpha <{\frac {\pi }{2}}}" /></span>. The input in this table provides various unit systems such as degree, radian, and so on. The angles other than those five can be obtained by using a calculator.<sup id="cite_ref-FOOTNOTEVarbergPurcellRigdon200742_3-0" class="reference"><a href="#cite_note-FOOTNOTEVarbergPurcellRigdon200742-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEYoung2017[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA37_37],_[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA78_78]_4-0" class="reference"><a href="#cite_note-FOOTNOTEYoung2017[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA37_37],_[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA78_78]-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <table class="wikitable" style="text-align:center;"> <tbody><tr> <th colspan="4" style="background:#ffdead;">Angle, <span class="texhtml mvar" style="font-style:italic;">x</span> </th> <th colspan="2" style="background:#ffdead;"><span class="texhtml">sin(<i>x</i>)</span> </th> <th colspan="2" style="background:#ffdead;"><span class="texhtml">cos(<i>x</i>)</span> </th></tr> <tr> <th style="background:#efefef;"><a href="/wiki/Degree_(angle)" title="Degree (angle)">Degrees</a> </th> <th style="background:#efefef;"><a href="/wiki/Radian" title="Radian">Radians</a> </th> <th style="background:#efefef;"><a href="/wiki/Gradian" title="Gradian">Gradians</a> </th> <th style="background:#efefef;"><a href="/wiki/Turn_(geometry)" class="mw-redirect" title="Turn (geometry)">Turns</a> </th> <th style="background:#efefef;">Exact </th> <th style="background:#efefef;">Decimal </th> <th style="background:#efefef;">Exact </th> <th style="background:#efefef;">Decimal </th></tr> <tr> <td>0° </td> <td>0 </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^{g}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0^{g}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec8a3ac7e88cac5b0efbd0ccee88f8fb38ddf6d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.184ex; height:2.343ex;" alt="{\displaystyle 0^{g}}" /></span> </td> <td>0 </td> <td>0 </td> <td>0 </td> <td>1 </td> <td>1 </td></tr> <tr> <td>30° </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 {1}{6}}\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{6}}\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97e405cc6714ea7bcd23cc0b93be4db2891a2a3e" 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 {1}{6}}\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 33{\frac {1}{3}}^{g}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>33</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 33{\frac {1}{3}}^{g}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03b18c8a4960058041de77fc9cad60f6a39d445c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:5.345ex; height:5.343ex;" alt="{\displaystyle 33{\frac {1}{3}}^{g}}" /></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 {1}{12}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>12</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{12}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff11356aa26897b393d8263bbc42082509b26e0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.161ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{12}}}" /></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 {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a11cfb2fdb143693b1daf78fcb5c11a023cb1c55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:1.999ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{2}}}" /></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>0.866 </td></tr> <tr> <td>45° </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 {1}{4}}\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{4}}\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9abf27bdbd2e4c3a20db2346924be65f99d1222c" 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 {1}{4}}\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 50^{g}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>50</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 50^{g}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf83158d702603d93ccf681b48d3a3a59e8c68eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.346ex; height:2.343ex;" alt="{\displaystyle 50^{g}}" /></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 {1}{8}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{8}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3f20dc5ae5815ab8628a1294c40639574e0c88e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:1.999ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{8}}}" /></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>0.707 </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>0.707 </td></tr> <tr> <td>60° </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 {1}{3}}\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{3}}\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b69e5105c103f9d81fd8b8a81dc45fdffc1ab3d" 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 {1}{3}}\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 66{\frac {2}{3}}^{g}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>66</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 66{\frac {2}{3}}^{g}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0382a4263948d597eac736f0c51c3606837144f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:5.345ex; height:5.343ex;" alt="{\displaystyle 66{\frac {2}{3}}^{g}}" /></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 {1}{6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{6}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba1caf4c96d913f6aafa9da0634f069fa42e0290" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:1.999ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{6}}}" /></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>0.866 </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 {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a11cfb2fdb143693b1daf78fcb5c11a023cb1c55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:1.999ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{2}}}" /></span> </td> <td>0.5 </td></tr> <tr> <td>90° </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 {1}{2}}\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db6c658a4c6c72f39ad03b95b0c4f4b44c4c0c2e" 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 {1}{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 100^{g}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>100</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 100^{g}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2263be1cdf87dc8c09883ea9a4c53ee3b90d891" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.509ex; height:2.343ex;" alt="{\displaystyle 100^{g}}" /></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 {1}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2dfb63ee75ec084f2abb25d248bc151a2687508" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:1.999ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{4}}}" /></span> </td> <td>1 </td> <td>1 </td> <td>0 </td> <td>0 </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Laws">Laws</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine_and_cosine&action=edit&section=4" title="Edit section: Laws"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Law_of_sines" title="Law of sines">Law of sines</a> and <a href="/wiki/Law_of_cosines" title="Law of cosines">Law of cosines</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Law_of_sines_(simple).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Law_of_sines_%28simple%29.svg/220px-Law_of_sines_%28simple%29.svg.png" decoding="async" width="220" height="251" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Law_of_sines_%28simple%29.svg/330px-Law_of_sines_%28simple%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Law_of_sines_%28simple%29.svg/440px-Law_of_sines_%28simple%29.svg.png 2x" data-file-width="350" data-file-height="400" /></a><figcaption>Law of sines and cosines' illustration</figcaption></figure> <p>The <a href="/wiki/Law_of_sines" title="Law of sines">law of sines</a> is useful for computing the lengths of the unknown sides in a triangle if two angles and one side are known.<sup id="cite_ref-FOOTNOTEAxler2012[httpsbooksgooglecombooksidB5RxDwAAQBAJpgPA634_634]_5-0" class="reference"><a href="#cite_note-FOOTNOTEAxler2012[httpsbooksgooglecombooksidB5RxDwAAQBAJpgPA634_634]-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Given a triangle <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 ABC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ABC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e55b44cfd965fbdc7a328d5db8a35a619db0971" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.273ex; height:2.176ex;" alt="{\displaystyle ABC}" /></span> with sides <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>, <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>, 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 c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" /></span>, and angles opposite those sides <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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" /></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 \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" /></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 \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>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/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" /></span>, the law states, <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 \alpha }{a}}={\frac {\sin \beta }{b}}={\frac {\sin \gamma }{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>α</mi> </mrow> <mi>a</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>β</mi> </mrow> <mi>b</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>γ</mi> </mrow> <mi>c</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\sin \alpha }{a}}={\frac {\sin \beta }{b}}={\frac {\sin \gamma }{c}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08ea457459d16cc937492f939b8e40e543078680" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:23.162ex; height:5.509ex;" alt="{\displaystyle {\frac {\sin \alpha }{a}}={\frac {\sin \beta }{b}}={\frac {\sin \gamma }{c}}.}" /></span> This is equivalent to the equality of the first three expressions below: <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 {a}{\sin \alpha }}={\frac {b}{\sin \beta }}={\frac {c}{\sin \gamma }}=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>α</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>β</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>γ</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 \alpha }}={\frac {b}{\sin \beta }}={\frac {c}{\sin \gamma }}=2R,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0db5ccda5a3e2e9a89f282c1fc4ef0ace7014bcc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:29.187ex; height:5.843ex;" alt="{\displaystyle {\frac {a}{\sin \alpha }}={\frac {b}{\sin \beta }}={\frac {c}{\sin \gamma }}=2R,}" /></span> where <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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" /></span> is the triangle's <a href="/wiki/Circumcircle" title="Circumcircle">circumradius</a>. </p><p>The <a href="/wiki/Law_of_cosines" title="Law of cosines">law of cosines</a> is useful for computing the length of an unknown side if two other sides and an angle are known.<sup id="cite_ref-FOOTNOTEAxler2012[httpsbooksgooglecombooksidB5RxDwAAQBAJpgPA634_634]_5-1" class="reference"><a href="#cite_note-FOOTNOTEAxler2012[httpsbooksgooglecombooksidB5RxDwAAQBAJpgPA634_634]-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> The law states, <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(\gamma )=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>γ</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(\gamma )=c^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f30e796ac2afe7443a406b49fa5a718eae315275" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.136ex; height:3.176ex;" alt="{\displaystyle a^{2}+b^{2}-2ab\cos(\gamma )=c^{2}}" /></span> In the case where <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 =\pi /2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</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 \gamma =\pi /2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/982a2789fde8a2fa65fd04c5e1bfec4dfecffc73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.018ex; height:2.843ex;" alt="{\displaystyle \gamma =\pi /2}" /></span> from which <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(\gamma )=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>γ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(\gamma )=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04328c6895a6a1741585425ddb9474ea27e5ea0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.444ex; height:2.843ex;" alt="{\displaystyle \cos(\gamma )=0}" /></span>, the resulting equation becomes the <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a>.<sup id="cite_ref-FOOTNOTEAxler2012[httpsbooksgooglecombooksidB5RxDwAAQBAJpgPA632_632]_6-0" class="reference"><a href="#cite_note-FOOTNOTEAxler2012[httpsbooksgooglecombooksidB5RxDwAAQBAJpgPA632_632]-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Vector_definition">Vector definition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine_and_cosine&action=edit&section=5" title="Edit section: Vector definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Cross_product" title="Cross product">cross product</a> and <a href="/wiki/Dot_product" title="Dot product">dot product</a> are operations on two <a href="/wiki/Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">vectors</a> in <a href="/wiki/Euclidean_vector_space" class="mw-redirect" title="Euclidean vector space">Euclidean vector space</a>. The sine and cosine functions can be defined in terms of the cross product and dot product. If <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 \mathbb {a} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">a</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {a} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4eceac218129ac559924499695b0fee74ac4341" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:1.676ex;" alt="{\displaystyle \mathbb {a} }" /></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 \mathbb {b} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">b</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {b} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b45ff930934e4f0a5476d53a1b3d71f7ab5d57ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \mathbb {b} }" /></span> are vectors, 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 \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> is the angle between <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 \mathbb {a} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">a</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {a} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4eceac218129ac559924499695b0fee74ac4341" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:1.676ex;" alt="{\displaystyle \mathbb {a} }" /></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 \mathbb {b} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">b</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {b} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b45ff930934e4f0a5476d53a1b3d71f7ab5d57ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \mathbb {b} }" /></span>, then sine and cosine can be defined as: <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 {\begin{aligned}\sin(\theta )&={\frac {|\mathbb {a} \times \mathbb {b} |}{|a||b|}},\\\cos(\theta )&={\frac {\mathbb {a} \cdot \mathbb {b} }{|a||b|}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">a</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">b</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sin(\theta )&={\frac {|\mathbb {a} \times \mathbb {b} |}{|a||b|}},\\\cos(\theta )&={\frac {\mathbb {a} \cdot \mathbb {b} }{|a||b|}}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bffda9a6c97544f7002e50bc1f7a0387faf5fdca" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:17.933ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}\sin(\theta )&={\frac {|\mathbb {a} \times \mathbb {b} |}{|a||b|}},\\\cos(\theta )&={\frac {\mathbb {a} \cdot \mathbb {b} }{|a||b|}}.\end{aligned}}}" /></span> </p> <div class="mw-heading mw-heading2"><h2 id="Analytic_descriptions">Analytic descriptions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine_and_cosine&action=edit&section=6" title="Edit section: Analytic descriptions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Unit_circle_definition">Unit circle definition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine_and_cosine&action=edit&section=7" title="Edit section: Unit circle definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The sine and cosine functions may also be defined in a more general way by using <a href="/wiki/Unit_circle#Trigonometric_functions_on_the_unit_circle" title="Unit circle">unit circle</a>, a circle of radius one centered at the origin <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 (0,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d630d3e781a53b0a3559ae7e5b45f9479a3141a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,0)}" /></span>, formulated as the equation of <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^{2}+y^{2}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+y^{2}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec84b90236512e8d27ff1a8f7707b60b63327de1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.7ex; height:3.009ex;" alt="{\displaystyle x^{2}+y^{2}=1}" /></span> in the <a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian coordinate system</a>. Let a line through the origin intersect the unit circle, making an angle of <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> with the positive half of the <span class="nowrap"><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>-</span>axis. The <span class="nowrap"><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>-</span> and <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" /></span>-</span>coordinates of this point of intersection are equal 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 \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> 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 \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>, respectively; that is,<sup id="cite_ref-FOOTNOTEVarbergPurcellRigdon200741_7-0" class="reference"><a href="#cite_note-FOOTNOTEVarbergPurcellRigdon200741-7"><span class="cite-bracket">[</span>7<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 \sin(\theta )=y,\qquad \cos(\theta )=x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>y</mi> <mo>,</mo> <mspace width="2em"></mspace> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(\theta )=y,\qquad \cos(\theta )=x.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e7decc76049a9174f255e3491d6e58bab1d4740" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.774ex; height:2.843ex;" alt="{\displaystyle \sin(\theta )=y,\qquad \cos(\theta )=x.}" /></span> </p><p>This definition is consistent with the right-angled triangle definition of sine and cosine when <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 0<\theta <{\frac {\pi }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>θ</mi> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle 0<\theta <{\frac {\pi }{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef78a27752a1f9a51d3b66c7863740f477aa4756" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:10.228ex; height:3.176ex;" alt="{\textstyle 0<\theta <{\frac {\pi }{2}}}" /></span> because the length of the hypotenuse of the unit circle is always 1; mathematically speaking, the sine of an angle equals the opposite side of the triangle, which is simply the <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" /></span>-</span>coordinate. A similar argument can be made for the cosine function to show that the cosine of an angle when <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 0<\theta <{\frac {\pi }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>θ</mi> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle 0<\theta <{\frac {\pi }{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef78a27752a1f9a51d3b66c7863740f477aa4756" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:10.228ex; height:3.176ex;" alt="{\textstyle 0<\theta <{\frac {\pi }{2}}}" /></span>, even under the new definition using the unit circle.