Trigonometric functions

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class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Simple_algebraic_values"> <div class="vector-toc-text"> 5.1 Simple algebraic values </div> </a> <ul id="toc-Simple_algebraic_values-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Definitions_in_analysis" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Definitions_in_analysis"> <div class="vector-toc-text"> 6 Definitions in analysis </div> </a> <button aria-controls="toc-Definitions_in_analysis-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Definitions in analysis subsection </button> <ul id="toc-Definitions_in_analysis-sublist" class="vector-toc-list"> <li id="toc-Definition_by_differential_equations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Definition_by_differential_equations"> <div class="vector-toc-text"> 6.1 Definition by differential equations </div> </a> <ul id="toc-Definition_by_differential_equations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Power_series_expansion" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Power_series_expansion"> <div class="vector-toc-text"> 6.2 Power series expansion </div> </a> <ul id="toc-Power_series_expansion-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Continued_fraction_expansion" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Continued_fraction_expansion"> <div class="vector-toc-text"> 6.3 Continued fraction expansion </div> </a> <ul id="toc-Continued_fraction_expansion-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Partial_fraction_expansion" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Partial_fraction_expansion"> <div class="vector-toc-text"> 6.4 Partial fraction expansion </div> </a> <ul id="toc-Partial_fraction_expansion-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Infinite_product_expansion" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Infinite_product_expansion"> <div class="vector-toc-text"> 6.5 Infinite product expansion </div> </a> <ul id="toc-Infinite_product_expansion-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Euler's_formula_and_the_exponential_function" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Euler's_formula_and_the_exponential_function"> <div class="vector-toc-text"> 6.6 Euler's formula and the exponential function </div> </a> <ul id="toc-Euler's_formula_and_the_exponential_function-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Definition_via_integration" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Definition_via_integration"> <div class="vector-toc-text"> 6.7 Definition via integration </div> </a> <ul id="toc-Definition_via_integration-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Definitions_using_functional_equations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Definitions_using_functional_equations"> <div class="vector-toc-text"> 6.8 Definitions using functional equations </div> </a> <ul id="toc-Definitions_using_functional_equations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_the_complex_plane" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_the_complex_plane"> <div class="vector-toc-text"> 6.9 In the complex plane </div> </a> <ul id="toc-In_the_complex_plane-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Periodicity_and_asymptotes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Periodicity_and_asymptotes"> <div class="vector-toc-text"> 7 Periodicity and asymptotes </div> </a> <ul id="toc-Periodicity_and_asymptotes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Basic_identities" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Basic_identities"> <div class="vector-toc-text"> 8 Basic identities </div> </a> <button aria-controls="toc-Basic_identities-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Basic identities subsection </button> <ul id="toc-Basic_identities-sublist" class="vector-toc-list"> <li id="toc-Parity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Parity"> <div class="vector-toc-text"> 8.1 Parity </div> </a> <ul id="toc-Parity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Periods" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Periods"> <div class="vector-toc-text"> 8.2 Periods </div> </a> <ul