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Available in 59 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-59" 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">59 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AF%D8%A7%D8%A6%D8%B1%D8%A9_%D9%88%D8%AD%D8%AF%D8%A9" title="دائرة وحدة – Arabic" lang="ar" hreflang="ar" data-title="دائرة وحدة" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%8F%E0%A6%95%E0%A6%95_%E0%A6%AC%E0%A7%83%E0%A6%A4%E0%A7%8D%E0%A6%A4" title="একক বৃত্ত – Bangla" lang="bn" hreflang="bn" data-title="একক বৃত্ত" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Tan-%C5%ABi-%C3%AE%E2%81%BF" title="Tan-ūi-îⁿ – Minnan" lang="nan" hreflang="nan" data-title="Tan-ūi-îⁿ" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%90%D0%B4%D0%B7%D1%96%D0%BD%D0%BA%D0%B0%D0%B2%D0%B0%D1%8F_%D0%B0%D0%BA%D1%80%D1%83%D0%B6%D0%BD%D0%B0%D1%81%D1%86%D1%8C" title="Адзінкавая акружнасць – Belarusian" lang="be" hreflang="be" data-title="Адзінкавая акружнасць" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%95%D0%B4%D0%B8%D0%BD%D0%B8%D1%87%D0%BD%D0%B0_%D0%BE%D0%BA%D1%80%D1%8A%D0%B6%D0%BD%D0%BE%D1%81%D1%82" title="Единична окръжност – Bulgarian" lang="bg" hreflang="bg" data-title="Единична окръжност" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Jedini%C4%8Dni_krug" title="Jedinični krug – Bosnian" lang="bs" hreflang="bs" data-title="Jedinični krug" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Circumfer%C3%A8ncia_goniom%C3%A8trica" title="Circumferència goniomètrica – Catalan" lang="ca" hreflang="ca" data-title="Circumferència goniomètrica" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9F%C4%95%D1%80%D1%87%C4%95%D0%BB%D0%BB%D0%B5_%C3%A7%D0%B0%D0%B2%D1%80%D0%B0%D0%BA%C4%83%D1%88" title="Пĕрчĕлле çавракăш – Chuvash" lang="cv" hreflang="cv" data-title="Пĕрчĕлле çавракăш" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Jednotkov%C3%A1_kru%C5%BEnice" title="Jednotková kružnice – Czech" lang="cs" hreflang="cs" data-title="Jednotková kružnice" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Enhedscirklen" title="Enhedscirklen – Danish" lang="da" hreflang="da" data-title="Enhedscirklen" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Einheitskreis" title="Einheitskreis – German" lang="de" hreflang="de" data-title="Einheitskreis" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/%C3%9Chikringjoon" title="Ühikringjoon – Estonian" lang="et" hreflang="et" data-title="Ühikringjoon" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9C%CE%BF%CE%BD%CE%B1%CE%B4%CE%B9%CE%B1%CE%AF%CE%BF%CF%82_%CE%BA%CF%8D%CE%BA%CE%BB%CE%BF%CF%82" title="Μοναδιαίος κύκλος – Greek" lang="el" hreflang="el" data-title="Μοναδιαίος κύκλος" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Circunferencia_goniom%C3%A9trica" title="Circunferencia goniométrica – Spanish" lang="es" hreflang="es" data-title="Circunferencia goniométrica" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Unuocirklo" title="Unuocirklo – Esperanto" lang="eo" hreflang="eo" data-title="Unuocirklo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AF%D8%A7%DB%8C%D8%B1%D9%87_%D9%88%D8%A7%D8%AD%D8%AF" title="دایره واحد – Persian" lang="fa" hreflang="fa" data-title="دایره واحد" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Cercle_unit%C3%A9" title="Cercle unité – French" lang="fr" hreflang="fr" data-title="Cercle unité" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Circunferencia_goniom%C3%A9trica" title="Circunferencia goniométrica – Galician" lang="gl" hreflang="gl" data-title="Circunferencia goniométrica" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%8B%A8%EC%9C%84%EC%9B%90" title="단위원 – Korean" lang="ko" hreflang="ko" data-title="단위원" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%84%D5%AB%D5%A1%D5%BE%D5%B8%D6%80_%D5%B7%D6%80%D5%BB%D5%A1%D5%B6%D5%A1%D5%A3%D5%AB%D5%AE" title="Միավոր շրջանագիծ – Armenian" lang="hy" hreflang="hy" data-title="Միավոր շրջանագիծ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Jedini%C4%8Dna_kru%C5%BEnica" title="Jedinična kružnica – Croatian" lang="hr" hreflang="hr" data-title="Jedinična kružnica" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Lingkaran_satuan" title="Lingkaran satuan – Indonesian" lang="id" hreflang="id" data-title="Lingkaran satuan" 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-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Circonferenza_unitaria" title="Circonferenza unitaria – Italian" lang="it" hreflang="it" data-title="Circonferenza unitaria" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A2%D7%92%D7%9C_%D7%94%D7%99%D7%97%D7%99%D7%93%D7%94" title="מעגל היחידה – Hebrew" lang="he" hreflang="he" data-title="מעגל