<sup id="cite_ref-FOOTNOTEYoung2017[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA68_68]_8-0" class="reference"><a href="#cite_note-FOOTNOTEYoung2017[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA68_68]-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEVarbergPurcellRigdon200747_9-0" class="reference"><a href="#cite_note-FOOTNOTEVarbergPurcellRigdon200747-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Graph_of_a_function_and_its_elementary_properties">Graph of a function and its elementary properties</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine_and_cosine&action=edit&section=8" title="Edit section: Graph of a function and its elementary properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Circle_cos_sin.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Circle_cos_sin.gif/500px-Circle_cos_sin.gif" decoding="async" width="440" height="264" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/3/3b/Circle_cos_sin.gif 1.5x" data-file-width="650" data-file-height="390" /></a><figcaption>Animation demonstrating how the sine function (in red) is graphed from the <span class="nowrap"><span class="texhtml"><i>y</i></span>-</span>coordinate (red dot) of a point on the <a href="/wiki/Unit_circle" title="Unit circle">unit circle</a> (in green), at an angle of <span class="texhtml"><i>θ</i></span>. The cosine (in blue) is the <span class="nowrap"><span class="texhtml"><i>x</i></span>-</span>coordinate.</figcaption></figure> <p>Using the unit circle definition has the advantage of drawing a graph of sine and cosine functions. This can be done by rotating counterclockwise a point along the circumference of a circle, depending on the input <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 >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b0ac07626379d065418cc158ce6be9aeccf33b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.351ex; height:2.176ex;" alt="{\displaystyle \theta >0}" /></span>. In a sine function, if the input is <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 \theta ={\frac {\pi }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \theta ={\frac {\pi }{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e6095ebb8d0b6505f453ae2da2fbd3477ecd3a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.967ex; height:3.176ex;" alt="{\textstyle \theta ={\frac {\pi }{2}}}" /></span>, the point is rotated counterclockwise and stopped exactly on the <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" /></span>-</span>axis. If <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 =\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab4db588619489e27efb50a1d0d5aa016c49ce15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.521ex; height:2.176ex;" alt="{\displaystyle \theta =\pi }" /></span>, the point is at the circle's halfway. If <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 =2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =2\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af3479f95999b7e195e5ecc2ee808bf02286332e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.683ex; height:2.176ex;" alt="{\displaystyle \theta =2\pi }" /></span>, the point returned to its origin. This results that both sine and cosine functions have the <a href="/wiki/Range_of_a_function" title="Range of a function">range</a> between <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\leq y\leq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> <mo>≤</mo> <mi>y</mi> <mo>≤</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1\leq y\leq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29d02e9a39a0ad4594157a2f8b82ad89863543c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.485ex; height:2.509ex;" alt="{\displaystyle -1\leq y\leq 1}" /></span>.<sup id="cite_ref-FOOTNOTEVarbergPurcellRigdon200741&ndash;42_10-0" class="reference"><a href="#cite_note-FOOTNOTEVarbergPurcellRigdon200741&ndash;42-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p>Extending the angle to any real domain, the point rotated counterclockwise continuously. This can be done similarly for the cosine function as well, although the point is rotated initially from the <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" /></span>-</span>coordinate. In other words, both sine and cosine functions are <a href="/wiki/Periodic_function" title="Periodic function">periodic</a>, meaning any angle added by the circumference's circle is the angle itself. Mathematically,<sup id="cite_ref-FOOTNOTEVarbergPurcellRigdon200741,_43_11-0" class="reference"><a href="#cite_note-FOOTNOTEVarbergPurcellRigdon200741,_43-11"><span class="cite-bracket">[</span>11<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 \sin(\theta +2\pi )=\sin(\theta ),\qquad \cos(\theta +2\pi )=\cos(\theta ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="2em"></mspace> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(\theta +2\pi )=\sin(\theta ),\qquad \cos(\theta +2\pi )=\cos(\theta ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d22ad562f48059fa7b33be5b34f0b0ca1979cd0d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.725ex; height:2.843ex;" alt="{\displaystyle \sin(\theta +2\pi )=\sin(\theta ),\qquad \cos(\theta +2\pi )=\cos(\theta ).}" /></span> </p><p>A function <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" /></span> is said to be <a href="/wiki/Odd_function" class="mw-redirect" title="Odd function">odd</a> if <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)=-f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(-x)=-f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b022ffe516cf5bc26a68fd954753aa2bddf508f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.55ex; height:2.843ex;" alt="{\displaystyle f(-x)=-f(x)}" /></span>, and is said to be <a href="/wiki/Even_function" class="mw-redirect" title="Even function">even</a> if <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)=f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(-x)=f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/185fd2e78903788bc5756b067d0ac6aae1846724" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.742ex; height:2.843ex;" alt="{\displaystyle f(-x)=f(x)}" /></span>. The sine function is odd, whereas the cosine function is even.<sup id="cite_ref-FOOTNOTEYoung2012[httpsbooksgooglecombooksidOMrcN0a3LxICpgRA1-PA165_165]_12-0" class="reference"><a href="#cite_note-FOOTNOTEYoung2012[httpsbooksgooglecombooksidOMrcN0a3LxICpgRA1-PA165_165]-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> Both sine and cosine functions are similar, with their difference being <a href="/wiki/Translation_(geometry)" title="Translation (geometry)">shifted</a> by <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>. This means,<sup id="cite_ref-FOOTNOTEVarbergPurcellRigdon200742,_47_13-0" class="reference"><a href="#cite_note-FOOTNOTEVarbergPurcellRigdon200742,_47-13"><span class="cite-bracket">[</span>13<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 {\begin{aligned}\sin(\theta )&=\cos \left({\frac {\pi }{2}}-\theta \right),\\\cos(\theta )&=\sin \left({\frac {\pi }{2}}-\theta \right).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <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> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <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> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sin(\theta )&=\cos \left({\frac {\pi }{2}}-\theta \right),\\\cos(\theta )&=\sin \left({\frac {\pi }{2}}-\theta \right).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4998408fbf02a4669c3e151a840a7bc314101ef" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:22.493ex; height:9.843ex;" alt="{\displaystyle {\begin{aligned}\sin(\theta )&=\cos \left({\frac {\pi }{2}}-\theta \right),\\\cos(\theta )&=\sin \left({\frac {\pi }{2}}-\theta \right).\end{aligned}}}" /></span> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Cosine_fixed_point.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Cosine_fixed_point.svg/250px-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/500px-Cosine_fixed_point.svg.png 2x" data-file-width="600" data-file-height="480" /></a><figcaption>The fixed point iteration <span class="texhtml"><i>x</i><sub><i>n</i>+1</sub> = cos(<i>x<sub>n</sub></i>)</span> with initial value <span class="texhtml"><i>x</i><sub>0</sub> = −1</span> converges to the Dottie number.</figcaption></figure> <p>Zero is the only real <a href="/wiki/Fixed_point_(mathematics)" title="Fixed point (mathematics)">fixed point</a> of the sine function; in other words the only intersection of the sine function and the <a href="/wiki/Identity_function" title="Identity function">identity function</a> is <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(0)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(0)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecb40badcf53b50e5d03c49c7a5e1977e3ff262e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.088ex; height:2.843ex;" alt="{\displaystyle \sin(0)=0}" /></span>. The only real fixed point of the cosine function is called the <a href="/wiki/Dottie_number" title="Dottie number">Dottie number</a>. The Dottie number is the unique real root of the equation <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> </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/3cf8688ffc25998040ed8bf59f0a6298233a143c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.678ex; height:2.843ex;" alt="{\displaystyle \cos(x)=x}" /></span>. The decimal expansion of the Dottie number is approximately 0.739085.<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> </p> <div class="mw-heading mw-heading4"><h4 id="Continuity_and_differentiation">Continuity and differentiation</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine_and_cosine&action=edit&section=9" title="Edit section: Continuity and differentiation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Differentiation_of_trigonometric_functions" title="Differentiation of trigonometric functions">Differentiation of trigonometric functions</a></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Sine_quads_01_Pengo.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Sine_quads_01_Pengo.svg/500px-Sine_quads_01_Pengo.svg.png" decoding="async" width="390" height="168" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Sine_quads_01_Pengo.svg/960px-Sine_quads_01_Pengo.svg.png 1.5x" data-file-width="1065" data-file-height="459" /></a><figcaption>The quadrants of the unit circle and of sin(<i>x</i>), using the <a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian coordinate system</a></figcaption></figure> <p>The sine and cosine functions are infinitely differentiable.<sup id="cite_ref-FOOTNOTEBourchteinBourchtein2022[httpsbooksgooglecombooksidnGxOEAAAQBAJpgPA294_294]_15-0" class="reference"><a href="#cite_note-FOOTNOTEBourchteinBourchtein2022[httpsbooksgooglecombooksidnGxOEAAAQBAJpgPA294_294]-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> The derivative of sine is cosine, and the derivative of cosine is negative sine:<sup id="cite_ref-FOOTNOTEVarbergPurcellRigdon2007115_16-0" class="reference"><a href="#cite_note-FOOTNOTEVarbergPurcellRigdon2007115-16"><span class="cite-bracket">[</span>16<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 {\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> <mo>,</mo> <mspace width="2em"></mspace> <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> <mo>.</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/3ae62cb5b610f8b5309ca193d299662fb18d4099" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:46.745ex; height:5.509ex;" alt="{\displaystyle {\frac {d}{dx}}\sin(x)=\cos(x),\qquad {\frac {d}{dx}}\cos(x)=-\sin(x).}" /></span> Continuing the process in higher-order derivative results in the repeated same functions; the fourth derivative of a sine is the sine itself.<sup id="cite_ref-FOOTNOTEBourchteinBourchtein2022[httpsbooksgooglecombooksidnGxOEAAAQBAJpgPA294_294]_15-1" class="reference"><a href="#cite_note-FOOTNOTEBourchteinBourchtein2022[httpsbooksgooglecombooksidnGxOEAAAQBAJpgPA294_294]-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> These derivatives can be applied to the <a href="/wiki/First_derivative_test" class="mw-redirect" title="First derivative test">first derivative test</a>, according to which the <a href="/wiki/Monotone_function" class="mw-redirect" title="Monotone function">monotonicity</a> of a function can be defined as the inequality of function's first derivative greater or less than equal to zero.<sup id="cite_ref-FOOTNOTEVarbergPurcellRigdon2007155_17-0" class="reference"><a href="#cite_note-FOOTNOTEVarbergPurcellRigdon2007155-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> It can also be applied to <a href="/wiki/Second_derivative_test" class="mw-redirect" title="Second derivative test">second derivative test</a>, according to which the <a href="/wiki/Concave_function" title="Concave function">concavity</a> of a function can be defined by applying the inequality of the function's second derivative greater or less than equal to zero.<sup id="cite_ref-FOOTNOTEVarbergPurcellRigdon2007157_18-0" class="reference"><a href="#cite_note-FOOTNOTEVarbergPurcellRigdon2007157-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> The following table shows that both sine and cosine functions have concavity and monotonicity—the positive sign (<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>) denotes a graph is increasing (going upward) and the negative sign (<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>) is decreasing (going downward)—in certain intervals.<sup id="cite_ref-FOOTNOTEVarbergRigdonPurcell200742_19-0" class="reference"><a href="#cite_note-FOOTNOTEVarbergRigdonPurcell200742-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> This information can be represented as a Cartesian coordinates system divided into four quadrants. </p> <table class="wikitable" style="text-align:center;"> <tbody><tr> <th rowspan="2"><a href="/wiki/Cartesian_coordinate_system#Quadrants_and_octants" title="Cartesian coordinate system">Quadrant</a> </th> <th colspan="2">Angle </th> <th colspan="3">Sine </th> <th colspan="3">Cosine </th></tr> <tr> <th><a href="/wiki/Degree_(angle)" title="Degree (angle)">Degrees</a> </th> <th><a href="/wiki/Radian" title="Radian">Radians</a> </th> <th><a href="/wiki/Sign_(mathematics)" title="Sign (mathematics)">Sign</a> </th> <th><a href="/wiki/Monotonic_function" title="Monotonic function">Monotony</a> </th> <th><a href="/wiki/Convex_function" title="Convex function">Convexity</a> </th> <th><a href="/wiki/Sign_(mathematics)" title="Sign (mathematics)">Sign</a> </th> <th><a href="/wiki/Monotonic_function" title="Monotonic function">Monotony</a> </th> <th><a href="/wiki/Convex_function" title="Convex function">Convexity</a> </th></tr> <tr> <td style="text-align:left;">1st quadrant, 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>Increasing </td> <td>Concave </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>Decreasing </td> <td>Concave </td></tr> <tr> <td style="text-align:left;">2nd quadrant, 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>Decreasing </td> <td>Concave </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>Decreasing </td> <td>Convex </td></tr> <tr> <td style="text-align:left;">3rd quadrant, 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>Decreasing </td> <td>Convex </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>Increasing </td> <td>Convex </td></tr> <tr> <td style="text-align:left;">4th quadrant, 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>Increasing </td> <td>Convex </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>Increasing </td> <td>Concave </td></tr></tbody></table> <p>Both sine and cosine functions can be defined by using differential equations. The pair of <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 \theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>,</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\cos \theta ,\sin \theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/591da286444bb19845241d047707dd8793d4b143" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.765ex; height:2.843ex;" alt="{\displaystyle (\cos \theta ,\sin \theta )}" /></span> is the solution <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(\theta ),y(\theta ))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>y</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x(\theta ),y(\theta ))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3761d6e3635ea327a1fea316311a9974ba2a2898" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.128ex; height:2.843ex;" alt="{\displaystyle (x(\theta ),y(\theta ))}" /></span> to the two-dimensional system of <a href="/wiki/Differential_equation" title="Differential equation">differential equations</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="{\displaystyle y'(\theta )=x(\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y'(\theta )=x(\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e24c7568490cc0b2b359757eab6411e071ea5a4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.073ex; height:3.009ex;" alt="{\displaystyle y'(\theta )=x(\theta )}" /></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 x'(\theta )=-y(\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>y</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x'(\theta )=-y(\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d2efd00b2708f705444970234d98bb745c37b4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.876ex; height:3.009ex;" alt="{\displaystyle x'(\theta )=-y(\theta )}" /></span> with the <a href="/wiki/Initial_conditions" class="mw-redirect" title="Initial conditions">initial conditions</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="{\displaystyle y(0)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y(0)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/343c32f38bb379b4b208477b130d8f522d3f0788" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.388ex; height:2.843ex;" alt="{\displaystyle y(0)=0}" /></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 x(0)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(0)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e502f85fa127369616c9d6cce7b0cfdfad2abbc1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.562ex; height:2.843ex;" alt="{\displaystyle x(0)=1}" /></span>. One could interpret the unit circle in the above definitions as defining the <a href="/wiki/Phase_space_trajectory" class="mw-redirect" title="Phase space trajectory">phase space trajectory</a> of the differential equation with the given initial conditions. It can be interpreted as a phase space trajectory of the system of differential equations <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y'(\theta )=x(\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y'(\theta )=x(\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e24c7568490cc0b2b359757eab6411e071ea5a4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.073ex; height:3.009ex;" alt="{\displaystyle y'(\theta )=x(\theta )}" /></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 x'(\theta )=-y(\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>y</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x'(\theta )=-y(\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d2efd00b2708f705444970234d98bb745c37b4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.876ex; height:3.009ex;" alt="{\displaystyle x'(\theta )=-y(\theta )}" /></span> starting from the initial conditions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y(0)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y(0)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/343c32f38bb379b4b208477b130d8f522d3f0788" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.388ex; height:2.843ex;" alt="{\displaystyle y(0)=0}" /></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 x(0)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(0)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e502f85fa127369616c9d6cce7b0cfdfad2abbc1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.562ex; height:2.843ex;" alt="{\displaystyle x(0)=1}" /></span>.