id="toc-Periods-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Pythagorean_identity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pythagorean_identity"> <div class="vector-toc-text"> 8.3 Pythagorean identity </div> </a> <ul id="toc-Pythagorean_identity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sum_and_difference_formulas" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sum_and_difference_formulas"> <div class="vector-toc-text"> 8.4 Sum and difference formulas </div> </a> <ul id="toc-Sum_and_difference_formulas-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Derivatives_and_antiderivatives" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Derivatives_and_antiderivatives"> <div class="vector-toc-text"> 8.5 Derivatives and antiderivatives </div> </a> <ul id="toc-Derivatives_and_antiderivatives-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Inverse_functions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Inverse_functions"> <div class="vector-toc-text"> 9 Inverse functions </div> </a> <ul id="toc-Inverse_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> 10 Applications </div> </a> <button aria-controls="toc-Applications-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Applications subsection </button> <ul id="toc-Applications-sublist" class="vector-toc-list"> <li id="toc-Angles_and_sides_of_a_triangle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Angles_and_sides_of_a_triangle"> <div class="vector-toc-text"> 10.1 Angles and sides of a triangle </div> </a> <ul id="toc-Angles_and_sides_of_a_triangle-sublist" class="vector-toc-list"> <li id="toc-Law_of_sines" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Law_of_sines"> <div class="vector-toc-text"> 10.1.1 Law of sines </div> </a> <ul id="toc-Law_of_sines-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Law_of_cosines" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Law_of_cosines"> <div class="vector-toc-text"> 10.1.2 Law of cosines </div> </a> <ul id="toc-Law_of_cosines-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Law_of_tangents" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Law_of_tangents"> <div class="vector-toc-text"> 10.1.3 Law of tangents </div> </a> <ul id="toc-Law_of_tangents-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Law_of_cotangents" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Law_of_cotangents"> <div class="vector-toc-text"> 10.1.4 Law of cotangents </div> </a> <ul id="toc-Law_of_cotangents-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Periodic_functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Periodic_functions"> <div class="vector-toc-text"> 10.2 Periodic functions </div> </a> <ul id="toc-Periodic_functions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> 11 History </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Etymology" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Etymology"> <div class="vector-toc-text"> 12 Etymology </div> </a> <ul id="toc-Etymology-sublist" class="vector-toc-list"> </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"> 13 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 14 Notes </div> </a> <ul id="toc-Notes-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"> 15 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </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"> 16 External links </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" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">Trigonometric functions</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 75 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-75" aria-hidden="true" > 75 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-smn mw-list-item"><a href="https://smn.wikipedia.org/wiki/Trigonometris%C3%A2%C5%A1_funktio" title="Trigonometrisâš funktio – Inari Sami" lang="smn" hreflang="smn" data-title="Trigonometrisâš funktio" data-language-autonym="Anarâškielâ" data-language-local-name="Inari Sami" class="interlanguage-link-target">Anarâškielâ</a></li><li class="interlanguage-link interwiki-ar badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://ar.wikipedia.org/wiki/%D8%AF%D9%88%D8%A7%D9%84_%D9%85%D8%AB%D9%84%D8%AB%D9%8A%D8%A9" title="دوال مثلثية – Arabic" lang="ar" hreflang="ar" data-title="دوال