היחידה" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%91%D0%B8%D1%80%D0%B4%D0%B8%D0%BA_%D0%B0%D0%B9%D0%BB%D0%B0%D0%BD%D0%B0" title="Бирдик айлана – Kyrgyz" lang="ky" hreflang="ky" data-title="Бирдик айлана" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Trigonometrisk%C4%81_ri%C5%86%C4%B7a_l%C4%ABnija" title="Trigonometriskā riņķa līnija – Latvian" lang="lv" hreflang="lv" data-title="Trigonometriskā riņķa līnija" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Vienetinis_apskritimas" title="Vienetinis apskritimas – Lithuanian" lang="lt" hreflang="lt" data-title="Vienetinis apskritimas" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%95%D0%B4%D0%B8%D0%BD%D0%B8%D1%87%D0%BD%D0%B0_%D0%BA%D1%80%D1%83%D0%B6%D0%BD%D0%B8%D1%86%D0%B0" title="Единична кружница – Macedonian" lang="mk" hreflang="mk" data-title="Единична кружница" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%AF%E0%B5%82%E0%B4%A3%E0%B4%BF%E0%B4%B1%E0%B5%8D%E0%B4%B1%E0%B5%8D_%E0%B4%B5%E0%B5%83%E0%B4%A4%E0%B5%8D%E0%B4%A4%E0%B4%82" title="യൂണിറ്റ് വൃത്തം – Malayalam" lang="ml" hreflang="ml" data-title="യൂണിറ്റ് വൃത്തം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%9D%D1%8D%D0%B3%D0%B6_%D1%82%D0%BE%D0%B9%D1%80%D0%BE%D0%B3" title="Нэгж тойрог – Mongolian" lang="mn" hreflang="mn" data-title="Нэгж тойрог" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Eenheidscirkel" title="Eenheidscirkel – Dutch" lang="nl" hreflang="nl" data-title="Eenheidscirkel" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%8D%98%E4%BD%8D%E5%86%86" title="単位円 – Japanese" lang="ja" hreflang="ja" data-title="単位円" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Enhetssirkelen" title="Enhetssirkelen – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Enhetssirkelen" 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-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Einingssirkel" title="Einingssirkel – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Einingssirkel" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%9A%E1%9E%84%E1%9F%92%E1%9E%9C%E1%9E%84%E1%9F%8B%E1%9E%8F%E1%9F%92%E1%9E%9A%E1%9E%B8%E1%9E%80%E1%9F%84%E1%9E%8E%E1%9E%98%E1%9E%B6%E1%9E%8F%E1%9F%92%E1%9E%9A" title="រង្វង់ត្រីកោណមាត្រ – Khmer" lang="km" hreflang="km" data-title="រង្វង់ត្រីកោណមាត្រ" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="Khmer" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Sirconferensa_goniom%C3%A9trica" title="Sirconferensa goniométrica – Piedmontese" lang="pms" hreflang="pms" data-title="Sirconferensa goniométrica" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Okr%C4%85g_jednostkowy" title="Okrąg jednostkowy – Polish" lang="pl" hreflang="pl" data-title="Okrąg jednostkowy" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/C%C3%ADrculo_unit%C3%A1rio" title="Círculo unitário – Portuguese" lang="pt" hreflang="pt" data-title="Círculo unitário" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Cerc_trigonometric" title="Cerc trigonometric – Romanian" lang="ro" hreflang="ro" data-title="Cerc trigonometric" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%95%D0%B4%D0%B8%D0%BD%D0%B8%D1%87%D0%BD%D0%B0%D1%8F_%D0%BE%D0%BA%D1%80%D1%83%D0%B6%D0%BD%D0%BE%D1%81%D1%82%D1%8C" title="Единичная окружность – Russian" lang="ru" hreflang="ru" data-title="Единичная окружность" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Rrethi_nj%C3%ABsi" title="Rrethi njësi – Albanian" lang="sq" hreflang="sq" data-title="Rrethi njësi" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Unit_circle" title="Unit circle – Simple English" lang="en-simple" hreflang="en-simple" data-title="Unit circle" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Jednotkov%C3%A1_kru%C5%BEnica" title="Jednotková kružnica – Slovak" lang="sk" hreflang="sk" data-title="Jednotková kružnica" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Enotska_kro%C5%BEnica" title="Enotska krožnica – Slovenian" lang="sl" hreflang="sl" data-title="Enotska krožnica" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%A8%D8%A7%D8%B2%D9%86%DB%95%DB%8C_%DB%8C%DB%95%DA%A9%DB%95" title="بازنەی یەکە – Central Kurdish" lang="ckb" hreflang="ckb" data-title="بازنەی یەکە" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%88%D0%B5%D0%B4%D0%B8%D0%BD%D0%B8%D1%87%D0%BD%D0%B8_%D0%BA%D1%80%D1%83%D0%B3" title="Јединични круг – Serbian" lang="sr" hreflang="sr" data-title="Јединични круг" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Jedini%C4%8Dni_krug" title="Jedinični krug – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Jedinični krug" data-language-autonym="Srpskohrvatski / српскохрватски" 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src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Unit_circle.svg/220px-Unit_circle.