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (August 2024)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading4"><h4 id="Integral_and_the_usage_in_mensuration">Integral and the usage in mensuration</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine_and_cosine&action=edit&section=10" title="Edit section: Integral and the usage in mensuration"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/List_of_integrals_of_trigonometric_functions" title="List of integrals of trigonometric functions">List of integrals of trigonometric functions</a></div> <p>Their area under a curve can be obtained by using the <a href="/wiki/Integral" title="Integral">integral</a> with a certain bounded interval. Their antiderivatives are: <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 \int \sin(x)\,dx=-\cos(x)+C\qquad \int \cos(x)\,dx=\sin(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"></mspace> <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> <mspace width="2em"></mspace> <mo>∫</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <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> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \sin(x)\,dx=-\cos(x)+C\qquad \int \cos(x)\,dx=\sin(x)+C,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e14210f7e049e74d1153559c98ea8fd4fb5efa0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:58.801ex; height:5.676ex;" alt="{\displaystyle \int \sin(x)\,dx=-\cos(x)+C\qquad \int \cos(x)\,dx=\sin(x)+C,}" /></span> where <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" /></span> denotes the <a href="/wiki/Constant_of_integration" title="Constant of integration">constant of integration</a>.<sup id="cite_ref-FOOTNOTEVarbergPurcellRigdon2007199_20-0" class="reference"><a href="#cite_note-FOOTNOTEVarbergPurcellRigdon2007199-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> These antiderivatives may be applied to compute the mensuration properties of both sine and cosine functions' curves with a given interval. For example, the <a href="/wiki/Arc_length" title="Arc length">arc length</a> of the sine curve between <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> 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 t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" /></span> is <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 \int _{0}^{t}\!{\sqrt {1+\cos ^{2}(x)}}\,dx={\sqrt {2}}\operatorname {E} \left(t,{\frac {1}{\sqrt {2}}}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <mspace width="negativethinmathspace"></mspace> <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"></mspace> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{0}^{t}\!{\sqrt {1+\cos ^{2}(x)}}\,dx={\sqrt {2}}\operatorname {E} \left(t,{\frac {1}{\sqrt {2}}}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4e732378f19b126018432e28c062594d9538420" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:37.896ex; height:6.676ex;" alt="{\displaystyle \int _{0}^{t}\!{\sqrt {1+\cos ^{2}(x)}}\,dx={\sqrt {2}}\operatorname {E} \left(t,{\frac {1}{\sqrt {2}}}\right),}" /></span> where <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> is the <a href="/wiki/Elliptic_integral#Incomplete_elliptic_integral_of_the_second_kind" title="Elliptic integral">incomplete elliptic integral of the second kind</a> with 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>. It cannot be expressed using <a href="/wiki/Elementary_function" title="Elementary function">elementary functions</a>.<sup id="cite_ref-FOOTNOTEVince2023[httpsbooksgooglecombooksidGnW6EAAAQBAJpgPA162_162]_21-0" class="reference"><a href="#cite_note-FOOTNOTEVince2023[httpsbooksgooglecombooksidGnW6EAAAQBAJpgPA162_162]-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> In the case of a full period, its arc length is <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 L={\frac {4{\sqrt {2\pi ^{3}}}}{\Gamma (1/4)^{2}}}+{\frac {\Gamma (1/4)^{2}}{\sqrt {2\pi }}}={\frac {2\pi }{\varpi }}+2\varpi \approx 7.6404\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <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> <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> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <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> </mrow> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mi>ϖ</mi> </mfrac> </mrow> <mo>+</mo> <mn>2</mn> <mi>ϖ</mi> <mo>≈</mo> <mn>7.6404</mn> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L={\frac {4{\sqrt {2\pi ^{3}}}}{\Gamma (1/4)^{2}}}+{\frac {\Gamma (1/4)^{2}}{\sqrt {2\pi }}}={\frac {2\pi }{\varpi }}+2\varpi \approx 7.6404\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ff41c7793fe9f289d455a188718fc398929f33a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:49.825ex; height:6.843ex;" alt="{\displaystyle L={\frac {4{\sqrt {2\pi ^{3}}}}{\Gamma (1/4)^{2}}}+{\frac {\Gamma (1/4)^{2}}{\sqrt {2\pi }}}={\frac {2\pi }{\varpi }}+2\varpi \approx 7.6404\ldots }" /></span> where <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> is the <a href="/wiki/Gamma_function" title="Gamma function">gamma function</a> 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 \varpi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϖ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varpi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e50d258418b5fa150a86b58f8d5eb40613e3ebf7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:1.676ex;" alt="{\displaystyle \varpi }" /></span> is the <a href="/wiki/Lemniscate_constant" title="Lemniscate constant">lemniscate constant</a>.<sup id="cite_ref-FOOTNOTEAdlaj2012_22-0" class="reference"><a href="#cite_note-FOOTNOTEAdlaj2012-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Inverse_functions">Inverse functions</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine_and_cosine&action=edit&section=11" title="Edit section: Inverse functions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File: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">arcsin(<i>x</i>)</span> and <span class="texhtml">arccos(<i>x</i>)</span> functions graphed on the Cartesian plane</figcaption></figure> <p>The <a href="/wiki/Inverse_function" title="Inverse function">inverse function</a> of sine is arcsine or inverse sine, denoted as "arcsin", "asin", or <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 ^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin ^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21bc4ca64ac415c9ae60fb4e60fe4bddee17b8ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.188ex; height:2.676ex;" alt="{\displaystyle \sin ^{-1}}" /></span>.<sup id="cite_ref-FOOTNOTEVarbergPurcellRigdon2007366_23-0" class="reference"><a href="#cite_note-FOOTNOTEVarbergPurcellRigdon2007366-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> The inverse function of cosine is arccosine, denoted as "arccos", "acos", or <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 ^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos ^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7eabf706b4642d521c6279a2f07ac9715c7679a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.444ex; height:2.676ex;" alt="{\displaystyle \cos ^{-1}}" /></span>.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> As sine and cosine are not <a href="/wiki/Injective" class="mw-redirect" title="Injective">injective</a>, their inverses are not exact inverse functions, but partial inverse functions. For example, <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(0)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(0)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecb40badcf53b50e5d03c49c7a5e1977e3ff262e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.088ex; height:2.843ex;" alt="{\displaystyle \sin(0)=0}" /></span>, but also <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 )=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(\pi )=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6e7e3130a6ba2043ae878c9b1dc7aa7998afb8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.258ex; height:2.843ex;" alt="{\displaystyle \sin(\pi )=0}" /></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 \sin(2\pi )=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(2\pi )=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55c160c102e7cd62b4a03d1df4761601235a2b3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.42ex; height:2.843ex;" alt="{\displaystyle \sin(2\pi )=0}" /></span>, and so on. It follows that the arcsine function is multivalued: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arcsin(0)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arcsin(0)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b60d669653eb991a58122581a70bfcb5972faa2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.195ex; height:2.843ex;" alt="{\displaystyle \arcsin(0)=0}" /></span>, but also <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arcsin(0)=\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arcsin(0)=\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0baa77dd89945de8bfba8932ed09518922043202" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.364ex; height:2.843ex;" alt="{\displaystyle \arcsin(0)=\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 \arcsin(0)=2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arcsin(0)=2\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9d2ea0077afd2a8800402ebb1177219694fdbf9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.527ex; height:2.843ex;" alt="{\displaystyle \arcsin(0)=2\pi }" /></span>, and so on. When only one value is desired, the function may be restricted to its <a href="/wiki/Principal_branch" title="Principal branch">principal branch</a>. With this restriction, for each <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> in the domain, the expression <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arcsin(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arcsin(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f111f8c1e5de2f92ee17eeabd64c4ea1bcd55196" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.101ex; height:2.843ex;" alt="{\displaystyle \arcsin(x)}" /></span> will evaluate only to a single value, called its <a href="/wiki/Principal_value" title="Principal value">principal value</a>. The standard range of principal values for arcsin is from <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"> <mo>−</mo> <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/492058e293760c15cb3009d69147178fbc72bcb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.586ex; height:3.176ex;" alt="{\textstyle -{\frac {\pi }{2}}}" /></span> 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="{\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>, and the standard range for arccos is from <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> 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 \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>.<sup id="cite_ref-FOOTNOTEVarbergPurcellRigdon2007365_25-0" class="reference"><a href="#cite_note-FOOTNOTEVarbergPurcellRigdon2007365-25"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </p><p>The inverse function of both sine and cosine are defined as:<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (August 2024)">citation needed</span></a></i>]</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 \theta =\arcsin \left({\frac {\text{opposite}}{\text{hypotenuse}}}\right)=\arccos \left({\frac {\text{adjacent}}{\text{hypotenuse}}}\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>opposite</mtext> <mtext>hypotenuse</mtext> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>arccos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mtext>adjacent</mtext> <mtext>hypotenuse</mtext> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =\arcsin \left({\frac {\text{opposite}}{\text{hypotenuse}}}\right)=\arccos \left({\frac {\text{adjacent}}{\text{hypotenuse}}}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a7fb2ae2f0c2ec3085667df4d1e54d48d67cd7a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:51.52ex; height:6.176ex;" alt="{\displaystyle \theta =\arcsin \left({\frac {\text{opposite}}{\text{hypotenuse}}}\right)=\arccos \left({\frac {\text{adjacent}}{\text{hypotenuse}}}\right),}" /></span> where for some integer <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>, <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 {\begin{aligned}\sin(y)=x\iff &y=\arcsin(x)+2\pi k,{\text{ or }}\\&y=\pi -\arcsin(x)+2\pi k\\\cos(y)=x\iff &y=\arccos(x)+2\pi k,{\text{ or }}\\&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"></mspace> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace"></mspace> </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> or </mtext> </mrow> </mtd> </mtr> <mtr> <mtd></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"></mspace> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace"></mspace> </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> or </mtext> </mrow> </mtd> </mtr> <mtr> <mtd></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{ or }}\\&y=\pi -\arcsin(x)+2\pi k\\\cos(y)=x\iff &y=\arccos(x)+2\pi k,{\text{ or }}\\&y=-\arccos(x)+2\pi k\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da20b32e7b46d5ccdf3f058b812c8ad9768fa678" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:41.933ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}\sin(y)=x\iff &y=\arcsin(x)+2\pi k,{\text{ or }}\\&y=\pi -\arcsin(x)+2\pi k\\\cos(y)=x\iff &y=\arccos(x)+2\pi k,{\text{ or }}\\&y=-\arccos(x)+2\pi k\end{aligned}}}" /></span> By definition, both functions satisfy the equations:<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (August 2024)">citation needed</span></a></i>]</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 \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"></mspace> <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-display 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> and <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 {\begin{aligned}\arcsin(\sin(\theta ))=\theta \quad &{\text{for}}\quad -{\frac {\pi }{2}}\leq \theta \leq {\frac {\pi }{2}}\\\arccos(\cos(\theta ))=\theta \quad &{\text{for}}\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"></mspace> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>for</mtext> </mrow> <mspace width="1em"></mspace> <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"></mspace> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>for</mtext> </mrow> <mspace width="1em"></mspace> <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{for}}\quad -{\frac {\pi }{2}}\leq \theta \leq {\frac {\pi }{2}}\\\arccos(\cos(\theta ))=\theta \quad &{\text{for}}\quad 0\leq \theta \leq \pi \end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11a64d948fdadfaf9eeb77c6fcd8c3f646f46cee" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.074ex; margin-bottom: -0.264ex; width:40.873ex; height:7.843ex;" alt="{\displaystyle {\begin{aligned}\arcsin(\sin(\theta ))=\theta \quad &{\text{for}}\quad -{\frac {\pi }{2}}\leq \theta \leq {\frac {\pi }{2}}\\\arccos(\cos(\theta ))=\theta \quad &{\text{for}}\quad 0\leq \theta \leq \pi \end{aligned}}}" /></span> </p> <div class="mw-heading mw-heading4"><h4 id="Other_identities">Other identities</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine_and_cosine&action=edit&section=12" title="Edit section: Other identities"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/List_of_trigonometric_identities" title="List of trigonometric identities">List of trigonometric identities</a></div> <p>According to <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a>, the squared hypotenuse is the sum of two squared legs of a right triangle. Dividing the formula on both sides with squared hypotenuse resulting in the <a href="/wiki/Pythagorean_trigonometric_identity" title="Pythagorean trigonometric identity">Pythagorean trigonometric identity</a>, the sum of a squared sine and a squared cosine equals 1:<sup id="cite_ref-FOOTNOTEYoung2017[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA99_99]_26-0" class="reference"><a href="#cite_note-FOOTNOTEYoung2017[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA99_99]-26"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>b<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 \sin ^{2}(\theta )+\cos ^{2}(\theta )=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <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> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin ^{2}(\theta )+\cos ^{2}(\theta )=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c2fa6795bad3d685ca0704a63f5771bda27d5ac" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.623ex; height:3.176ex;" alt="{\displaystyle \sin ^{2}(\theta )+\cos ^{2}(\theta )=1.}" /></span> </p><p>Sine and cosine satisfy the following double-angle formulas:<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>26<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 {\begin{aligned}\sin(2\theta )&=2\sin(\theta )\cos(\theta ),\\\cos(2\theta )&=\cos ^{2}(\theta )-\sin ^{2}(\theta )\\&=2\cos ^{2}(\theta )-1\\&=1-2\sin ^{2}(\theta )\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mi>sin</mi> <mo>⁡</mo> <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> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <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> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sin(2\theta )&=2\sin(\theta )\cos(\theta ),\\\cos(2\theta )&=\cos ^{2}(\theta )-\sin ^{2}(\theta )\\&=2\cos ^{2}(\theta )-1\\&=1-2\sin ^{2}(\theta )\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/497be0073c63433342edee671020c7aa109e0e6e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:27.738ex; height:12.843ex;" alt="{\displaystyle {\begin{aligned}\sin(2\theta )&=2\sin(\theta )\cos(\theta ),\\\cos(2\theta )&=\cos ^{2}(\theta )-\sin ^{2}(\theta )\\&=2\cos ^{2}(\theta )-1\\&=1-2\sin ^{2}(\theta )\end{aligned}}}" /></span> </p><p><span class="anchor" id="Sine_squared_function"></span> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File: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>Sine function in blue and sine squared function in red. The <span class="nowrap"><span class="texhtml"><i>x</i></span>-</span>axis is in radians.</figcaption></figure> <p>The cosine double angle formula implies that sin<sup>2</sup> and cos<sup>2</sup> are, themselves, shifted and scaled sine waves. Specifically,<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>27<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 \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"></mspace> <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-display 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> The graph shows both sine and sine squared functions, with the sine in blue and the sine squared in red. Both graphs have the same shape but with different ranges of values and different periods. Sine squared has only positive values, but twice the number of periods.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (August 2024)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading3"><h3 id="Series_and_polynomials">Series and polynomials</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine_and_cosine&action=edit&section=13" title="Edit section: Series and polynomials"><span>edit</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/File: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>This animation shows how including more and more terms in the partial sum of its Taylor series approaches a sine curve.