مثلثية" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Funci%C3%B3n_trigonom%C3%A9trica" title="Función trigonométrica – Asturian" lang="ast" hreflang="ast" data-title="Función trigonométrica" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target">Asturianu</a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Triqonometrik_funksiyalar" title="Triqonometrik funksiyalar – Azerbaijani" lang="az" hreflang="az" data-title="Triqonometrik funksiyalar" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target">Azərbaycanca</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%A4%E0%A7%8D%E0%A6%B0%E0%A6%BF%E0%A6%95%E0%A7%8B%E0%A6%A3%E0%A6%AE%E0%A6%BF%E0%A6%A4%E0%A6%BF%E0%A6%95_%E0%A6%85%E0%A6%AA%E0%A7%87%E0%A6%95%E0%A7%8D%E0%A6%B7%E0%A6%95" title="ত্রিকোণমিতিক অপেক্ষক – Bangla" lang="bn" hreflang="bn" data-title="ত্রিকোণমিতিক অপেক্ষক" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Sa%E2%81%BF-kak_h%C3%A2m-s%C3%B2%CD%98" title="Saⁿ-kak hâm-sò͘ – Minnan" lang="nan" hreflang="nan" data-title="Saⁿ-kak hâm-sò͘" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target">閩南語 / Bân-lâm-gú</a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%A2%D1%80%D0%B8%D0%B3%D0%BE%D0%BD%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D0%BA_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F%D0%BB%D0%B0%D1%80" title="Тригонометрик функциялар – Bashkir" lang="ba" hreflang="ba" data-title="Тригонометрик функциялар" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target">Башҡортса</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A2%D1%80%D1%8B%D0%B3%D0%B0%D0%BD%D0%B0%D0%BC%D0%B5%D1%82%D1%80%D1%8B%D1%87%D0%BD%D1%8B%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D1%8B%D1%96" title="Трыганаметрычныя функцыі – Belarusian" lang="be" hreflang="be" data-title="Трыганаметрычныя функцыі" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A2%D1%80%D0%B8%D0%B3%D0%BE%D0%BD%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%87%D0%BD%D0%B0_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Тригонометрична функция – Bulgarian" lang="bg" hreflang="bg" data-title="Тригонометрична функция" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Trigonometrijska_funkcija" title="Trigonometrijska funkcija – Bosnian" lang="bs" hreflang="bs" data-title="Trigonometrijska funkcija" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target">Bosanski</a></li><li class="interlanguage-link interwiki-ca badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://ca.wikipedia.org/wiki/Funci%C3%B3_trigonom%C3%A8trica" title="Funció trigonomètrica – Catalan" lang="ca" hreflang="ca" data-title="Funció trigonomètrica" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A2%D1%80%D0%B8%D0%B3%D0%BE%D0%BD%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D0%BB%D0%BB%D0%B5_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%81%D0%B5%D0%BC" title="Тригонометрилле функцисем – Chuvash" lang="cv" hreflang="cv" data-title="Тригонометрилле функцисем" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target">Чӑвашла</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Goniometrick%C3%A1_funkce" title="Goniometrická funkce – Czech" lang="cs" hreflang="cs" data-title="Goniometrická funkce" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Ffwythiannau_trigonometrig" title="Ffwythiannau trigonometrig – Welsh" lang="cy" hreflang="cy" data-title="Ffwythiannau trigonometrig" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target">Cymraeg</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Trigonometrisk_funktion" title="Trigonometrisk funktion – Danish" lang="da" hreflang="da" data-title="Trigonometrisk funktion" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Trigonometrische_Funktion" title="Trigonometrische Funktion – German" lang="de" hreflang="de" data-title="Trigonometrische Funktion" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Trigonomeetrilised_funktsioonid" title="Trigonomeetrilised funktsioonid – Estonian" lang="et" hreflang="et" data-title="Trigonomeetrilised funktsioonid" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%B9%CE%BA%CE%AE_%CF%83%CF%85%CE%BD%CE%AC%CF%81%CF%84%CE%B7%CF%83%CE%B7" title="Τριγωνομετρική συνάρτηση – Greek" lang="el" hreflang="el" data-title="Τριγωνομετρική