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Unit_circle.svg/330px-Unit_circle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Unit_circle.svg/440px-Unit_circle.svg.png 2x" data-file-width="352" data-file-height="352" /></a><figcaption>Illustration of a unit circle. The variable <i>t</i> is an <a href="/wiki/Angle" title="Angle">angle</a> measure.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:2pi-unrolled.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/2pi-unrolled.gif/220px-2pi-unrolled.gif" decoding="async" width="220" height="76" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/2pi-unrolled.gif/330px-2pi-unrolled.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/67/2pi-unrolled.gif/440px-2pi-unrolled.gif 2x" data-file-width="870" data-file-height="300" /></a><figcaption>Animation of the act of unrolling the circumference of a unit circle, a circle with radius of 1. Since <span class="texhtml"><i>C</i> = 2<i>πr</i></span>, the circumference of a unit circle is <span class="texhtml">2π</span>.</figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a <b>unit circle</b> is a <a href="/wiki/Circle" title="Circle">circle</a> of unit <a href="/wiki/Radius" title="Radius">radius</a>—that is, a radius of 1.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Frequently, especially in <a href="/wiki/Trigonometry" title="Trigonometry">trigonometry</a>, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the <a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian coordinate system</a> in the <a href="/wiki/Euclidean_plane" title="Euclidean plane">Euclidean plane</a>. In <a href="/wiki/Topology" title="Topology">topology</a>, it is often denoted as <span class="texhtml"><i>S</i><sup>1</sup></span> because it is a one-dimensional unit <a href="/wiki/N-sphere" title="N-sphere"><span class="texhtml"><i>n</i></span>-sphere</a>.<sup id="cite_ref-Unit_sphere_2-0" class="reference"><a href="#cite_note-Unit_sphere-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>note 1<span class="cite-bracket">]</span></a></sup> </p><p>If <span class="texhtml">(<i>x</i>, <i>y</i>)</span> is a point on the unit circle's <a href="/wiki/Circumference" title="Circumference">circumference</a>, then <span class="texhtml">|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>x</i></span>|</span> and <span class="texhtml">|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>y</i></span>|</span> are the lengths of the legs of a <a href="/wiki/Right_triangle" title="Right triangle">right triangle</a> whose hypotenuse has length 1. Thus, by the <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a>, <span class="texhtml"><i>x</i></span> and <span class="texhtml"><i>y</i></span> satisfy the equation <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 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/bd4ba9a57bd82841e5889575e2ff3e1ef5fad8e9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.347ex; height:3.009ex;" alt="{\displaystyle x^{2}+y^{2}=1.}"></span> </p><p>Since <span class="texhtml"><i>x</i><sup>2</sup> = (−<i>x</i>)<sup>2</sup></span> for all <span class="texhtml"><i>x</i></span>, and since the reflection of any point on the unit circle about the <span class="texhtml"><i>x</i></span>- or <span class="texhtml"><i>y</i></span>-axis is also on the unit circle, the above equation holds for all points <span class="texhtml">(<i>x</i>, <i>y</i>)</span> on the unit circle, not only those in the first quadrant. </p><p>The interior of the unit circle is called the open <a href="/wiki/Unit_disk" title="Unit disk">unit disk</a>, while the interior of the unit circle combined with the unit circle itself is called the closed unit disk. </p><p>One may also use other notions of "distance" to define other "unit circles", such as the <a href="/wiki/Riemannian_circle" class="mw-redirect" title="Riemannian circle">Riemannian circle</a>; see the article on <a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)">mathematical norms</a> for additional examples. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="In_the_complex_plane">In the complex plane</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unit_circle&action=edit&section=1" title="Edit section: In the complex plane"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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">Main article: <a href="/wiki/Unit_complex_numbers" class="mw-redirect" title="Unit complex numbers">unit complex numbers</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Unitycircle-complex.