</figcaption></figure> <p>Both sine and cosine functions can be defined by using a <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a>, a <a href="/wiki/Power_series" title="Power series">power series</a> involving the higher-order derivatives. As mentioned in <a href="#Continuity_and_differentiation">§ Continuity and differentiation</a>, the <a href="/wiki/Derivative" title="Derivative">derivative</a> of sine is cosine and that the derivative of cosine is the negative of sine. This means the successive derivatives of <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> are <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>, <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"> <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 -\sin(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7326163fe774b61667587334208ecaef5798056" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.19ex; height:2.843ex;" alt="{\displaystyle -\sin(x)}" /></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 -\cos(x)}"> <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> </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/3f5dca2087cf314bb30116832bea1df5e00086be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.445ex; height:2.843ex;" alt="{\displaystyle -\cos(x)}" /></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 \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>, continuing to repeat those four functions. The <span class="nowrap"><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 (4n+k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>4</mn> <mi>n</mi> <mo>+</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (4n+k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6d26c80aefe4f8e27c3b52e0b0375980f8addba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.418ex; height:2.843ex;" alt="{\displaystyle (4n+k)}" /></span>-</span>th derivative, evaluated at the point 0: <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 \sin ^{(4n+k)}(0)={\begin{cases}0&{\text{when }}k=0\\1&{\text{when }}k=1\\0&{\text{when }}k=2\\-1&{\text{when }}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>when </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>when </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>when </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>when </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{when }}k=0\\1&{\text{when }}k=1\\0&{\text{when }}k=2\\-1&{\text{when }}k=3\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d46c038e501fe9a5f4ca170b57667a244b9b77b2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:33.84ex; height:11.176ex;" alt="{\displaystyle \sin ^{(4n+k)}(0)={\begin{cases}0&{\text{when }}k=0\\1&{\text{when }}k=1\\0&{\text{when }}k=2\\-1&{\text{when }}k=3\end{cases}}}" /></span> where the superscript represents repeated differentiation. This implies the following Taylor series expansion at <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>. One can then use the theory of <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a> to show that the following identities hold for all <a href="/wiki/Real_number" title="Real number">real numbers</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="{\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>—where <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> is the angle in radians.<sup id="cite_ref-FOOTNOTEVarbergPurcellRigdon2007491&ndash;492_30-0" class="reference"><a href="#cite_note-FOOTNOTEVarbergPurcellRigdon2007491&ndash;492-30"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> More generally, for all <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>:<sup id="cite_ref-FOOTNOTEAbramowitzStegun1970[httpsbooksgooglecombooksidMtU8uP7XMvoCpgPA74_74]_31-0" class="reference"><a href="#cite_note-FOOTNOTEAbramowitzStegun1970[httpsbooksgooglecombooksidMtU8uP7XMvoCpgPA74_74]-31"><span class="cite-bracket">[</span>29<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 {\begin{aligned}\sin(x)&=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots \\&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}x^{2n+1}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>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> <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 \\&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}x^{2n+1}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f13ee50efedabff2d208516e6a31ea495a9c843d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:34.919ex; height:12.843ex;" alt="{\displaystyle {\begin{aligned}\sin(x)&=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots \\&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}x^{2n+1}\end{aligned}}}" /></span> Taking the derivative of each term gives the Taylor series for cosine:<sup id="cite_ref-FOOTNOTEVarbergPurcellRigdon2007491&ndash;492_30-1" class="reference"><a href="#cite_note-FOOTNOTEVarbergPurcellRigdon2007491&ndash;492-30"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEAbramowitzStegun1970[httpsbooksgooglecombooksidMtU8uP7XMvoCpgPA74_74]_31-1" class="reference"><a href="#cite_note-FOOTNOTEAbramowitzStegun1970[httpsbooksgooglecombooksidMtU8uP7XMvoCpgPA74_74]-31"><span class="cite-bracket">[</span>29<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 {\begin{aligned}\cos(x)&=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots \\&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}x^{2n}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <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> <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 \\&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}x^{2n}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/742bd8177eff1a0186a49c19c6a0c626c7fe3563" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:35.007ex; height:12.843ex;" alt="{\displaystyle {\begin{aligned}\cos(x)&=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots \\&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}x^{2n}\end{aligned}}}" /></span> </p><p>Both sine and cosine functions with multiple angles may appear as their <a href="/wiki/Linear_combination" title="Linear combination">linear combination</a>, resulting in a polynomial. Such a polynomial is known as the <a href="/wiki/Trigonometric_polynomial" title="Trigonometric polynomial">trigonometric polynomial</a>. The trigonometric polynomial's ample applications may be acquired in <a href="/wiki/Trigonometric_interpolation" title="Trigonometric interpolation">its interpolation</a>, and its extension of a periodic function known as the <a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a>. Let <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_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/790f9209748c2dca7ed7b81932c37c02af1dbc31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.448ex; height:2.009ex;" alt="{\displaystyle a_{n}}" /></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 b_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28e2d72f6dd9375c8f1f59f1effd9b4e5492ac97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.216ex; height:2.509ex;" alt="{\displaystyle b_{n}}" /></span> be any coefficients, then the trigonometric polynomial of a degree <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 N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" /></span>—denoted as <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 T(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1171c29b4c2b5575f50a4ea9313f90448a2cbe05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.775ex; height:2.843ex;" alt="{\displaystyle T(x)}" /></span>—is defined as:<sup id="cite_ref-FOOTNOTEPowell1981150_32-0" class="reference"><a href="#cite_note-FOOTNOTEPowell1981150-32"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTERudin198788_33-0" class="reference"><a href="#cite_note-FOOTNOTERudin198788-33"><span class="cite-bracket">[</span>31<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 T(x)=a_{0}+\sum _{n=1}^{N}a_{n}\cos(nx)+\sum _{n=1}^{N}b_{n}\sin(nx).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(x)=a_{0}+\sum _{n=1}^{N}a_{n}\cos(nx)+\sum _{n=1}^{N}b_{n}\sin(nx).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d91df6657ed384066a15aee6f8f0ef01c9a0787" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:44.442ex; height:7.343ex;" alt="{\displaystyle T(x)=a_{0}+\sum _{n=1}^{N}a_{n}\cos(nx)+\sum _{n=1}^{N}b_{n}\sin(nx).}" /></span> </p><p>The <a href="/wiki/Trigonometric_series" title="Trigonometric series">trigonometric series</a> can be defined similarly analogous to the trigonometric polynomial, its infinite inversion. Let <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_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/730f6906700685b6d52f3958b1c2ae659d2d97d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.962ex; height:2.509ex;" alt="{\displaystyle A_{n}}" /></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 B_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f568bf6d34e97b9fdda0dc7e276d6c4501d2045" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.982ex; height:2.509ex;" alt="{\displaystyle B_{n}}" /></span> be any coefficients, then the trigonometric series can be defined as:<sup id="cite_ref-FOOTNOTEZygmund19681_34-0" class="reference"><a href="#cite_note-FOOTNOTEZygmund19681-34"><span class="cite-bracket">[</span>32<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 {\frac {1}{2}}A_{0}+\sum _{n=1}^{\infty }A_{n}\cos(nx)+B_{n}\sin(nx).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}A_{0}+\sum _{n=1}^{\infty }A_{n}\cos(nx)+B_{n}\sin(nx).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a20bbca53c25bda2d3a375f83924db3637e013b7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:36.618ex; height:6.843ex;" alt="{\displaystyle {\frac {1}{2}}A_{0}+\sum _{n=1}^{\infty }A_{n}\cos(nx)+B_{n}\sin(nx).}" /></span> In the case of a Fourier series with a given integrable function <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" /></span>, the coefficients of a trigonometric series are:<sup id="cite_ref-FOOTNOTEZygmund196811_35-0" class="reference"><a href="#cite_note-FOOTNOTEZygmund196811-35"><span class="cite-bracket">[</span>33<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 {\begin{aligned}A_{n}&={\frac {1}{\pi }}\int _{0}^{2\pi }f(x)\cos(nx)\,dx,\\B_{n}&={\frac {1}{\pi }}\int _{0}^{2\pi }f(x)\sin(nx)\,dx.\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> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>π</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>x</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>π</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>x</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}A_{n}&={\frac {1}{\pi }}\int _{0}^{2\pi }f(x)\cos(nx)\,dx,\\B_{n}&={\frac {1}{\pi }}\int _{0}^{2\pi }f(x)\sin(nx)\,dx.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71e46a4445a4bb3a0d5933f7f542c39673dafef1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:30.264ex; height:12.843ex;" alt="{\displaystyle {\begin{aligned}A_{n}&={\frac {1}{\pi }}\int _{0}^{2\pi }f(x)\cos(nx)\,dx,\\B_{n}&={\frac {1}{\pi }}\int _{0}^{2\pi }f(x)\sin(nx)\,dx.\end{aligned}}}" /></span> </p> <div class="mw-heading mw-heading2"><h2 id="Complex_numbers_relationship">Complex numbers relationship</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine_and_cosine&action=edit&section=14" title="Edit section: Complex numbers relationship"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_citations_needed_section plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Sine_and_cosine" title="Special:EditPage/Sine and cosine">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a> in this section. Unsourced material may be challenged and removed.</span> <span class="date-container"><i>(<span class="date">August 2024</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Complex_exponential_function_definitions">Complex exponential function definitions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine_and_cosine&action=edit&section=15" title="Edit section: Complex exponential function definitions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Both sine and cosine can be extended further via <a href="/wiki/Complex_number" title="Complex number">complex number</a>, a set of numbers composed of both <a href="/wiki/Real_number" title="Real number">real</a> and <a href="/wiki/Imaginary_number" title="Imaginary number">imaginary numbers</a>. For real number <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>, the definition of both sine and cosine functions can be extended in a <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a> in terms of an <a href="/wiki/Exponential_function" title="Exponential function">exponential function</a> as follows:<sup id="cite_ref-FOOTNOTEHowie2003[httpsbooksgooglecombooksid0FZDBAAAQBAJpgPA24_24]_36-0" class="reference"><a href="#cite_note-FOOTNOTEHowie2003[httpsbooksgooglecombooksid0FZDBAAAQBAJpgPA24_24]-36"><span class="cite-bracket">[</span>34<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 {\begin{aligned}\sin(\theta )&={\frac {e^{i\theta }-e^{-i\theta }}{2i}},\\\cos(\theta )&={\frac {e^{i\theta }+e^{-i\theta }}{2}},\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>θ</mi> <mo stretchy="false">)</mo> </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>θ</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>θ</mi> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>i</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </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>θ</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>θ</mi> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sin(\theta )&={\frac {e^{i\theta }-e^{-i\theta }}{2i}},\\\cos(\theta )&={\frac {e^{i\theta }+e^{-i\theta }}{2}},\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/725f29f76be948091e80a1467b067a8ca9e9dbd3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:20.771ex; height:11.509ex;" alt="{\displaystyle {\begin{aligned}\sin(\theta )&={\frac {e^{i\theta }-e^{-i\theta }}{2i}},\\\cos(\theta )&={\frac {e^{i\theta }+e^{-i\theta }}{2}},\end{aligned}}}" /></span> </p><p>Alternatively, both functions can be defined in terms of <a href="/wiki/Euler%27s_formula" title="Euler's formula">Euler's formula</a>:<sup id="cite_ref-FOOTNOTEHowie2003[httpsbooksgooglecombooksid0FZDBAAAQBAJpgPA24_24]_36-1" class="reference"><a href="#cite_note-FOOTNOTEHowie2003[httpsbooksgooglecombooksid0FZDBAAAQBAJpgPA24_24]-36"><span class="cite-bracket">[</span>34<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 {\begin{aligned}e^{i\theta }&=\cos(\theta )+i\sin(\theta ),\\e^{-i\theta }&=\cos(\theta )-i\sin(\theta ).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <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> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>θ</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <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> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}e^{i\theta }&=\cos(\theta )+i\sin(\theta ),\\e^{-i\theta }&=\cos(\theta )-i\sin(\theta ).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77db983756d03fe6b65cfb8af534812004ebfbbb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:24.226ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}e^{i\theta }&=\cos(\theta )+i\sin(\theta ),\\e^{-i\theta }&=\cos(\theta )-i\sin(\theta ).\end{aligned}}}" /></span> </p><p>When plotted on the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a>, the function <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^{ix}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{ix}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5271aa5bef13f6bd715afeda45bc59ae37d7c6d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.823ex; height:2.676ex;" alt="{\displaystyle e^{ix}}" /></span> for real values of <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> traces out the <a href="/wiki/Unit_circle" title="Unit circle">unit circle</a> in the complex plane. Both sine and cosine functions may be simplified to the imaginary and real parts of <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> as:<sup id="cite_ref-FOOTNOTERudin19872_37-0" class="reference"><a href="#cite_note-FOOTNOTERudin19872-37"><span class="cite-bracket">[</span>35<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 {\begin{aligned}\sin \theta &=\operatorname {Im} (e^{i\theta }),\\\cos \theta &=\operatorname {Re} (e^{i\theta }).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>Im</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sin \theta &=\operatorname {Im} (e^{i\theta }),\\\cos \theta &=\operatorname {Re} (e^{i\theta }).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/646f096950ba96ce2f21de18bfa047b6421b21e0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:16.324ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}\sin \theta &=\operatorname {Im} (e^{i\theta }),\\\cos \theta &=\operatorname {Re} (e^{i\theta }).\end{aligned}}}" /></span> </p><p>When <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=x+iy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=x+iy}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08e90bb6b36fef59c6113eed2a08f10d77240741" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.315ex; height:2.509ex;" alt="{\displaystyle z=x+iy}" /></span> for real values <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> 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 y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" /></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i={\sqrt {-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i={\sqrt {-1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/370c8cebe9634fbfc84c29ea61680b0ad4a1ae0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.807ex; height:3.009ex;" alt="{\displaystyle i={\sqrt {-1}}}" /></span>, both sine and cosine functions can be expressed in terms of real sines, cosines, and <a href="/wiki/Hyperbolic_function" class="mw-redirect" title="Hyperbolic function">hyperbolic functions</a> as:<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (August 2024)">citation needed</span></a></i>]</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 {\begin{aligned}\sin z&=\sin x\cosh y+i\cos x\sinh y,\\\cos z&=\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> <mi>z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mi>cosh</mi> <mo>⁡</mo> <mi>y</mi> <mo>+</mo> <mi>i</mi> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mi>sinh</mi> <mo>⁡</mo> <mi>y</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mi>cosh</mi> <mo>⁡</mo> <mi>y</mi> <mo>−</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mi>sinh</mi> <mo>⁡</mo> <mi>y</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sin z&=\sin x\cosh y+i\cos x\sinh y,\\\cos z&=\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/47251493866932a39ed455153827afe21a781596" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:34.924ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}\sin z&=\sin x\cosh y+i\cos x\sinh y,\\\cos z&=\cos x\cosh y-i\sin x\sinh y.