συνάρτηση" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Funci%C3%B3n_trigonom%C3%A9trica" title="Función trigonométrica – Spanish" lang="es" hreflang="es" data-title="Función trigonométrica" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Trigonometria_funkcio" title="Trigonometria funkcio – Esperanto" lang="eo" hreflang="eo" data-title="Trigonometria funkcio" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Funtzio_trigonometriko" title="Funtzio trigonometriko – Basque" lang="eu" hreflang="eu" data-title="Funtzio trigonometriko" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D9%88%D8%A7%D8%A8%D8%B9_%D9%85%D8%AB%D9%84%D8%AB%D8%A7%D8%AA%DB%8C" title="توابع مثلثاتی – Persian" lang="fa" hreflang="fa" data-title="توابع مثلثاتی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Fonction_trigonom%C3%A9trique" title="Fonction trigonométrique – French" lang="fr" hreflang="fr" data-title="Fonction trigonométrique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Funci%C3%B3n_trigonom%C3%A9trica" title="Función trigonométrica – Galician" lang="gl" hreflang="gl" data-title="Función trigonométrica" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%82%BC%EA%B0%81_%ED%95%A8%EC%88%98" title="삼각 함수 – Korean" lang="ko" hreflang="ko" data-title="삼각 함수" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B5%D5%BC%D5%A1%D5%B6%D5%AF%D5%B5%D5%B8%D6%82%D5%B6%D5%A1%D5%B9%D5%A1%D6%83%D5%A1%D5%AF%D5%A1%D5%B6_%D6%86%D5%B8%D6%82%D5%B6%D5%AF%D6%81%D5%AB%D5%A1%D5%B6%D5%A5%D6%80" title="Եռանկյունաչափական ֆունկցիաներ – Armenian" lang="hy" hreflang="hy" data-title="Եռանկյունաչափական ֆունկցիաներ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%95%E0%A5%8B%E0%A4%A3%E0%A4%AE%E0%A4%BF%E0%A4%A4%E0%A5%80%E0%A4%AF_%E0%A4%AB%E0%A4%B2%E0%A4%A8" title="त्रिकोणमितीय फलन – Hindi" lang="hi" hreflang="hi" data-title="त्रिकोणमितीय फलन" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Trigonometrijske_funkcije" title="Trigonometrijske funkcije – Croatian" lang="hr" hreflang="hr" data-title="Trigonometrijske funkcije" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Trigonometriala_funciono" title="Trigonometriala funciono – Ido" lang="io" hreflang="io" data-title="Trigonometriala funciono" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target">Ido</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Fungsi_trigonometri" title="Fungsi trigonometri – Indonesian" lang="id" hreflang="id" data-title="Fungsi trigonometri" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Hornafall" title="Hornafall – Icelandic" lang="is" hreflang="is" data-title="Hornafall" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target">Íslenska</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Funzione_trigonometrica" title="Funzione trigonometrica – Italian" lang="it" hreflang="it" data-title="Funzione trigonometrica" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%95%D7%AA_%D7%98%D7%A8%D7%99%D7%92%D7%95%D7%A0%D7%95%D7%9E%D7%98%D7%A8%D7%99%D7%95%D7%AA" title="פונקציות טריגונומטריות – Hebrew" lang="he" hreflang="he" data-title="פונקציות טריגונומטריות" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%A2%E1%83%A0%E1%83%98%E1%83%92%E1%83%9D%E1%83%9C%E1%83%9D%E1%83%9B%E1%83%94%E1%83%A2%E1%83%A0%E1%83%98%E1%83%A3%E1%83%9A%E1%83%98_%E1%83%A4%E1%83%A3%E1%83%9C%E1%83%A5%E1%83%AA%E1%83%98%E1%83%94%E1%83%91%E1%83%98" title="ტრიგონომეტრიული ფუნქციები – Georgian" lang="ka" hreflang="ka" data-title="ტრიგონომეტრიული ფუნქციები" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target">ქართული</a></li><li class="interlanguage-link interwiki-lo mw-list-item"><a href="https://lo.wikipedia.org/wiki/%E0%BA%95%E0%BA%B3%E0%BA%A5%E0%BA%B2%E0%BB%84%E0%BA%95%E0%BA%A1%E0%BA%B8%E0%BA%A1" title="ຕຳລາໄຕມຸມ – Lao" lang="lo" hreflang="lo" data-title="ຕຳລາໄຕມຸມ" data-language-autonym="ລາວ" data-language-local-name="Lao" class="interlanguage-link-target">ລາວ</a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Functiones_trigonometricae" title="Functiones trigonometricae – Latin" lang="la" hreflang="la" data-title="Functiones trigonometricae" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target">Latina</a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Trigonometrisk%C4%81s_funkcijas" title="Trigonometriskās funkcijas – Latvian" lang="lv" hreflang="lv" data-title="Trigonometriskās