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Unitycircle-complex.gif/220px-Unitycircle-complex.gif" decoding="async" width="220" height="218" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Unitycircle-complex.gif/330px-Unitycircle-complex.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Unitycircle-complex.gif/440px-Unitycircle-complex.gif 2x" data-file-width="500" data-file-height="495" /></a><figcaption>Animation of the unit circle with angles</figcaption></figure> <p>In the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a>, numbers of unit magnitude are called the <a href="/wiki/Unit_complex_numbers" class="mw-redirect" title="Unit complex numbers">unit complex numbers</a>. This is the set of <a href="/wiki/Complex_number" title="Complex number">complex numbers</a> <span class="texhtml mvar" style="font-style:italic;">z</span> such that <span class="mwe-math-element"><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|=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |z|=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abfbcdd6b1afc76fb8a016f500ba46153841a4c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.29ex; height:2.843ex;" alt="{\displaystyle |z|=1.}"></span> When broken into real and imaginary components <span class="mwe-math-element"><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> <mo>,</mo> </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/8ee21a775fda65273b6e1a6786b7f3d51a85de65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.961ex; height:2.509ex;" alt="{\displaystyle z=x+iy,}"></span> this condition 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 |z|^{2}=z{\bar {z}}=x^{2}+y^{2}=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>=</mo> <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 |z|^{2}=z{\bar {z}}=x^{2}+y^{2}=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3974daa6bb5176c5e3b709ccd0a0946872d9da0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.364ex; height:3.343ex;" alt="{\displaystyle |z|^{2}=z{\bar {z}}=x^{2}+y^{2}=1.}"></span> </p><p>The complex unit circle can be parametrized by angle 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 \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> from the positive real axis using the complex <a href="/wiki/Exponential_function" title="Exponential function">exponential function</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 z=e^{i\theta }=\cos \theta +i\sin \theta .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=e^{i\theta }=\cos \theta +i\sin \theta .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4d0745efb7e728dc6e5d566e49bc5c2894a2422" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:23.538ex; height:2.843ex;" alt="{\displaystyle z=e^{i\theta }=\cos \theta +i\sin \theta .}"></span> (See <a href="/wiki/Euler%27s_formula" title="Euler's formula">Euler's formula</a>.) </p><p>Under the complex multiplication operation, the unit complex numbers form a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> called the <i><a href="/wiki/Circle_group" title="Circle group">circle group</a></i>, usually denoted <span class="mwe-math-element"><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 {T} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">T</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {T} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23bfcd3d5e94ad3ec191037d3e330ea77943d9cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.197ex; height:2.176ex;" alt="{\displaystyle \mathbb {T} .}"></span> In <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, a unit complex number is called a <a href="/wiki/Phase_factor" title="Phase factor">phase factor</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Trigonometric_functions_on_the_unit_circle">Trigonometric functions on the unit circle</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unit_circle&action=edit&section=2" title="Edit section: Trigonometric functions on the unit circle"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Unit-circle_sin_cos_tan_cot_exsec_excsc_versin_vercos_coversin_covercos.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Unit-circle_sin_cos_tan_cot_exsec_excsc_versin_vercos_coversin_covercos.svg/350px-Unit-circle_sin_cos_tan_cot_exsec_excsc_versin_vercos_coversin_covercos.svg.png" decoding="async" width="350" height="239" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Unit-circle_sin_cos_tan_cot_exsec_excsc_versin_vercos_coversin_covercos.svg/525px-Unit-circle_sin_cos_tan_cot_exsec_excsc_versin_vercos_coversin_covercos.