\end{aligned}}}" /></span> </p> <div class="mw-heading mw-heading3"><h3 id="Polar_coordinates">Polar coordinates</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine_and_cosine&action=edit&section=16" title="Edit section: Polar coordinates"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File: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/250px-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> 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 \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> are the real and imaginary parts of <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>Sine and cosine are used to connect the real and imaginary parts of a <a href="/wiki/Complex_number" title="Complex number">complex number</a> with its <a href="/wiki/Polar_coordinates" class="mw-redirect" title="Polar coordinates">polar coordinates</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="{\displaystyle (r,\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (r,\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed8396fdc359fb06c93722137c959e7496e47ed6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.982ex; height:2.843ex;" alt="{\displaystyle (r,\theta )}" /></span>: <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(\theta )+i\sin(\theta )),}"> <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>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> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=r(\cos(\theta )+i\sin(\theta )),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8523d84328f079c3a2b8c3444dc071cbc043689" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.488ex; height:2.843ex;" alt="{\displaystyle z=r(\cos(\theta )+i\sin(\theta )),}" /></span> and the real and imaginary parts are <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 {\begin{aligned}\operatorname {Re} (z)&=r\cos(\theta ),\\\operatorname {Im} (z)&=r\sin(\theta ),\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>r</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>Im</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>r</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {Re} (z)&=r\cos(\theta ),\\\operatorname {Im} (z)&=r\sin(\theta ),\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abf44a594aabf7817998c91793f805ecdb366282" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.616ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}\operatorname {Re} (z)&=r\cos(\theta ),\\\operatorname {Im} (z)&=r\sin(\theta ),\end{aligned}}}" /></span> where <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 r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" /></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 \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> represent the magnitude and angle of the complex number <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" /></span>. </p><p>For any real number <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>, Euler's formula in terms of polar coordinates is stated as <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 z=re^{i\theta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>r</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle z=re^{i\theta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95946f451434232fc04452bd3b3e4251de1a9d13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.889ex; height:2.676ex;" alt="{\textstyle z=re^{i\theta }}" /></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Complex_arguments">Complex arguments</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine_and_cosine&action=edit&section=17" title="Edit section: Complex arguments"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Complex_sin.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/45/Complex_sin.jpg/250px-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/500px-Complex_sin.jpg 2x" data-file-width="944" data-file-height="1079" /></a><figcaption><a href="/wiki/Domain_coloring" title="Domain coloring">Domain coloring</a> of sin(<i>z</i>) in the complex plane. Brightness indicates absolute magnitude, hue represents complex argument.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File: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/250px-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/500px-Sin_z_vector_field_02_Pengo.svg.png 2x" data-file-width="720" data-file-height="540" /></a><figcaption>Vector field rendering of sin(<i>z</i>)</figcaption></figure> <p>Applying the series definition of the sine and cosine to a complex argument, <i>z</i>, gives: </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> <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> <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> <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> <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> <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>where sinh and cosh are the <a href="/wiki/Hyperbolic_function" class="mw-redirect" title="Hyperbolic function">hyperbolic sine and cosine</a>. These are <a href="/wiki/Entire_function" title="Entire function">entire functions</a>. </p><p>It is also sometimes useful to express the complex sine and cosine functions in terms of the real and imaginary parts of its argument: </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> <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> <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-heading4"><h4 id="Partial_fraction_and_product_expansions_of_complex_sine">Partial fraction and product expansions of complex sine</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine_and_cosine&action=edit&section=18" title="Edit section: Partial fraction and product expansions of complex sine"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Using the partial fraction expansion technique in <a href="/wiki/Complex_analysis" title="Complex analysis">complex analysis</a>, one can find that the infinite series <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 \sum _{n=-\infty }^{\infty }{\frac {(-1)^{n}}{z-n}}={\frac {1}{z}}-2z\sum _{n=1}^{\infty }{\frac {(-1)^{n}}{n^{2}-z^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </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> <mi>z</mi> <mo>−</mo> <mi>n</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>z</mi> </mfrac> </mrow> <mo>−</mo> <mn>2</mn> <mi>z</mi> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow 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> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=-\infty }^{\infty }{\frac {(-1)^{n}}{z-n}}={\frac {1}{z}}-2z\sum _{n=1}^{\infty }{\frac {(-1)^{n}}{n^{2}-z^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/767d43fdad5d7a47e46f1433d20d909d91af8883" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:34.995ex; height:6.843ex;" alt="{\displaystyle \sum _{n=-\infty }^{\infty }{\frac {(-1)^{n}}{z-n}}={\frac {1}{z}}-2z\sum _{n=1}^{\infty }{\frac {(-1)^{n}}{n^{2}-z^{2}}}}" /></span> both converge and are equal 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="{\textstyle {\frac {\pi }{\sin(\pi z)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {\pi }{\sin(\pi z)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02adeb7acbfc515905a0c72f7130ae5d2589df72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:5.846ex; height:3.843ex;" alt="{\textstyle {\frac {\pi }{\sin(\pi z)}}}" /></span>. Similarly, one can show that <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 {\pi ^{2}}{\sin ^{2}(\pi z)}}=\sum _{n=-\infty }^{\infty }{\frac {1}{(z-n)^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo>−</mo> <mi>n</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\pi ^{2}}{\sin ^{2}(\pi z)}}=\sum _{n=-\infty }^{\infty }{\frac {1}{(z-n)^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76b4bc619e6aa45ebdcdb9e88551598f69931529" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:27.317ex; height:6.843ex;" alt="{\displaystyle {\frac {\pi ^{2}}{\sin ^{2}(\pi z)}}=\sum _{n=-\infty }^{\infty }{\frac {1}{(z-n)^{2}}}.}" /></span> </p><p>Using product expansion technique, one can derive <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 \sin(\pi z)=\pi z\prod _{n=1}^{\infty }\left(1-{\frac {z^{2}}{n^{2}}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>π</mi> <mi>z</mi> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(\pi z)=\pi z\prod _{n=1}^{\infty }\left(1-{\frac {z^{2}}{n^{2}}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17ce4c82316483de357bac1133ac3d7af0092707" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:28.208ex; height:6.843ex;" alt="{\displaystyle \sin(\pi z)=\pi z\prod _{n=1}^{\infty }\left(1-{\frac {z^{2}}{n^{2}}}\right).}" /></span> </p> <div class="mw-heading mw-heading4"><h4 id="Usage_of_complex_sine">Usage of complex sine</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine_and_cosine&action=edit&section=19" title="Edit section: Usage of complex sine"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>sin(<i>z</i>) is found in the <a href="/wiki/Functional_equation" title="Functional equation">functional equation</a> for the <a href="/wiki/Gamma_function" title="Gamma function">Gamma function</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 \Gamma (s)\Gamma (1-s)={\pi \over \sin(\pi s)},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma (s)\Gamma (1-s)={\pi \over \sin(\pi s)},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07f53255b83f8d3db49cf23173da760f7048b689" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:24.377ex; height:5.509ex;" alt="{\displaystyle \Gamma (s)\Gamma (1-s)={\pi \over \sin(\pi s)},}" /></span></dd></dl> <p>which in turn is found in the <a href="/wiki/Functional_equation" title="Functional equation">functional equation</a> for the <a href="/wiki/Riemann_zeta-function" class="mw-redirect" title="Riemann zeta-function">Riemann zeta-function</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 \zeta (s)=2(2\pi )^{s-1}\Gamma (1-s)\sin \left({\frac {\pi }{2}}s\right)\zeta (1-s).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mi>s</mi> </mrow> <mo>)</mo> </mrow> <mi>ζ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (s)=2(2\pi )^{s-1}\Gamma (1-s)\sin \left({\frac {\pi }{2}}s\right)\zeta (1-s).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f170929c08858f62da589f8fccb2565c43c769a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:41.94ex; height:4.843ex;" alt="{\displaystyle \zeta (s)=2(2\pi )^{s-1}\Gamma (1-s)\sin \left({\frac {\pi }{2}}s\right)\zeta (1-s).}" /></span></dd></dl> <p>As a <a href="/wiki/Holomorphic_function" title="Holomorphic function">holomorphic function</a>, sin <i>z</i> is a 2D solution of <a href="/wiki/Laplace%27s_equation" title="Laplace's equation">Laplace's equation</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 \Delta u(x_{1},x_{2})=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta u(x_{1},x_{2})=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdaaf4948b285dfd3deb916716fc5daa6d9d3d22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.784ex; height:2.843ex;" alt="{\displaystyle \Delta u(x_{1},x_{2})=0.}" /></span></dd></dl> <p>The complex sine function is also related to the level curves of <a href="/wiki/Pendulums" class="mw-redirect" title="Pendulums">pendulums</a>.<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="Please clarify the preceding statement or statements with a good explanation from a reliable source. (August 2019)">how?</span></a></i>]</sup><sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup><sup class="noprint Inline-Template noprint noexcerpt Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:NOTRS" class="mw-redirect" title="Wikipedia:NOTRS"><span title="This claim needs references to better sources. (August 2019)">better source needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading3"><h3 id="Complex_graphs">Complex graphs</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine_and_cosine&action=edit&section=20" title="Edit section: Complex graphs"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table style="text-align:center"> <caption><b>Sine function in the complex plane</b> </caption> <tbody><tr> <td><figure class="mw-halign-none" typeof="mw:File"><a href="/wiki/File: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/347px-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/File: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/347px-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/File: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/347px-Complex_sin_abs_01_Pengo.svg.png 2x" data-file-width="720" data-file-height="540" /></a><figcaption></figcaption></figure> </td></tr> <tr> <td>Real component </td> <td>Imaginary component </td> <td>Magnitude </td></tr></tbody></table> <p><br /> </p> <table style="text-align:center"> <caption><b>Arcsine function in the complex plane</b> </caption> <tbody><tr> <td><figure class="mw-halign-none" typeof="mw:File"><a href="/wiki/File: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/347px-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/File: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/250px-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/330px-Complex_arcsin_imag_01_Pengo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Complex_arcsin_imag_01_Pengo.svg/500px-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/File: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/347px-Complex_arcsin_abs_01_Pengo.svg.png 2x" data-file-width="720" data-file-height="540" /></a><figcaption></figcaption></figure> </td></tr> <tr> <td>Real component </td> <td>Imaginary component </td> <td>Magnitude </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Background">Background</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine_and_cosine&action=edit&section=21" title="Edit section: Background"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Etymology">Etymology</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine_and_cosine&action=edit&section=22" title="Edit section: Etymology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/History_of_trigonometry#Etymology" title="History of trigonometry">History of trigonometry § Etymology</a></div> <p>The word <i>sine</i> is derived, indirectly, from the <a href="/wiki/Sanskrit" title="Sanskrit">Sanskrit</a> word <span title="Sanskrit-language text"><i lang="sa">jyā</i></span> 'bow-string' or more specifically its synonym <span title="Sanskrit-language text"><i lang="sa">jīvá</i></span> (both adopted from <a href="/wiki/Ancient_Greek_language" class="mw-redirect" title="Ancient Greek language">Ancient Greek</a> <span title="Ancient Greek (to 1453)-language text"><span lang="grc">χορδή</span></span> 'string'), due to visual similarity between the arc of a circle with its corresponding chord and a bow with its string (see <a href="/wiki/Jy%C4%81,_koti-jy%C4%81_and_utkrama-jy%C4%81" title="Jyā, koti-jyā and utkrama-jyā">jyā, koti-jyā and utkrama-jyā</a>). This was <a href="/wiki/Transliteration" title="Transliteration">transliterated</a> in <a href="/wiki/Arabic_language" class="mw-redirect" title="Arabic language">Arabic</a> as <span title="Arabic-language romanization"><i lang="ar-Latn">jība</i></span>, which is meaningless in that language and written as <span title="Arabic-language romanization"><i lang="ar-Latn">jb</i></span> (<span title="Arabic-language text"><span lang="ar" dir="rtl">جب</span></span>). Since Arabic is written without short vowels, <span title="Arabic-language romanization"><i lang="ar-Latn">jb</i></span> was interpreted as the <a href="/wiki/Homograph" title="Homograph">homograph</a> <span title="Arabic-language romanization"><i lang="ar-Latn">jayb</i></span> (<a href="https://en.wiktionary.org/wiki/%D8%AC%D9%8A%D8%A8" class="extiw" title="wikt:جيب">جيب</a>), which means 'bosom', 'pocket', or 'fold'.<sup id="cite_ref-FOOTNOTEPlofker2009[httpsbooksgooglecombooksidDHvThPNp9yMCpgPA257_257]_39-0" class="reference"><a href="#cite_note-FOOTNOTEPlofker2009[httpsbooksgooglecombooksidDHvThPNp9yMCpgPA257_257]-39"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEMaor1998[httpsbooksgooglecombooksidr9aMrneWFpUCpgPA35_35]_40-0" class="reference"><a href="#cite_note-FOOTNOTEMaor1998[httpsbooksgooglecombooksidr9aMrneWFpUCpgPA35_35]-40"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> When the Arabic texts of <a href="/wiki/Al-Battani" title="Al-Battani">Al-Battani</a> and <a href="/wiki/Mu%E1%B8%A5ammad_ibn_M%C5%ABs%C4%81_al-Khw%C4%81rizm%C4%AB" class="mw-redirect" title="Muḥammad ibn Mūsā al-Khwārizmī">al-Khwārizmī</a> were translated into <a href="/wiki/Medieval_Latin" title="Medieval Latin">Medieval Latin</a> in the 12th century by <a href="/wiki/Gerard_of_Cremona" title="Gerard of Cremona">Gerard of Cremona</a>, he used the Latin equivalent <a href="https://en.wiktionary.org/wiki/sinus" class="extiw" title="wikt:sinus"><i>sinus</i></a> (which also means 'bay' or 'fold', and more specifically 'the hanging fold of a <a href="/wiki/Toga" title="Toga">toga</a> over the breast').<sup id="cite_ref-FOOTNOTEMerzbachBoyer2011_41-0" class="reference"><a href="#cite_note-FOOTNOTEMerzbachBoyer2011-41"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEMaor199835&ndash;36_42-0" class="reference"><a href="#cite_note-FOOTNOTEMaor199835&ndash;36-42"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEKatz2008253_43-0" class="reference"><a href="#cite_note-FOOTNOTEKatz2008253-43"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> Gerard was probably not the first scholar to use this translation; Robert of Chester appears to have preceded him and there is evidence of even earlier usage.<sup id="cite_ref-FOOTNOTESmith1958202_44-0" class="reference"><a href="#cite_note-FOOTNOTESmith1958202-44"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-45" class="reference"><a href="#cite_note-45"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup> The English form <i>sine</i> was introduced in <a href="/wiki/Thomas_Fale" title="Thomas Fale">Thomas Fale</a>'s 1593 <i>Horologiographia</i>.<sup id="cite_ref-46" class="reference"><a href="#cite_note-46"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup> </p><p>The word <i>cosine</i> derives from an abbreviation of the Latin <span title="Latin-language text"><i lang="la">complementi sinus</i></span> 'sine of the <a href="/wiki/Complementary_angle" class="mw-redirect" title="Complementary angle">complementary angle</a>' as <i>cosinus</i> in <a href="/wiki/Edmund_Gunter" title="Edmund Gunter">Edmund Gunter</a>'s <i>Canon triangulorum</i> (1620), which also includes a similar definition of <i>cotangens</i>.<sup id="cite_ref-FOOTNOTEGunter1620_47-0" class="reference"><a href="#cite_note-FOOTNOTEGunter1620-47"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="History">History</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine_and_cosine&action=edit&section=23" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/History_of_trigonometry" title="History of trigonometry">History of trigonometry</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File: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" title="Ottoman Empire">Ottoman Turkey</a> with axes for looking up the sine and <a href="/wiki/Versine" title="Versine">versine</a> of angles</figcaption></figure> <p>While the early study of trigonometry can be traced to antiquity, the <a href="/wiki/Trigonometric_functions" title="Trigonometric functions">trigonometric functions</a> as they are in use today were developed in the medieval period. The <a href="/wiki/Chord_(geometry)" title="Chord (geometry)">chord</a> function was discovered by <a href="/wiki/Hipparchus" title="Hipparchus">Hipparchus</a> of <a href="/wiki/%C4%B0znik" title="İznik">Nicaea</a> (180–125 BCE) and <a href="/wiki/Ptolemy" title="Ptolemy">Ptolemy</a> of <a href="/wiki/Egypt_(Roman_province)" class="mw-redirect" title="Egypt (Roman province)">Roman Egypt</a> (90–165 CE).<sup id="cite_ref-48" class="reference"><a href="#cite_note-48"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> </p><p>The sine and cosine functions are closely related to the <a href="/wiki/Jy%C4%81,_koti-jy%C4%81_and_utkrama-jy%C4%81" title="Jyā, koti-jyā and utkrama-jyā"><span title="Sanskrit-language romanization"><i lang="sa-Latn">jyā</i></span> and <span title="Sanskrit-language romanization"><i lang="sa-Latn">koṭi-jyā</i></span></a> functions used in <a href="/wiki/Indian_astronomy" title="Indian astronomy">Indian astronomy</a> during the <a href="/wiki/Gupta_period" class="mw-redirect" title="Gupta period">Gupta period</a> (<i><a href="/wiki/Aryabhatiya" title="Aryabhatiya">Aryabhatiya</a></i> and <i><a href="/wiki/Surya_Siddhanta" title="Surya Siddhanta">Surya Siddhanta</a></i>), via translation from Sanskrit to Arabic and then from Arabic to Latin.