funkcijas" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target">Latviešu</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Trigonometrin%C4%97s_funkcijos" title="Trigonometrinės funkcijos – Lithuanian" lang="lt" hreflang="lt" data-title="Trigonometrinės funkcijos" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Funsionas_trigonometrial" title="Funsionas trigonometrial – Lingua Franca Nova" lang="lfn" hreflang="lfn" data-title="Funsionas trigonometrial" data-language-autonym="Lingua Franca Nova" data-language-local-name="Lingua Franca Nova" class="interlanguage-link-target">Lingua Franca Nova</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Sz%C3%B6gf%C3%BCggv%C3%A9nyek" title="Szögfüggvények – Hungarian" lang="hu" hreflang="hu" data-title="Szögfüggvények" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A2%D1%80%D0%B8%D0%B3%D0%BE%D0%BD%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%81%D0%BA%D0%B8_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D0%B8" title="Тригонометриски функции – Macedonian" lang="mk" hreflang="mk" data-title="Тригонометриски функции" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target">Македонски</a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%B8%E0%A5%8D%E0%A4%AA%E0%A4%B0%E0%A5%8D%E0%A4%B6_(%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%95%E0%A5%8B%E0%A4%A3%E0%A4%AE%E0%A4%BF%E0%A4%A4%E0%A5%80%E0%A4%AF_%E0%A4%AB%E0%A4%B2)" title="स्पर्श (त्रिकोणमितीय फल) – Marathi" lang="mr" hreflang="mr" data-title="स्पर्श (त्रिकोणमितीय फल)" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target">मराठी</a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Fungsi_trigonometri" title="Fungsi trigonometri – Malay" lang="ms" hreflang="ms" data-title="Fungsi trigonometri" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target">Bahasa Melayu</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Goniometrische_functie" title="Goniometrische functie – Dutch" lang="nl" hreflang="nl" data-title="Goniometrische functie" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E4%B8%89%E8%A7%92%E9%96%A2%E6%95%B0" title="三角関数 – Japanese" lang="ja" hreflang="ja" data-title="三角関数" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Trigonometrisk_funksjon" title="Trigonometrisk funksjon – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Trigonometrisk funksjon" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Trigonometrisk_funksjon" title="Trigonometrisk funksjon – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Trigonometrisk funksjon" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Foncions_trigonometricas" title="Foncions trigonometricas – Occitan" lang="oc" hreflang="oc" data-title="Foncions trigonometricas" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target">Occitan</a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Trigonometrik_funksiyalar" title="Trigonometrik funksiyalar – Uzbek" lang="uz" hreflang="uz" data-title="Trigonometrik funksiyalar" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target">Oʻzbekcha / ўзбекча</a></li><li class="interlanguage-link interwiki-km badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://km.wikipedia.org/wiki/%E1%9E%A2%E1%9E%93%E1%9E%BB%E1%9E%82%E1%9E%98%E1%9E%93%E1%9F%8D%E1%9E%8F%E1%9F%92%E1%9E%9A%E1%9E%B8%E1%9E%80%E1%9F%84%E1%9E%8E%E1%9E%98%E1%9E%B6%E1%9E%8F%E1%9F%92%E1%9E%9A" title="អនុគមន៍ត្រីកោណមាត្រ – Khmer" lang="km" hreflang="km" data-title="អនុគមន៍ត្រីកោណមាត្រ" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="Khmer" class="interlanguage-link-target">ភាសាខ្មែរ</a></li><li class="interlanguage-link interwiki-pl badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://pl.wikipedia.org/wiki/Funkcje_trygonometryczne" title="Funkcje trygonometryczne – Polish" lang="pl" hreflang="pl" data-title="Funkcje trygonometryczne" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Fun%C3%A7%C3%A3o_trigonom%C3%A9trica" title="Função trigonométrica – Portuguese" lang="pt" hreflang="pt" data-title="Função trigonométrica" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Func%C8%9Bie_trigonometric%C4%83" title="Funcție trigonometrică – Romanian" lang="ro" hreflang="ro" data-title="Funcție