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/af/Unit-circle_sin_cos_tan_cot_exsec_excsc_versin_vercos_coversin_covercos.svg/700px-Unit-circle_sin_cos_tan_cot_exsec_excsc_versin_vercos_coversin_covercos.svg.png 2x" data-file-width="1265" data-file-height="865" /></a><figcaption>All of the trigonometric functions of the angle <span class="texhtml"><i>θ</i></span> (theta) can be constructed geometrically in terms of a unit circle centered at <i>O</i>.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Periodic_sine.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Periodic_sine.svg/220px-Periodic_sine.svg.png" decoding="async" width="220" height="301" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Periodic_sine.svg/330px-Periodic_sine.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Periodic_sine.svg/440px-Periodic_sine.svg.png 2x" data-file-width="405" data-file-height="555" /></a><figcaption>Sine function on unit circle (top) and its graph (bottom)</figcaption></figure> <p>The <a href="/wiki/Trigonometric_functions" title="Trigonometric functions">trigonometric functions</a> cosine and sine of angle <span class="texhtml"><i>θ</i></span> may be defined on the unit circle as follows: If <span class="texhtml">(<i>x</i>, <i>y</i>)</span> is a point on the unit circle, and if the ray from the origin <span class="texhtml">(0, 0)</span> to <span class="texhtml">(<i>x</i>, <i>y</i>)</span> makes an <a href="/wiki/Angle" title="Angle">angle</a> <span class="texhtml"><i>θ</i></span> from the positive <span class="texhtml"><i>x</i></span>-axis, (where counterclockwise turning is positive), then <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 \cos \theta =x\quad {\text{and}}\quad \sin \theta =y.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>x</mi> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> <mspace width="1em" /> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>y</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \theta =x\quad {\text{and}}\quad \sin \theta =y.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94f54d9af620968ed7a309f243df9a718ca6243d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:27.031ex; height:2.509ex;" alt="{\displaystyle \cos \theta =x\quad {\text{and}}\quad \sin \theta =y.}"></span> </p><p>The equation <span class="texhtml"><i>x</i><sup>2</sup> + <i>y</i><sup>2</sup> = 1</span> gives the relation <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 \cos ^{2}\theta +\sin ^{2}\theta =1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos ^{2}\theta +\sin ^{2}\theta =1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e79d40676a67b3463099809cd6bd9e046d5781f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.778ex; height:2.843ex;" alt="{\displaystyle \cos ^{2}\theta +\sin ^{2}\theta =1.}"></span> </p><p>The unit circle also demonstrates that <a href="/wiki/Sine" class="mw-redirect" title="Sine">sine</a> and <a href="/wiki/Cosine" class="mw-redirect" title="Cosine">cosine</a> are <a href="/wiki/Periodic_function" title="Periodic function">periodic functions</a>, with the identities <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 \cos \theta =\cos(2\pi k+\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>k</mi> <mo>+</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \theta =\cos(2\pi k+\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65c6c880d7c4ed64cc3db288af123fec3f026e79" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.244ex; height:2.843ex;" alt="{\displaystyle \cos \theta =\cos(2\pi k+\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 \sin \theta =\sin(2\pi k+\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>k</mi> <mo>+</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \theta =\sin(2\pi k+\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42ba79dfa8eb6f67f6fe249e49f13f1a59da8460" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.733ex; height:2.843ex;" alt="{\displaystyle \sin \theta =\sin(2\pi k+\theta )}"></span> for any <a href="/wiki/Integer" title="Integer">integer</a> <span class="texhtml"><i>k</i></span>. </p><p>Triangles constructed on the unit circle can also be used to illustrate the periodicity of the trigonometric functions. First, construct a radius <span class="texhtml">OP</span> from the origin <span class="texhtml">O</span> to a point <span class="texhtml">P(<i>x</i><sub>1</sub>,<i>y</i><sub>1</sub>)</span> on the unit circle such that an angle <span class="texhtml"><i>t</i></span> with <span class="texhtml">0 < <i>t</i> < <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">π</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span> is formed with the positive arm of the <span class="texhtml"><i>x</i></span>-axis. Now consider a point <span class="texhtml">Q(<i>x</i><sub>1</sub>,0)</span> and line segments <span class="texhtml">PQ ⊥ OQ</span>. The result is a right triangle <span class="texhtml">△OPQ</span> with <span class="texhtml">∠QOP = <i>t</i></span>. Because <span class="texhtml">PQ</span> has length <span class="texhtml"><i>y</i><sub>1</sub></span>, <span class="texhtml">OQ</span> length <span class="texhtml"><i>x</i><sub>1</sub></span>, and <span class="texhtml">OP</span> has length 1 as a radius on the unit circle, <span class="texhtml">sin(<i>t</i>) = <i>y</i><sub>1</sub></span> and <span class="texhtml">cos(<i>t</i>) = <i>x</i><sub>1</sub></span>. Having established these equivalences, take another radius <span class="texhtml">OR</span> from the origin to a point <span class="texhtml">R(−<i>x</i><sub>1</sub>,<i>y</i><sub>1</sub>)</span> on the circle such that the same angle <span class="texhtml"><i>t</i></span> is formed with the negative arm of the <span class="texhtml"><i>x</i></span>-axis. Now consider a point <span class="texhtml">S(−<i>x</i><sub>1</sub>,0)</span> and line segments <span class="texhtml">RS ⊥ OS</span>. The result is a right triangle <span class="texhtml">△ORS</span> with <span class="texhtml">∠SOR = <i>t</i></span>. It can hence be seen that, because <span class="texhtml">∠ROQ = π − <i>t</i></span>, <span class="texhtml">R</span> is at <span class="texhtml">(cos(π − <i>t</i>), sin(π − <i>t</i>))</span> in the same way that P is at <span class="texhtml">(cos(<i>t</i>), sin(<i>t</i>))</span>. The conclusion is that, since <span class="texhtml">(−<i>x</i><sub>1</sub>, <i>y</i><sub>1</sub>)</span> is the same as <span class="texhtml">(cos(π − <i>t</i>), sin(π − <i>t</i>))</span> and <span class="texhtml">(<i>x</i><sub>1</sub>,<i>y</i><sub>1</sub>)</span> is the same as <span class="texhtml">(cos(<i>t</i>),sin(<i>t</i>))</span>, it is true that <span class="texhtml">sin(<i>t</i>) = sin(π − <i>t</i>)</span> and <span class="texhtml">−cos(<i>t</i>) = cos(π − <i>t</i>)</span>. It may be inferred in a similar manner that <span class="texhtml">tan(π − <i>t</i>) = −tan(<i>t</i>)</span>, since <span class="texhtml">tan(<i>t</i>) = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>y</i><sub>1</sub></span><span class="sr-only">/</span><span class="den"><i>x</i><sub>1</sub></span></span>⁠</span></span> and <span class="texhtml">tan(π − <i>t</i>) = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>y</i><sub>1</sub></span><span class="sr-only">/</span><span class="den">−<i>x</i><sub>1</sub></span></span>⁠</span></span>. A simple demonstration of the above can be seen in the equality <span class="texhtml">sin(<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">π</span><span class="sr-only">/</span><span class="den">4</span></span>⁠</span>) = sin(<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">3π</span><span class="sr-only">/</span><span class="den">4</span></span>⁠</span>) = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den"><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span></span></span>⁠</span></span>. </p><p>When working with right triangles, sine, cosine, and other trigonometric functions only make sense for angle measures more than zero and less than <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><span class="texhtml mvar" style="font-style:italic;">π</span></span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>. However, when defined with the unit circle, these functions produce meaningful values for any <a href="/wiki/Real_number" title="Real number">real</a>-valued angle measure – even those greater than 2<span class="texhtml mvar" style="font-style:italic;">π</span>. In fact, all six standard trigonometric functions – sine, cosine, tangent, cotangent, secant, and cosecant, as well as archaic functions like <a href="/wiki/Versine" title="Versine">versine</a> and <a href="/wiki/Exsecant" title="Exsecant">exsecant</a> – can be defined geometrically in terms of a unit circle, as shown at right. </p><p>Using the unit circle, the values of any trigonometric function for many angles other than those labeled can be easily calculated by hand using the <a href="/wiki/Trigonometric_identity#Angle_sum_and_difference_identities" class="mw-redirect" title="Trigonometric identity">angle sum and difference formulas</a>. </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Unit_circle_angles_color.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Unit_circle_angles_color.svg/300px-Unit_circle_angles_color.svg.png" decoding="async" width="300" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Unit_circle_angles_color.svg/450px-Unit_circle_angles_color.