<sup id="cite_ref-FOOTNOTEMerzbachBoyer2011_41-1" class="reference"><a href="#cite_note-FOOTNOTEMerzbachBoyer2011-41"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-49" class="reference"><a href="#cite_note-49"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup> </p><p>All six trigonometric functions in current use were known in <a href="/wiki/Islamic_mathematics" class="mw-redirect" title="Islamic mathematics">Islamic mathematics</a> by the 9th century, as was the <a href="/wiki/Law_of_sines" title="Law of sines">law of sines</a>, used in <a href="/wiki/Solving_triangles" class="mw-redirect" title="Solving triangles">solving triangles</a>.<sup id="cite_ref-Gingerich_1986_50-0" class="reference"><a href="#cite_note-Gingerich_1986-50"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Al-Khw%C4%81rizm%C4%AB" class="mw-redirect" title="Al-Khwārizmī">Al-Khwārizmī</a> (c. 780–850) produced tables of sines, cosines and tangents.<sup id="cite_ref-Sesiano_51-0" class="reference"><a href="#cite_note-Sesiano-51"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Britannica_52-0" class="reference"><a href="#cite_note-Britannica-52"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Muhammad_ibn_J%C4%81bir_al-Harr%C4%81n%C4%AB_al-Batt%C4%81n%C4%AB" class="mw-redirect" title="Muhammad ibn Jābir al-Harrānī al-Battānī">Muhammad ibn Jābir al-Harrānī al-Battānī</a> (853–929) discovered the reciprocal functions of secant and cosecant, and produced the first table of cosecants for each degree from 1° to 90°.<sup id="cite_ref-Britannica_52-1" class="reference"><a href="#cite_note-Britannica-52"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup> </p><p>In the early 17th-century, the French mathematician <a href="/wiki/Albert_Girard" title="Albert Girard">Albert Girard</a> published the first use of the abbreviations <i>sin</i>, <i>cos</i>, and <i>tan</i>; these were further promulgated by Euler (see below). The <i>Opus palatinum de triangulis</i> of <a href="/wiki/Georg_Joachim_Rheticus" title="Georg Joachim Rheticus">Georg Joachim Rheticus</a>, a student of <a href="/wiki/Copernicus" class="mw-redirect" title="Copernicus">Copernicus</a>, was probably the first in Europe to define trigonometric functions directly in terms of right triangles instead of circles, with tables for all six trigonometric functions; this work was finished by Rheticus' student Valentin Otho in 1596. </p><p>In a paper published in 1682, <a href="/wiki/Gottfried_Leibniz" class="mw-redirect" title="Gottfried Leibniz">Leibniz</a> proved that sin <i>x</i> is not an <a href="/wiki/Algebraic_function" title="Algebraic function">algebraic function</a> of <i>x</i>.<sup id="cite_ref-53" class="reference"><a href="#cite_note-53"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Roger_Cotes" title="Roger Cotes">Roger Cotes</a> computed the derivative of sine in his <i>Harmonia Mensurarum</i> (1722).<sup id="cite_ref-54" class="reference"><a href="#cite_note-54"><span class="cite-bracket">[</span>52<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a>'s <i>Introductio in analysin infinitorum</i> (1748) was mostly responsible for establishing the analytic treatment of trigonometric functions in Europe, also defining them as infinite series and presenting "<a href="/wiki/Euler%27s_formula" title="Euler's formula">Euler's formula</a>", as well as the near-modern abbreviations <i>sin.</i>, <i>cos.</i>, <i>tang.</i>, <i>cot.</i>, <i>sec.</i>, and <i>cosec.</i><sup id="cite_ref-FOOTNOTEMerzbachBoyer2011_41-2" class="reference"><a href="#cite_note-FOOTNOTEMerzbachBoyer2011-41"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Software_implementations">Software implementations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine_and_cosine&action=edit&section=24" title="Edit section: Software implementations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444" /><table class="box-More_citations_needed_section plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Sine_and_cosine" title="Special:EditPage/Sine and cosine">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a> in this section. Unsourced material may be challenged and removed.</span> <span class="date-container"><i>(<span class="date">August 2024</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Lookup_table#Computing_sines" title="Lookup table">Lookup table § Computing sines</a></div> <p>There is no standard algorithm for calculating sine and cosine. <a href="/wiki/IEEE_754" title="IEEE 754">IEEE 754</a>, the most widely used standard for the specification of reliable floating-point computation, does not address calculating trigonometric functions such as sine. The reason is that no efficient algorithm is known for computing sine and cosine with a specified accuracy, especially for large inputs.<sup id="cite_ref-FOOTNOTEZimmermann2006_55-0" class="reference"><a href="#cite_note-FOOTNOTEZimmermann2006-55"><span class="cite-bracket">[</span>53<span class="cite-bracket">]</span></a></sup> </p><p>Algorithms for calculating sine may be balanced for such constraints as speed, accuracy, portability, or range of input values accepted. This can lead to different results for different algorithms, especially for special circumstances such as very large inputs, e.g. <code>sin(10<sup>22</sup>)</code>. </p><p>A common programming optimization, used especially in 3D graphics, is to pre-calculate a table of sine values, for example one value per degree, then for values in-between pick the closest pre-calculated value, or <a href="/wiki/Linear_interpolation" title="Linear interpolation">linearly interpolate</a> between the 2 closest values to approximate it. This allows results to be looked up from a table rather than being calculated in real time. With modern CPU architectures this method may offer no advantage.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (October 2012)">citation needed</span></a></i>]</sup> </p><p>The <a href="/wiki/CORDIC" title="CORDIC">CORDIC</a> algorithm is commonly used in scientific calculators. </p><p>The sine and cosine functions, along with other trigonometric functions, are widely available across programming languages and platforms. In computing, they are typically abbreviated to <code>sin</code> and <code>cos</code>. </p><p>Some CPU architectures have a built-in instruction for sine, including the Intel x87 FPUs since the 80387. </p><p>In programming languages, <code>sin</code> and <code>cos</code> are typically either a built-in function or found within the language's standard math library. For example, the <a href="/wiki/C_standard_library" title="C standard library">C standard library</a> defines sine functions within <a href="/wiki/C_mathematical_functions" title="C mathematical functions">math.h</a>: <code>sin(<a href="/wiki/Double-precision_floating-point_format" title="Double-precision floating-point format">double</a>)</code>, <code>sinf(<a href="/wiki/Single-precision_floating-point_format" title="Single-precision floating-point format">float</a>)</code>, and <code>sinl(<a href="/wiki/Long_double" title="Long double">long double</a>)</code>. The parameter of each is a <a href="/wiki/Floating_point" class="mw-redirect" title="Floating point">floating point</a> value, specifying the angle in radians. Each function returns the same <a href="/wiki/Data_type" title="Data type">data type</a> as it accepts. Many other trigonometric functions are also defined in <a href="/wiki/C_mathematical_functions" title="C mathematical functions">math.h</a>, such as for cosine, arc sine, and hyperbolic sine (sinh). Similarly, <a href="/wiki/Python_(programming_language)" title="Python (programming language)">Python</a> defines <code>math.sin(x)</code> and <code>math.cos(x)</code> within the built-in <code>math</code> module. Complex sine and cosine functions are also available within the <code>cmath</code> module, e.g. <code>cmath.sin(z)</code>. <a href="/wiki/CPython" title="CPython">CPython</a>'s math functions call the <a href="/wiki/C_(programming_language)" title="C (programming language)">C</a> <code>math</code> library, and use a <a href="/wiki/Double-precision_floating-point_format" title="Double-precision floating-point format">double-precision floating-point format</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Turns_based_implementations">Turns based implementations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine_and_cosine&action=edit&section=25" title="Edit section: Turns based implementations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Some software libraries provide implementations of sine and cosine using the input angle in half-<a href="/wiki/Turn_(angle)" title="Turn (angle)">turns</a>, a half-turn being an angle of 180 degrees or <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> radians. Representing angles in turns or half-turns has accuracy advantages and efficiency advantages in some cases.<sup id="cite_ref-matlab_56-0" class="reference"><a href="#cite_note-matlab-56"><span class="cite-bracket">[</span>54<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-r_57-0" class="reference"><a href="#cite_note-r-57"><span class="cite-bracket">[</span>55<span class="cite-bracket">]</span></a></sup> These functions are called <code>sinpi</code> and <code>cospi</code> in MATLAB,<sup id="cite_ref-matlab_56-1" class="reference"><a href="#cite_note-matlab-56"><span class="cite-bracket">[</span>54<span class="cite-bracket">]</span></a></sup> OpenCL,<sup id="cite_ref-58" class="reference"><a href="#cite_note-58"><span class="cite-bracket">[</span>56<span class="cite-bracket">]</span></a></sup> R,<sup id="cite_ref-r_57-1" class="reference"><a href="#cite_note-r-57"><span class="cite-bracket">[</span>55<span class="cite-bracket">]</span></a></sup> Julia,<sup id="cite_ref-59" class="reference"><a href="#cite_note-59"><span class="cite-bracket">[</span>57<span class="cite-bracket">]</span></a></sup> CUDA,<sup id="cite_ref-60" class="reference"><a href="#cite_note-60"><span class="cite-bracket">[</span>58<span class="cite-bracket">]</span></a></sup> and ARM.<sup id="cite_ref-61" class="reference"><a href="#cite_note-61"><span class="cite-bracket">[</span>59<span class="cite-bracket">]</span></a></sup> For example, <code>sinpi(x)</code> would evaluate 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> where <i>x</i> is expressed in half-turns, and consequently the final input to the function, <span class="texhtml"><i>πx</i></span> can be interpreted in radians by <span class="texhtml">sin</span>. </p><p>The accuracy advantage stems from the ability to perfectly represent key angles like full-turn, half-turn, and quarter-turn losslessly in binary floating-point or fixed-point. In contrast, representing <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>, 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="{\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> in binary floating-point or binary scaled fixed-point always involves a loss of accuracy since irrational numbers cannot be represented with finitely many binary digits. </p><p>Turns also have an accuracy advantage and efficiency advantage for computing modulo to one period. Computing modulo 1 turn or modulo 2 half-turns can be losslessly and efficiently computed in both floating-point and fixed-point. For example, computing modulo 1 or modulo 2 for a binary point scaled fixed-point value requires only a bit shift or bitwise AND operation. In contrast, computing 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> involves inaccuracies in representing <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>For applications involving angle sensors, the sensor typically provides angle measurements in a form directly compatible with turns or half-turns. For example, an angle sensor may count from 0 to 4096 over one complete revolution.<sup id="cite_ref-62" class="reference"><a href="#cite_note-62"><span class="cite-bracket">[</span>60<span class="cite-bracket">]</span></a></sup> If half-turns are used as the unit for angle, then the value provided by the sensor directly and losslessly maps to a fixed-point data type with 11 bits to the right of the binary point. In contrast, if radians are used as the unit for storing the angle, then the inaccuracies and cost of multiplying the raw sensor integer by an approximation 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="{\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> would be incurred. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine_and_cosine&action=edit&section=26" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 20em;"> <ul><li><a href="/wiki/%C4%80ryabha%E1%B9%ADa%27s_sine_table" title="Āryabhaṭa's sine table">Āryabhaṭa's sine table</a></li> <li><a href="/wiki/Bhaskara_I%27s_sine_approximation_formula" class="mw-redirect" title="Bhaskara I's sine approximation formula">Bhaskara I's sine approximation formula</a></li> <li><a href="/wiki/Discrete_sine_transform" title="Discrete sine transform">Discrete sine transform</a></li> <li><a href="/wiki/Dixon_elliptic_functions" title="Dixon elliptic functions">Dixon elliptic functions</a></li> <li><a href="/wiki/Euler%27s_formula" title="Euler's formula">Euler's formula</a></li> <li><a href="/wiki/Generalized_trigonometry" title="Generalized trigonometry">Generalized trigonometry</a></li> <li><a href="/wiki/Hyperbolic_function" class="mw-redirect" title="Hyperbolic function">Hyperbolic function</a></li> <li><a href="/wiki/Lemniscate_elliptic_functions" title="Lemniscate elliptic functions">Lemniscate elliptic functions</a></li> <li><a href="/wiki/Law_of_sines" title="Law of sines">Law of sines</a></li> <li><a href="/wiki/List_of_periodic_functions" title="List of periodic functions">List of periodic functions</a></li> <li><a href="/wiki/List_of_trigonometric_identities" title="List of trigonometric identities">List of trigonometric identities</a></li> <li><a href="/wiki/Madhava_series" title="Madhava series">Madhava series</a></li> <li><a href="/wiki/Madhava%27s_sine_table" title="Madhava's sine table">Madhava's sine table</a></li> <li><a href="/wiki/Optical_sine_theorem" title="Optical sine theorem">Optical sine theorem</a></li> <li><a href="/wiki/Polar_sine" title="Polar sine">Polar sine</a>—a generalization to vertex angles</li> <li><a href="/wiki/Proofs_of_trigonometric_identities" title="Proofs of trigonometric identities">Proofs of trigonometric identities</a></li> <li><a href="/wiki/Sinc_function" title="Sinc function">Sinc function</a></li> <li><a href="/wiki/Sine_and_cosine_transforms" title="Sine and cosine transforms">Sine and cosine transforms</a></li> <li><a href="/wiki/Sine_integral" class="mw-redirect" title="Sine integral">Sine integral</a></li> <li><a href="/wiki/Sine_quadrant" title="Sine quadrant">Sine quadrant</a></li> <li><a href="/wiki/Sine_wave" title="Sine wave">Sine wave</a></li> <li><a href="/wiki/Sine%E2%80%93Gordon_equation" class="mw-redirect" title="Sine–Gordon equation">Sine–Gordon equation</a></li> <li><a href="/wiki/Sinusoidal_model" title="Sinusoidal model">Sinusoidal model</a></li> <li><a href="/wiki/Mnemonics_in_trigonometry#SOH-CAH-TOA" title="Mnemonics in trigonometry">SOH-CAH-TOA</a></li> <li><a href="/wiki/Trigonometric_functions" title="Trigonometric functions">Trigonometric functions</a></li> <li><a href="/wiki/Trigonometric_integral" title="Trigonometric integral">Trigonometric integral</a></li></ul> </div> <p><br /> </p> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine_and_cosine&action=edit&section=27" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Footnotes">Footnotes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine_and_cosine&action=edit&section=28" title="Edit section: Footnotes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text">The superscript of −1 in <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 ^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin ^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21bc4ca64ac415c9ae60fb4e60fe4bddee17b8ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.188ex; height:2.676ex;" alt="{\displaystyle \sin ^{-1}}" /></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 ^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos ^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7eabf706b4642d521c6279a2f07ac9715c7679a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.444ex; height:2.676ex;" alt="{\displaystyle \cos ^{-1}}" /></span> denotes the inverse of a function, instead of <a href="/wiki/Exponentiation" title="Exponentiation">exponentiation</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">Here, <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}(x)}"> <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>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin ^{2}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53a0a5d7313c2ee4d060de4479eb4d418ce73310" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.049ex; height:3.176ex;" alt="{\displaystyle \sin ^{2}(x)}" /></span> means the squared sine function <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)\cdot \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> <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 \sin(x)\cdot \sin(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6abd8bbecb1c69b243659fadd4125845de237b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.668ex; height:2.843ex;" alt="{\displaystyle \sin(x)\cdot \sin(x)}" /></span>.</span> </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="Citations">Citations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine_and_cosine&action=edit&section=29" title="Edit section: Citations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626" /><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-FOOTNOTEYoung2017[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA27_27]-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEYoung2017[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA27_27]_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEYoung2017[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA27_27]_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-FOOTNOTEYoung2017[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA27_27]_1-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFYoung2017">Young (2017)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=476ZDwAAQBAJ&pg=PA27">27</a>.</span> </li> <li id="cite_note-FOOTNOTEYoung2017[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA36_36]-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEYoung2017[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA36_36]_2-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFYoung2017">Young (2017)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=476ZDwAAQBAJ&pg=PA36">36</a>.</span> </li> <li id="cite_note-FOOTNOTEVarbergPurcellRigdon200742-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEVarbergPurcellRigdon200742_3-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFVarbergPurcellRigdon2007">Varberg, Purcell & Rigdon (2007)</a>, p. 42.</span> </li> <li id="cite_note-FOOTNOTEYoung2017[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA37_37],_[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA78_78]-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEYoung2017[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA37_37],_[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA78_78]_4-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFYoung2017">Young (2017)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=476ZDwAAQBAJ&pg=PA37">37</a>, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=476ZDwAAQBAJ&pg=PA78">78</a>.</span> </li> <li id="cite_note-FOOTNOTEAxler2012[httpsbooksgooglecombooksidB5RxDwAAQBAJpgPA634_634]-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEAxler2012[httpsbooksgooglecombooksidB5RxDwAAQBAJpgPA634_634]_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEAxler2012[httpsbooksgooglecombooksidB5RxDwAAQBAJpgPA634_634]_5-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFAxler2012">Axler (2012)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=B5RxDwAAQBAJ&pg=PA634">634</a>.