trigonometrică" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A2%D1%80%D0%B8%D0%B3%D0%BE%D0%BD%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B8%D0%B5_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D0%B8" title="Тригонометрические функции – Russian" lang="ru" hreflang="ru" data-title="Тригонометрические функции" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Funksionet_trigonometrike" title="Funksionet trigonometrike – Albanian" lang="sq" hreflang="sq" data-title="Funksionet trigonometrike" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target">Shqip</a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%AD%E0%B7%8A%E2%80%8D%E0%B6%BB%E0%B7%92%E0%B6%9A%E0%B7%9D%E0%B6%AB%E0%B6%B8%E0%B7%92%E0%B6%AD%E0%B7%92%E0%B6%9A_%E0%B7%81%E0%B7%8A%E2%80%8D%E0%B6%BB%E0%B7%92%E0%B6%AD" title="ත්‍රිකෝණමිතික ශ්‍රිත – Sinhala" lang="si" hreflang="si" data-title="ත්‍රිකෝණමිතික ශ්‍රිත" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target">සිංහල</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Trigonometric_function" title="Trigonometric function – Simple English" lang="en-simple" hreflang="en-simple" data-title="Trigonometric function" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Goniometrick%C3%A1_funkcia" title="Goniometrická funkcia – Slovak" lang="sk" hreflang="sk" data-title="Goniometrická funkcia" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Trigonometri%C4%8Dna_funkcija" title="Trigonometrična funkcija – Slovenian" lang="sl" hreflang="sl" data-title="Trigonometrična funkcija" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%81%D8%A7%D9%86%DA%A9%D8%B4%D9%86%DB%95_%D8%B3%DB%8E%DA%AF%DB%86%D8%B4%DB%95%DB%8C%DB%8C%DB%8C%DB%95%DA%A9%D8%A7%D9%86" title="فانکشنە سێگۆشەیییەکان – Central Kurdish" lang="ckb" hreflang="ckb" data-title="فانکشنە سێگۆشەیییەکان" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target">کوردی</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A2%D1%80%D0%B8%D0%B3%D0%BE%D0%BD%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%98%D1%81%D0%BA%D0%B5_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%98%D0%B5" title="Тригонометријске функције – Serbian" lang="sr" hreflang="sr" data-title="Тригонометријске функције" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Trigonometrijske_funkcije" title="Trigonometrijske funkcije – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Trigonometrijske funkcije" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Trigonometrinen_funktio" title="Trigonometrinen funktio – Finnish" lang="fi" hreflang="fi" data-title="Trigonometrinen funktio" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Trigonometrisk_funktion" title="Trigonometrisk funktion – Swedish" lang="sv" hreflang="sv" data-title="Trigonometrisk funktion" data-language-autonym="Svenska" 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For the Clausen-related function, see <a href="/wiki/Log_sine_function" class="mw-redirect" title="Log sine function">log sine function</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">"Logarithmic cosine" redirects here. For the Clausen-related function, see <a href="/wiki/Log_cosine_function" class="mw-redirect" title="Log cosine function">log cosine function</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist 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screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar nomobile nowraplinks"><tbody><tr><th class="sidebar-title" style="background:#ccccff;"><a href="/wiki/Trigonometry" title="Trigonometry">Trigonometry</a></th></tr><tr><td class="sidebar-image"><a href="/wiki/File:Sinus_und_Kosinus_am_Einheitskreis_1.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Sinus_und_Kosinus_am_Einheitskreis_1.svg/250px-Sinus_und_Kosinus_am_Einheitskreis_1.svg.png" decoding="async" width="250" height="250" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Sinus_und_Kosinus_am_Einheitskreis_1.svg/375px-Sinus_und_Kosinus_am_Einheitskreis_1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/72/Sinus_und_Kosinus_am_Einheitskreis_1.svg/500px-Sinus_und_Kosinus_am_Einheitskreis_1.svg.png 2x" data-file-width="410" data-file-height="410" /></a></td></tr><tr><td class="sidebar-content hlist" style="padding-top:0.2em;padding-bottom:0.7em;"> <ul><li><a href="/wiki/Outline_of_trigonometry" title="Outline of trigonometry">Outline</a></li> <li><a href="/wiki/History_of_trigonometry" title="History