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Unit_circle_angles_color.svg/600px-Unit_circle_angles_color.svg.png 2x" data-file-width="720" data-file-height="720" /></a><figcaption>The unit circle, showing <a href="/wiki/Exact_trigonometric_constants" class="mw-redirect" title="Exact trigonometric constants">coordinates of certain points</a></figcaption></figure> <div class="mw-heading mw-heading2"><h2 id="Complex_dynamics">Complex dynamics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unit_circle&action=edit&section=3" title="Edit section: Complex dynamics"><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/Complex_dynamics" title="Complex dynamics">Complex dynamics</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Erays.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/Erays.svg/220px-Erays.svg.png" decoding="async" width="220" height="110" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/Erays.svg/330px-Erays.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/79/Erays.svg/440px-Erays.svg.png 2x" data-file-width="1000" data-file-height="500" /></a><figcaption>Unit circle in complex dynamics</figcaption></figure> <p>The <a href="/wiki/Julia_set" title="Julia set">Julia set</a> of <a href="/wiki/Dynamical_system_(definition)" class="mw-redirect" title="Dynamical system (definition)">discrete nonlinear dynamical system</a> with <a href="/wiki/Evolution_function" class="mw-redirect" title="Evolution function">evolution function</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{0}(x)=x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{0}(x)=x^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a1e3fcab1bf54980595879e55fc3cbbba91952c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.815ex; height:3.176ex;" alt="{\displaystyle f_{0}(x)=x^{2}}"></span> is a unit circle. It is a simplest case so it is widely used in the study of dynamical systems. </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=Unit_circle&action=edit&section=4" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Angle_measure" class="mw-redirect" title="Angle measure">Angle measure</a></li> <li><a href="/wiki/Pythagorean_trigonometric_identity" title="Pythagorean trigonometric identity">Pythagorean trigonometric identity</a></li> <li><a href="/wiki/Riemannian_circle" class="mw-redirect" title="Riemannian circle">Riemannian circle</a></li> <li><a href="/wiki/Radian" title="Radian">Radian</a></li> <li><a href="/wiki/Unit_disk" title="Unit disk">Unit disk</a></li> <li><a href="/wiki/Unit_sphere" title="Unit sphere">Unit sphere</a></li> <li><a href="/wiki/Unit_hyperbola" title="Unit hyperbola">Unit hyperbola</a></li> <li><a href="/wiki/Unit_square" title="Unit square">Unit square</a></li> <li><a href="/wiki/Turn_(angle)" title="Turn (angle)">Turn (angle)</a></li> <li><a href="/wiki/Z-transform" title="Z-transform">z-transform</a></li> <li><a href="/wiki/Smith_chart" title="Smith chart">Smith chart</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unit_circle&action=edit&section=5" title="Edit section: Notes"><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"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">For further discussion, see the <a href="/wiki/Circle" title="Circle">technical distinction between a circle and a disk</a>.<sup id="cite_ref-Unit_sphere_2-1" class="reference"><a href="#cite_note-Unit_sphere-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></span> </li> </ol></div></div> <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=Unit_circle&action=edit&section=6" title="Edit section: References"><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"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</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 id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/UnitCircle.html">"Unit Circle"</a>. <i>mathworld.wolfram.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-05-05</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=mathworld.wolfram.com&rft.atitle=Unit+Circle&rft.aulast=Weisstein&rft.aufirst=Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FUnitCircle.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUnit+circle" class="Z3988"></span></span> </li> <li id="cite_note-Unit_sphere-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Unit_sphere_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Unit_sphere_2-1"><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 id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Hypersphere.html">"Hypersphere"</a>. <i>mathworld.wolfram.com</i><span class="reference-accessdate">. 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