</span> </li> <li id="cite_note-FOOTNOTEAxler2012[httpsbooksgooglecombooksidB5RxDwAAQBAJpgPA632_632]-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEAxler2012[httpsbooksgooglecombooksidB5RxDwAAQBAJpgPA632_632]_6-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFAxler2012">Axler (2012)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=B5RxDwAAQBAJ&pg=PA632">632</a>.</span> </li> <li id="cite_note-FOOTNOTEVarbergPurcellRigdon200741-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEVarbergPurcellRigdon200741_7-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFVarbergPurcellRigdon2007">Varberg, Purcell & Rigdon (2007)</a>, p. 41.</span> </li> <li id="cite_note-FOOTNOTEYoung2017[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA68_68]-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEYoung2017[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA68_68]_8-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFYoung2017">Young (2017)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=476ZDwAAQBAJ&pg=PA68">68</a>.</span> </li> <li id="cite_note-FOOTNOTEVarbergPurcellRigdon200747-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEVarbergPurcellRigdon200747_9-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFVarbergPurcellRigdon2007">Varberg, Purcell & Rigdon (2007)</a>, p. 47.</span> </li> <li id="cite_note-FOOTNOTEVarbergPurcellRigdon200741&ndash;42-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEVarbergPurcellRigdon200741&ndash;42_10-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFVarbergPurcellRigdon2007">Varberg, Purcell & Rigdon (2007)</a>, p. 41–42.</span> </li> <li id="cite_note-FOOTNOTEVarbergPurcellRigdon200741,_43-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEVarbergPurcellRigdon200741,_43_11-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFVarbergPurcellRigdon2007">Varberg, Purcell & Rigdon (2007)</a>, p. 41, 43.</span> </li> <li id="cite_note-FOOTNOTEYoung2012[httpsbooksgooglecombooksidOMrcN0a3LxICpgRA1-PA165_165]-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEYoung2012[httpsbooksgooglecombooksidOMrcN0a3LxICpgRA1-PA165_165]_12-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFYoung2012">Young (2012)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=OMrcN0a3LxIC&pg=RA1-PA165">165</a>.</span> </li> <li id="cite_note-FOOTNOTEVarbergPurcellRigdon200742,_47-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEVarbergPurcellRigdon200742,_47_13-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFVarbergPurcellRigdon2007">Varberg, Purcell & Rigdon (2007)</a>, p. 42, 47.</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"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://oeis.org/A003957">"OEIS A003957"</a>. <i>oeis.org</i><span class="reference-accessdate">. Retrieved <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%2Fen.wikipedia.org%3ASine+and+cosine" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEBourchteinBourchtein2022[httpsbooksgooglecombooksidnGxOEAAAQBAJpgPA294_294]-15"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEBourchteinBourchtein2022[httpsbooksgooglecombooksidnGxOEAAAQBAJpgPA294_294]_15-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEBourchteinBourchtein2022[httpsbooksgooglecombooksidnGxOEAAAQBAJpgPA294_294]_15-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFBourchteinBourchtein2022">Bourchtein & Bourchtein (2022)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=nGxOEAAAQBAJ&pg=PA294">294</a>.</span> </li> <li id="cite_note-FOOTNOTEVarbergPurcellRigdon2007115-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEVarbergPurcellRigdon2007115_16-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFVarbergPurcellRigdon2007">Varberg, Purcell & Rigdon (2007)</a>, p. 115.</span> </li> <li id="cite_note-FOOTNOTEVarbergPurcellRigdon2007155-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEVarbergPurcellRigdon2007155_17-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFVarbergPurcellRigdon2007">Varberg, Purcell & Rigdon (2007)</a>, p. 155.</span> </li> <li id="cite_note-FOOTNOTEVarbergPurcellRigdon2007157-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEVarbergPurcellRigdon2007157_18-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFVarbergPurcellRigdon2007">Varberg, Purcell & Rigdon (2007)</a>, p. 157.</span> </li> <li id="cite_note-FOOTNOTEVarbergRigdonPurcell200742-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEVarbergRigdonPurcell200742_19-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFVarbergRigdonPurcell2007">Varberg, Rigdon & Purcell (2007)</a>, p. 42.<span class="error harv-error" style="display: none; font-size:100%"> sfnp error: no target: CITEREFVarbergRigdonPurcell2007 (<a href="/wiki/Category:Harv_and_Sfn_template_errors" title="Category:Harv and Sfn template errors">help</a>)</span></span> </li> <li id="cite_note-FOOTNOTEVarbergPurcellRigdon2007199-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEVarbergPurcellRigdon2007199_20-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFVarbergPurcellRigdon2007">Varberg, Purcell & Rigdon (2007)</a>, p. 199.</span> </li> <li id="cite_note-FOOTNOTEVince2023[httpsbooksgooglecombooksidGnW6EAAAQBAJpgPA162_162]-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEVince2023[httpsbooksgooglecombooksidGnW6EAAAQBAJpgPA162_162]_21-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFVince2023">Vince (2023)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=GnW6EAAAQBAJ&pg=PA162">162</a>.</span> </li> <li id="cite_note-FOOTNOTEAdlaj2012-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEAdlaj2012_22-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFAdlaj2012">Adlaj (2012)</a>.</span> </li> <li id="cite_note-FOOTNOTEVarbergPurcellRigdon2007366-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEVarbergPurcellRigdon2007366_23-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFVarbergPurcellRigdon2007">Varberg, Purcell & Rigdon (2007)</a>, p. 366.</span> </li> <li id="cite_note-FOOTNOTEVarbergPurcellRigdon2007365-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEVarbergPurcellRigdon2007365_25-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFVarbergPurcellRigdon2007">Varberg, Purcell & Rigdon (2007)</a>, p. 365.</span> </li> <li id="cite_note-FOOTNOTEYoung2017[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA99_99]-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEYoung2017[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA99_99]_26-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFYoung2017">Young (2017)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=476ZDwAAQBAJ&pg=PA99">99</a>.</span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDennis_G._Zill2013" class="citation book cs1">Dennis G. Zill (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=dtS5M4lx7scC"><i>Precalculus with Calculus Previews</i></a>. Jones & Bartlett Publishers. p. 238. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4496-4515-1" title="Special:BookSources/978-1-4496-4515-1"><bdi>978-1-4496-4515-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Precalculus+with+Calculus+Previews&rft.pages=238&rft.pub=Jones+%26+Bartlett+Publishers&rft.date=2013&rft.isbn=978-1-4496-4515-1&rft.au=Dennis+G.+Zill&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DdtS5M4lx7scC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASine+and+cosine" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=dtS5M4lx7scC&pg=PA238">Extract of page 238</a></span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://calculus.subwiki.org/wiki/Sine-squared_function#Identities">"Sine-squared function"</a><span class="reference-accessdate">. Retrieved <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%2Fen.wikipedia.org%3ASine+and+cosine" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEVarbergPurcellRigdon2007491&ndash;492-30"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEVarbergPurcellRigdon2007491&ndash;492_30-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEVarbergPurcellRigdon2007491&ndash;492_30-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFVarbergPurcellRigdon2007">Varberg, Purcell & Rigdon (2007)</a>, p. 491–492.</span> </li> <li id="cite_note-FOOTNOTEAbramowitzStegun1970[httpsbooksgooglecombooksidMtU8uP7XMvoCpgPA74_74]-31"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEAbramowitzStegun1970[httpsbooksgooglecombooksidMtU8uP7XMvoCpgPA74_74]_31-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEAbramowitzStegun1970[httpsbooksgooglecombooksidMtU8uP7XMvoCpgPA74_74]_31-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFAbramowitzStegun1970">Abramowitz & Stegun (1970)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=MtU8uP7XMvoC&pg=PA74">74</a>.</span> </li> <li id="cite_note-FOOTNOTEPowell1981150-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEPowell1981150_32-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFPowell1981">Powell (1981)</a>, p. 150.</span> </li> <li id="cite_note-FOOTNOTERudin198788-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTERudin198788_33-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFRudin1987">Rudin (1987)</a>, p. 88.</span> </li> <li id="cite_note-FOOTNOTEZygmund19681-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEZygmund19681_34-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFZygmund1968">Zygmund (1968)</a>, p. 1.</span> </li> <li id="cite_note-FOOTNOTEZygmund196811-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEZygmund196811_35-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFZygmund1968">Zygmund (1968)</a>, p. 11.</span> </li> <li id="cite_note-FOOTNOTEHowie2003[httpsbooksgooglecombooksid0FZDBAAAQBAJpgPA24_24]-36"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEHowie2003[httpsbooksgooglecombooksid0FZDBAAAQBAJpgPA24_24]_36-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEHowie2003[httpsbooksgooglecombooksid0FZDBAAAQBAJpgPA24_24]_36-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFHowie2003">Howie (2003)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=0FZDBAAAQBAJ&pg=PA24">24</a>.</span> </li> <li id="cite_note-FOOTNOTERudin19872-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTERudin19872_37-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFRudin1987">Rudin (1987)</a>, p. 2.</span> </li> <li id="cite_note-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-38">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://math.stackexchange.com/q/220418">"Why are the phase portrait of the simple plane pendulum and a domain coloring of sin(z) so similar?"</a>. <i>math.stackexchange.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2019-08-12</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=math.stackexchange.com&rft.atitle=Why+are+the+phase+portrait+of+the+simple+plane+pendulum+and+a+domain+coloring+of+sin%28z%29+so+similar%3F&rft_id=https%3A%2F%2Fmath.stackexchange.com%2Fq%2F220418&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASine+and+cosine" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEPlofker2009[httpsbooksgooglecombooksidDHvThPNp9yMCpgPA257_257]-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEPlofker2009[httpsbooksgooglecombooksidDHvThPNp9yMCpgPA257_257]_39-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFPlofker2009">Plofker (2009)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=DHvThPNp9yMC&pg=PA257">257</a>.</span> </li> <li id="cite_note-FOOTNOTEMaor1998[httpsbooksgooglecombooksidr9aMrneWFpUCpgPA35_35]-40"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEMaor1998[httpsbooksgooglecombooksidr9aMrneWFpUCpgPA35_35]_40-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFMaor1998">Maor (1998)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=r9aMrneWFpUC&pg=PA35">35</a>.</span> </li> <li id="cite_note-FOOTNOTEMerzbachBoyer2011-41"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEMerzbachBoyer2011_41-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEMerzbachBoyer2011_41-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-FOOTNOTEMerzbachBoyer2011_41-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFMerzbachBoyer2011">Merzbach & Boyer (2011)</a>.</span> </li> <li id="cite_note-FOOTNOTEMaor199835&ndash;36-42"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEMaor199835&ndash;36_42-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFMaor1998">Maor (1998)</a>, p. 35–36.</span> </li> <li id="cite_note-FOOTNOTEKatz2008253-43"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEKatz2008253_43-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFKatz2008">Katz (2008)</a>, p. 253.</span> </li> <li id="cite_note-FOOTNOTESmith1958202-44"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTESmith1958202_44-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFSmith1958">Smith (1958)</a>, p. 202.</span> </li> <li id="cite_note-45"><span class="mw-cite-backlink"><b><a href="#cite_ref-45">^</a></b></span> <span class="reference-text">Various sources credit the first use of <span title="Medieval Latin-language text"><i lang="la">sinus</i></span> to either <ul><li><a href="/wiki/Plato_Tiburtinus" title="Plato Tiburtinus">Plato Tiburtinus</a>'s 1116 translation of the <i>Astronomy</i> of <a href="/wiki/Al-Battani" title="Al-Battani">Al-Battani</a></li> <li><a href="/wiki/Gerard_of_Cremona" title="Gerard of Cremona">Gerard of Cremona</a>'s translation of the <i>Algebra</i> of <a href="/wiki/Mu%E1%B8%A5ammad_ibn_M%C5%ABs%C4%81_al-Khw%C4%81rizm%C4%AB" class="mw-redirect" title="Muḥammad ibn Mūsā al-Khwārizmī">al-Khwārizmī</a></li> <li><a href="/wiki/Robert_of_Chester" title="Robert of Chester">Robert of Chester</a>'s 1145 translation of the tables of al-Khwārizmī</li></ul> See <a href="#CITEREFMerlet2004">Merlet (2004)</a>. See <a href="#CITEREFMaor1998">Maor (1998)</a>, Chapter 3, for an earlier etymology crediting Gerard. See <a href="#CITEREFKatz2008">Katz (2008)</a>, p. 210.</span> </li> <li id="cite_note-46"><span class="mw-cite-backlink"><b><a href="#cite_ref-46">^</a></b></span> <span class="reference-text">Fale's book alternately uses the spellings "sine", "signe", or "sign". <div class="paragraphbreak" style="margin-top:0.5em"></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFFale1593" class="citation book cs1">Fale, Thomas (1593). <a rel="nofollow" class="external text" href="https://archive.org/details/b30333106/page/19/mode/1up"><i>Horologiographia. The Art of Dialling: Teaching, an Easie and Perfect Way to make all Kindes of Dials ...</i></a> London: F. Kingston. p. 11, for example.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Horologiographia.+The+Art+of+Dialling%3A+Teaching%2C+an+Easie+and+Perfect+Way+to+make+all+Kindes+of+Dials+....&rft.place=London&rft.pages=11%2C+for+example&rft.pub=F.+Kingston&rft.date=1593&rft.aulast=Fale&rft.aufirst=Thomas&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fb30333106%2Fpage%2F19%2Fmode%2F1up&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASine+and+cosine" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEGunter1620-47"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEGunter1620_47-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFGunter1620">Gunter (1620)</a>.</span> </li> <li id="cite_note-48"><span class="mw-cite-backlink"><b><a href="#cite_ref-48">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBrendan1965" class="citation journal cs1">Brendan, T. (February 1965). "How Ptolemy constructed trigonometry tables". <i>The Mathematics Teacher</i>. <b>58</b> (2): <span class="nowrap">141–</span>149. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.5951%2FMT.58.2.0141">10.5951/MT.58.2.0141</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/27967990">27967990</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Mathematics+Teacher&rft.atitle=How+Ptolemy+constructed+trigonometry+tables&rft.volume=58&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E141-%3C%2Fspan%3E149&rft.date=1965-02&rft_id=info%3Adoi%2F10.5951%2FMT.58.2.0141&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F27967990%23id-name%3DJSTOR&rft.aulast=Brendan&rft.aufirst=T.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASine+and+cosine" class="Z3988"></span></span> </li> <li id="cite_note-49"><span class="mw-cite-backlink"><b><a href="#cite_ref-49">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFVan_Brummelen2009" class="citation book cs1"><a href="/wiki/Glen_Van_Brummelen" title="Glen Van Brummelen">Van Brummelen, Glen</a> (2009). "India". <i>The Mathematics of the Heavens and the Earth</i>. Princeton University Press. Ch. 3, pp. 94–134. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-12973-0" title="Special:BookSources/978-0-691-12973-0"><bdi>978-0-691-12973-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=India&rft.btitle=The+Mathematics+of+the+Heavens+and+the+Earth&rft.pages=Ch.+3%2C+pp.-94-134&rft.pub=Princeton+University+Press&rft.date=2009&rft.isbn=978-0-691-12973-0&rft.aulast=Van+Brummelen&rft.aufirst=Glen&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASine+and+cosine" class="Z3988"></span></span> </li> <li id="cite_note-Gingerich_1986-50"><span class="mw-cite-backlink"><b><a href="#cite_ref-Gingerich_1986_50-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGingerich1986" class="citation magazine cs1">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. p. 74. Archived from <a rel="nofollow" class="external text" href="http://faculty.kfupm.edu.sa/PHYS/alshukri/PHYS215/Islamic_astronomy.htm">the original</a> on 2013-10-19<span class="reference-accessdate">. Retrieved <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%2Fen.wikipedia.org%3ASine+and+cosine" class="Z3988"></span></span> </li> <li id="cite_note-Sesiano-51"><span class="mw-cite-backlink"><b><a href="#cite_ref-Sesiano_51-0">^</a></b></span> <span class="reference-text">Jacques Sesiano, "Islamic mathematics", p. 157, in <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSelinD'Ambrosio2000" class="citation book cs1"><a href="/wiki/Helaine_Selin" title="Helaine Selin">Selin, Helaine</a>; <a href="/wiki/Ubiratan_D%27Ambrosio" title="Ubiratan D'Ambrosio">D'Ambrosio, Ubiratan</a>, eds. (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/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4020-0260-1" title="Special:BookSources/978-1-4020-0260-1"><bdi>978-1-4020-0260-1</bdi></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%2Fen.wikipedia.org%3ASine+and+cosine" class="Z3988"></span></span> </li> <li id="cite_note-Britannica-52"><span class="mw-cite-backlink">^ <a href="#cite_ref-Britannica_52-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Britannica_52-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.britannica.com/EBchecked/topic/605281/trigonometry">"trigonometry"</a>. Encyclopedia Britannica. 17 June 2024.</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.date=2024-06-17&rft_id=http%3A%2F%2Fwww.britannica.com%2FEBchecked%2Ftopic%2F605281%2Ftrigonometry&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASine+and+cosine" class="Z3988"></span></span> </li> <li id="cite_note-53"><span class="mw-cite-backlink"><b><a href="#cite_ref-53">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFNicolás_Bourbaki1994" class="citation book cs1">Nicolás Bourbaki (1994). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/elementsofhistor0000bour"><i>Elements of the History of Mathematics</i></a></span>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9783540647676" title="Special:BookSources/9783540647676"><bdi>9783540647676</bdi></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%2Fen.wikipedia.org%3ASine+and+cosine" class="Z3988"></span></span> </li> <li id="cite_note-54"><span class="mw-cite-backlink"><b><a href="#cite_ref-54">^</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">Archived</a> 2011-07-20 at the <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">Archived</a> 2011-07-20 at the <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/">Archived</a> 2011-07-20 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></span> </li> <li id="cite_note-FOOTNOTEZimmermann2006-55"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEZimmermann2006_55-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFZimmermann2006">Zimmermann (2006)</a>.