of trigonometry">History</a></li> <li><a href="/wiki/Uses_of_trigonometry" title="Uses of trigonometry">Usage</a></li></ul> <ul><li><a class="mw-selflink selflink">Functions</a> (<a href="/wiki/Sine_and_cosine" title="Sine and cosine">sin</a>, <a href="/wiki/Sine_and_cosine" title="Sine and cosine">cos</a>, <a class="mw-selflink-fragment" href="#tangent">tan</a>, <a href="/wiki/Inverse_trigonometric_functions" title="Inverse trigonometric functions">inverse</a>)</li> <li><a href="/wiki/Generalized_trigonometry" title="Generalized trigonometry">Generalized trigonometry</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#ddddff;"> Reference</th></tr><tr><td class="sidebar-content hlist" style="padding-top:0.2em;padding-bottom:0.7em;"> <ul><li><a href="/wiki/List_of_trigonometric_identities" title="List of trigonometric identities">Identities</a></li> <li><a href="/wiki/Exact_trigonometric_values" title="Exact trigonometric values">Exact constants</a></li> <li><a href="/wiki/Trigonometric_tables" title="Trigonometric tables">Tables</a></li> <li><a href="/wiki/Unit_circle" title="Unit circle">Unit circle</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#ddddff;"> Laws and theorems</th></tr><tr><td class="sidebar-content hlist" style="padding-top:0.2em;padding-bottom:0.7em;"> <ul><li><a href="/wiki/Law_of_sines" title="Law of sines">Sines</a></li> <li><a href="/wiki/Law_of_cosines" title="Law of cosines">Cosines</a></li> <li><a href="/wiki/Law_of_tangents" title="Law of tangents">Tangents</a></li> <li><a href="/wiki/Law_of_cotangents" title="Law of cotangents">Cotangents</a></li></ul> <ul><li><a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#ddddff;"> <a href="/wiki/Calculus" title="Calculus">Calculus</a></th></tr><tr><td class="sidebar-content hlist" style="padding-top:0.2em;padding-bottom:0.7em;"> <ul><li><a href="/wiki/Trigonometric_substitution" title="Trigonometric substitution">Trigonometric substitution</a></li> <li><a href="/wiki/List_of_integrals_of_trigonometric_functions" title="List of integrals of trigonometric functions">Integrals</a> (<a href="/wiki/List_of_integrals_of_inverse_trigonometric_functions" title="List of integrals of inverse trigonometric functions">inverse functions</a>)</li> <li><a href="/wiki/Differentiation_of_trigonometric_functions" title="Differentiation of trigonometric functions">Derivatives</a></li> <li><a href="/wiki/Trigonometric_series" title="Trigonometric series">Trigonometric series</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#ddddff;"> Mathematicians</th></tr><tr><td class="sidebar-content hlist" style="padding-top:0.2em;padding-bottom:0.7em;"> <ul><li><a href="/wiki/Hipparchus" title="Hipparchus">Hipparchus</a></li> <li><a href="/wiki/Ptolemy" title="Ptolemy">Ptolemy</a></li> <li><a href="/wiki/Brahmagupta" title="Brahmagupta">Brahmagupta</a></li> <li><a href="/wiki/Habash_al-Hasib_al-Marwazi" class="mw-redirect" title="Habash al-Hasib al-Marwazi">al-Hasib</a></li> <li><a href="/wiki/Al-Battani" title="Al-Battani">al-Battani</a></li> <li><a href="/wiki/Regiomontanus" title="Regiomontanus">Regiomontanus</a></li> <li><a href="/wiki/Fran%C3%A7ois_Vi%C3%A8te" title="François Viète">Viète</a></li> <li><a href="/wiki/Abraham_de_Moivre" title="Abraham de Moivre">de Moivre</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euler</a></li> <li><a href="/wiki/Joseph_Fourier" title="Joseph Fourier">Fourier</a></li></ul></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Trigonometry" title="Template:Trigonometry"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Trigonometry" title="Template talk:Trigonometry"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Trigonometry" title="Special:EditPage/Template:Trigonometry"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Academ_Base_of_trigonometry.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/Academ_Base_of_trigonometry.svg/300px-Academ_Base_of_trigonometry.svg.png" decoding="async" width="300" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/Academ_Base_of_trigonometry.svg/450px-Academ_Base_of_trigonometry.