</span> </li> <li id="cite_note-matlab-56"><span class="mw-cite-backlink">^ <a href="#cite_ref-matlab_56-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-matlab_56-1"><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-57"><span class="mw-cite-backlink">^ <a href="#cite_ref-r_57-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-r_57-1"><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-58"><span class="mw-cite-backlink"><b><a href="#cite_ref-58">^</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-59"><span class="mw-cite-backlink"><b><a href="#cite_ref-59">^</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-60"><span class="mw-cite-backlink"><b><a href="#cite_ref-60">^</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-61"><span class="mw-cite-backlink"><b><a href="#cite_ref-61">^</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-62"><span class="mw-cite-backlink"><b><a href="#cite_ref-62">^</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" class="external text" href="https://web.archive.org/web/20190417143715/https://www.allegromicro.com/en/Products/Magnetic-Linear-And-Angular-Position-Sensor-ICs/Angular-Position-Sensor-ICs/AAS33051.aspx">Archived</a> 2019-04-17 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></span> </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="Works_cited">Works cited</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine_and_cosine&action=edit&section=30" title="Edit section: Works cited"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin refbegin-columns references-column-width" style="column-width: 30em"> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAbramowitzStegun1970" class="citation cs2"><a href="/wiki/Milton_Abramowitz" title="Milton Abramowitz">Abramowitz, Milton</a>; <a href="/wiki/Irene_Stegun" title="Irene Stegun">Stegun, Irene A.</a> (1970), <i><a href="/wiki/Abramowitz_and_Stegun" title="Abramowitz and Stegun">Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables</a></i>, New York: <a href="/wiki/Dover_Publications" title="Dover Publications">Dover Publications</a>, Ninth printing</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+Mathematical+Functions+with+Formulas%2C+Graphs%2C+and+Mathematical+Tables&rft.place=New+York&rft.pub=Dover+Publications&rft.date=1970&rft.aulast=Abramowitz&rft.aufirst=Milton&rft.au=Stegun%2C+Irene+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASine+and+cosine" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAdlaj2012" class="citation cs2">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 class="cs1-format">(PDF)</span>, <i>American Mathematical Society</i>, <b>59</b> (8): 1097</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Mathematical+Society&rft.atitle=An+Eloquent+Formula+for+the+Perimeter+of+an+Ellipse&rft.volume=59&rft.issue=8&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%2Fen.wikipedia.org%3ASine+and+cosine" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAxler2012" class="citation cs2">Axler, Sheldon (2012), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=B5RxDwAAQBAJ"><i>Algebra and Trigonometry</i></a>, <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0470-58579-5" title="Special:BookSources/978-0470-58579-5"><bdi>978-0470-58579-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebra+and+Trigonometry&rft.pub=John+Wiley+%26+Sons&rft.date=2012&rft.isbn=978-0470-58579-5&rft.aulast=Axler&rft.aufirst=Sheldon&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DB5RxDwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASine+and+cosine" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBourchteinBourchtein2022" class="citation cs2">Bourchtein, Ludmila; Bourchtein, Andrei (2022), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=nGxOEAAAQBAJ"><i>Theory of Infinite Sequences and Series</i></a>, Springer, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-030-79431-6">10.1007/978-3-030-79431-6</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-030-79431-6" title="Special:BookSources/978-3-030-79431-6"><bdi>978-3-030-79431-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theory+of+Infinite+Sequences+and+Series&rft.pub=Springer&rft.date=2022&rft_id=info%3Adoi%2F10.1007%2F978-3-030-79431-6&rft.isbn=978-3-030-79431-6&rft.aulast=Bourchtein&rft.aufirst=Ludmila&rft.au=Bourchtein%2C+Andrei&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DnGxOEAAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASine+and+cosine" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGunter1620" class="citation cs2"><a href="/wiki/Edmund_Gunter" title="Edmund Gunter">Gunter, Edmund</a> (1620), <i>Canon triangulorum</i></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Canon+triangulorum&rft.date=1620&rft.aulast=Gunter&rft.aufirst=Edmund&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASine+and+cosine" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHowie2003" class="citation cs2">Howie, John M. (2003), <i>Complex Analysis</i>, Springer Undergraduate Mathematics Series, Springer, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4471-0027-0">10.1007/978-1-4471-0027-0</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4471-0027-0" title="Special:BookSources/978-1-4471-0027-0"><bdi>978-1-4471-0027-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Complex+Analysis&rft.series=Springer+Undergraduate+Mathematics+Series&rft.pub=Springer&rft.date=2003&rft_id=info%3Adoi%2F10.1007%2F978-1-4471-0027-0&rft.isbn=978-1-4471-0027-0&rft.aulast=Howie&rft.aufirst=John+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASine+and+cosine" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFTraupman,_Ph.D.1966" class="citation cs2">Traupman, Ph.D., John C. (1966), <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/boysgirlsbookabo00gard_0"><i>The New College Latin & English Dictionary</i></a></span>, Toronto: Bantam, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-553-27619-0" title="Special:BookSources/0-553-27619-0"><bdi>0-553-27619-0</bdi></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%2Fen.wikipedia.org%3ASine+and+cosine" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKatz2008" class="citation cs2">Katz, Victor J. (2008), <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"><i>A History of Mathematics</i></a> <span class="cs1-format">(PDF)</span> (3rd ed.), Boston: Addison-Wesley, <q>The English word "sine" comes from a series of mistranslations of the Sanskrit <span title="Sanskrit-language text"><i lang="sa">jyā-ardha</i></span> (chord-half). Āryabhaṭa frequently abbreviated this term to <span title="Sanskrit-language text"><i lang="sa">jyā</i></span> or its synonym <span title="Sanskrit-language text"><i lang="sa">jīvá</i></span>. When some of the Hindu works were later translated into Arabic, the word was simply transcribed phonetically into an otherwise meaningless Arabic word <span title="Arabic-language romanization"><i lang="ar-Latn">jiba</i></span>. But since Arabic is written without vowels, later writers interpreted the consonants <span title="Arabic-language romanization"><i lang="ar-Latn">jb</i></span> as <span title="Arabic-language text"><i lang="ar">jaib</i></span>, which means bosom or breast. In the twelfth century, when an Arabic trigonometry work was translated into Latin, the translator used the equivalent Latin word <span title="Latin-language text"><i lang="la">sinus</i></span>, which also meant bosom, and by extension, fold (as in a toga over a breast), or a bay or gulf.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Mathematics&rft.place=Boston&rft.edition=3rd&rft.pub=Addison-Wesley&rft.date=2008&rft.aulast=Katz&rft.aufirst=Victor+J.&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%2Fen.wikipedia.org%3ASine+and+cosine" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMaor1998" class="citation cs2">Maor, Eli (1998), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=r9aMrneWFpUC"><i>Trigonometric Delights</i></a>, <a href="/wiki/Princeton_University_Press" title="Princeton University Press">Princeton University Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-4008-4282-4" title="Special:BookSources/1-4008-4282-4"><bdi>1-4008-4282-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Trigonometric+Delights&rft.pub=Princeton+University+Press&rft.date=1998&rft.isbn=1-4008-4282-4&rft.aulast=Maor&rft.aufirst=Eli&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dr9aMrneWFpUC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASine+and+cosine" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMerlet2004" class="citation cs2">Merlet, Jean-Pierre (2004), "A Note on the History of the Trigonometric Functions", in Ceccarelli, Marco (ed.), <i>International Symposium on History of Machines and Mechanisms</i>, Springer, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F1-4020-2204-2">10.1007/1-4020-2204-2</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4020-2203-6" title="Special:BookSources/978-1-4020-2203-6"><bdi>978-1-4020-2203-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=A+Note+on+the+History+of+the+Trigonometric+Functions&rft.btitle=International+Symposium+on+History+of+Machines+and+Mechanisms&rft.pub=Springer&rft.date=2004&rft_id=info%3Adoi%2F10.1007%2F1-4020-2204-2&rft.isbn=978-1-4020-2203-6&rft.aulast=Merlet&rft.aufirst=Jean-Pierre&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASine+and+cosine" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMerzbachBoyer2011" class="citation cs2"><a href="/wiki/Uta_Merzbach" title="Uta Merzbach">Merzbach, Uta C.</a>; <a href="/wiki/Carl_B._Boyer" class="mw-redirect" title="Carl B. Boyer">Boyer, Carl B.</a> (2011), <i>A History of Mathematics</i> (3rd ed.), <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>, <q>It was Robert of Chester's translation from Arabic that resulted in our word "sine". The Hindus had given the name jiva to the half-chord in trigonometry, and the Arabs had taken this over as jiba. In the Arabic language, there is also the word jaib meaning "bay" or "inlet". When Robert of Chester came to translate the technical word jiba, he seems to have confused this with the word jaib (perhaps because vowels were omitted); hence, he used the word sinus, the Latin word for "bay" or "inlet".<i></i></q></cite><i><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Mathematics&rft.edition=3rd&rft.pub=John+Wiley+%26+Sons&rft.date=2011&rft.aulast=Merzbach&rft.aufirst=Uta+C.&rft.au=Boyer%2C+Carl+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASine+and+cosine" class="Z3988"></span></i></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPlofker2009" class="citation cs2">Plofker (2009), <a href="/wiki/Mathematics_in_India_(book)" title="Mathematics in India (book)"><i>Mathematics in India</i></a>, <a href="/wiki/Princeton_University_Press" title="Princeton University Press">Princeton University Press</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+in+India&rft.pub=Princeton+University+Press&rft.date=2009&rft.au=Plofker&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASine+and+cosine" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPowell1981" class="citation cs2"><a href="/wiki/Michael_J._D._Powell" title="Michael J. D. Powell">Powell, Michael J. D.</a> (1981), <i>Approximation Theory and Methods</i>, <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-29514-7" title="Special:BookSources/978-0-521-29514-7"><bdi>978-0-521-29514-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Approximation+Theory+and+Methods&rft.pub=Cambridge+University+Press&rft.date=1981&rft.isbn=978-0-521-29514-7&rft.aulast=Powell&rft.aufirst=Michael+J.+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASine+and+cosine" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRudin1987" class="citation cs2"><a href="/wiki/Walter_Rudin" title="Walter Rudin">Rudin, Walter</a> (1987), <i>Real and complex analysis</i> (3rd ed.), New York: <a href="/wiki/McGraw-Hill" class="mw-redirect" title="McGraw-Hill">McGraw-Hill</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-054234-1" title="Special:BookSources/978-0-07-054234-1"><bdi>978-0-07-054234-1</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0924157">0924157</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Real+and+complex+analysis&rft.place=New+York&rft.edition=3rd&rft.pub=McGraw-Hill&rft.date=1987&rft.isbn=978-0-07-054234-1&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D924157%23id-name%3DMR&rft.aulast=Rudin&rft.aufirst=Walter&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASine+and+cosine" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSmith1958" class="citation cs2">Smith, D. E. (1958) [1925], <i>History of Mathematics</i>, vol. I, <a href="/wiki/Dover_Publications" title="Dover Publications">Dover Publications</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-20429-4" title="Special:BookSources/0-486-20429-4"><bdi>0-486-20429-4</bdi></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.pub=Dover+Publications&rft.date=1958&rft.isbn=0-486-20429-4&rft.aulast=Smith&rft.aufirst=D.+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASine+and+cosine" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFVarbergPurcellRigdon2007" class="citation cs2">Varberg, Dale E.; Purcell, Edwin J.; Rigdon, Steven E. (2007), <a rel="nofollow" class="external text" href="https://archive.org/details/matematika-a-purcell-calculus-9th-ed/mode/2up"><i>Calculus</i></a> (9th ed.), <a href="/wiki/Pearson_Prentice_Hall" class="mw-redirect" title="Pearson Prentice Hall">Pearson Prentice Hall</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0131469686" title="Special:BookSources/978-0131469686"><bdi>978-0131469686</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus&rft.edition=9th&rft.pub=Pearson+Prentice+Hall&rft.date=2007&rft.isbn=978-0131469686&rft.aulast=Varberg&rft.aufirst=Dale+E.&rft.au=Purcell%2C+Edwin+J.&rft.au=Rigdon%2C+Steven+E.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmatematika-a-purcell-calculus-9th-ed%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASine+and+cosine" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFVince2023" class="citation cs2">Vince, John (2023), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=GnW6EAAAQBAJ"><i>Calculus for Computer Graphics</i></a>, Springer, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-031-28117-4">10.1007/978-3-031-28117-4</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-031-28117-4" title="Special:BookSources/978-3-031-28117-4"><bdi>978-3-031-28117-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus+for+Computer+Graphics&rft.pub=Springer&rft.date=2023&rft_id=info%3Adoi%2F10.1007%2F978-3-031-28117-4&rft.isbn=978-3-031-28117-4&rft.aulast=Vince&rft.aufirst=John&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DGnW6EAAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASine+and+cosine" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFYoung2012" class="citation cs2"><a href="/wiki/Cynthia_Y._Young" title="Cynthia Y. Young">Young, Cynthia</a> (2012), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=OMrcN0a3LxIC"><i>Trigonometry</i></a> (3rd ed.), John Wiley & Sons, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-119-32113-2" title="Special:BookSources/978-1-119-32113-2"><bdi>978-1-119-32113-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Trigonometry&rft.edition=3rd&rft.pub=John+Wiley+%26+Sons&rft.date=2012&rft.isbn=978-1-119-32113-2&rft.aulast=Young&rft.aufirst=Cynthia&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DOMrcN0a3LxIC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASine+and+cosine" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFYoung2017" class="citation cs2">——— (2017), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=476ZDwAAQBAJ"><i>Trigonometry</i></a> (4th ed.), John Wiley & Sons, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-119-32113-2" title="Special:BookSources/978-1-119-32113-2"><bdi>978-1-119-32113-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Trigonometry&rft.edition=4th&rft.pub=John+Wiley+%26+Sons&rft.date=2017&rft.isbn=978-1-119-32113-2&rft.aulast=Young&rft.aufirst=Cynthia&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D476ZDwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASine+and+cosine" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFZimmermann2006" class="citation cs2">Zimmermann, Paul (2006), "Can we trust floating-point numbers?", <a rel="nofollow" class="external text" href="http://www.jaist.ac.jp/~bjorner/ae-is-budapest/talks/Sept20pm2_Zimmermann.pdf"><i>Grand Challenges of Informatics</i></a> <span class="cs1-format">(PDF)</span>, p. 14/31</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Can+we+trust+floating-point+numbers%3F&rft.btitle=Grand+Challenges+of+Informatics&rft.pages=14%2F31&rft.date=2006&rft.aulast=Zimmermann&rft.aufirst=Paul&rft_id=http%3A%2F%2Fwww.jaist.ac.jp%2F~bjorner%2Fae-is-budapest%2Ftalks%2FSept20pm2_Zimmermann.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASine+and+cosine" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFZygmund1968" class="citation cs2"><a href="/wiki/Antoni_Zygmund" title="Antoni Zygmund">Zygmund, Antoni</a> (1968), <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/trigonometricser0012azyg/"><i>Trigonometric Series</i></a></span> (2nd, reprinted ed.), <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0236587">0236587</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Trigonometric+Series&rft.edition=2nd%2C+reprinted&rft.pub=Cambridge+University+Press&rft.date=1968&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0236587%23id-name%3DMR&rft.aulast=Zygmund&rft.aufirst=Antoni&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ftrigonometricser0012azyg%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASine+and+cosine" class="Z3988"></span></li></ul> </div> <div class="mw-heading mw-heading2"><h2 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.navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Trigonometric_and_hyperbolic_functions" title="Template:Trigonometric and hyperbolic functions"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Trigonometric_and_hyperbolic_functions" title="Template talk:Trigonometric and hyperbolic functions"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Trigonometric_and_hyperbolic_functions" title="Special:EditPage/Template:Trigonometric and hyperbolic functions"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Trigonometric_and_hyperbolic_functions49" style="font-size:114%;margin:0 4em">Trigonometric and hyperbolic functions</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Groups</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/Trigonometric_functions" title="Trigonometric functions">Trigonometric</a> <ul><li><a class="mw-selflink selflink">Sine and cosine</a></li></ul></li> <li><a href="/wiki/Inverse_trigonometric_functions" title="Inverse trigonometric functions">Inverse trigonometric</a></li> <li><a href="/wiki/Hyperbolic_functions" title="Hyperbolic functions">Hyperbolic</a></li> <li><a href="/wiki/Inverse_hyperbolic_functions" title="Inverse hyperbolic functions">Inverse hyperbolic</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</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/Versine" title="Versine">Versine</a></li> <li><a href="/wiki/Exsecant" title="Exsecant">Exsecant</a></li> <li><a href="/wiki/Jy%C4%81,_koti-jy%C4%81_and_utkrama-jy%C4%81" title="Jyā, koti-jyā and utkrama-jyā">Jyā, koti-jyā and utkrama-jyā</a></li> <li><a href="/wiki/Atan2" title="Atan2">atan2</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Sine_and_cosine&oldid=1282465886">https://en.wikipedia.org/w/index.php?title=Sine_and_cosine&oldid=1282465886</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Angle" title="Category:Angle">Angle</a></li><li><a href="/wiki/Category:Trigonometric_functions" title="Category:Trigonometric functions">Trigonometric 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