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/11/Academ_Base_of_trigonometry.svg/600px-Academ_Base_of_trigonometry.svg.png 2x" data-file-width="600" data-file-height="600" /></a><figcaption>Basis of trigonometry: if two <a href="/wiki/Right_triangle" title="Right triangle">right triangles</a> have equal <a href="/wiki/Acute_angle" class="mw-redirect" title="Acute angle">acute angles</a>, they are <a href="/wiki/Similarity_(geometry)" title="Similarity (geometry)">similar</a>, so their corresponding side lengths are <a href="/wiki/Proportionality_(mathematics)" title="Proportionality (mathematics)">proportional</a>.</figcaption></figure> In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, the trigonometric functions (also called circular functions, angle functions or goniometric functions)<a href="#cite_note-klein-1">[1]</a> are <a href="/wiki/Real_function" class="mw-redirect" title="Real function">real functions</a> which relate an angle of a <a href="/wiki/Right-angled_triangle" class="mw-redirect" title="Right-angled triangle">right-angled triangle</a> to ratios of two side lengths. They are widely used in all sciences that are related to <a href="/wiki/Geometry" title="Geometry">geometry</a>, such as <a href="/wiki/Navigation" title="Navigation">navigation</a>, <a href="/wiki/Solid_mechanics" title="Solid mechanics">solid mechanics</a>, <a href="/wiki/Celestial_mechanics" title="Celestial mechanics">celestial mechanics</a>, <a href="/wiki/Geodesy" title="Geodesy">geodesy</a>, and many others. They are among the simplest <a href="/wiki/Periodic_function" title="Periodic function">periodic functions</a>, and as such are also widely used for studying periodic phenomena through <a href="/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a>. The trigonometric functions most widely used in modern mathematics are the <a href="/wiki/Sine" class="mw-redirect" title="Sine">sine</a>, the <a href="/wiki/Cosine" class="mw-redirect" title="Cosine">cosine</a>, and the tangent functions. Their <a href="/wiki/Multiplicative_inverse" title="Multiplicative inverse">reciprocals</a> are respectively the cosecant, the secant, and the cotangent functions, which are less used. Each of these six trigonometric functions has a corresponding <a href="/wiki/Inverse_trigonometric_functions" title="Inverse trigonometric functions">inverse function</a>, and an analog among the <a href="/wiki/Hyperbolic_functions" title="Hyperbolic functions">hyperbolic functions</a>. The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for <a href="/wiki/Acute_angle" class="mw-redirect" title="Acute angle">acute angles</a>. To extend the sine and cosine functions to functions whose <a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a> is the whole <a href="/wiki/Real_line" class="mw-redirect" title="Real line">real line</a>, geometrical definitions using the standard <a href="/wiki/Unit_circle" title="Unit circle">unit circle</a> (i.e., a circle with <a href="/wiki/Radius" title="Radius">radius</a> 1 unit) are often used; then the domain of the other functions is the real line with some isolated points removed. Modern definitions express trigonometric functions as <a href="/wiki/Series_(mathematics)" title="Series (mathematics)">infinite series</a> or as solutions of <a href="/wiki/Differential_equation" title="Differential equation">differential equations</a>. This allows extending the domain of sine and cosine functions to the whole <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a>, and the domain of the other trigonometric functions to the complex plane with some isolated points removed. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Notation">Notation</h2>[<a href="/w/index.php?title=Trigonometric_functions&action=edit&section=1" title="Edit section: Notation">edit</a>]</div> Conventionally, an abbreviation of each trigonometric function's name is used as its symbol in formulas. Today, the most common versions of these abbreviations are "sin" for sine, "cos" for cosine, "tan" or "tg" for tangent, "sec" for secant, "csc" or "cosec" for cosecant, and "cot" or "ctg" for cotangent. Historically, these abbreviations were first used in prose sentences to indicate particular <a href="/wiki/Line_segment" title="Line segment">line segments</a> or their lengths related to an <a href="/wiki/Circular_arc" title="Circular arc">arc</a> of an arbitrary circle, and later to indicate ratios of lengths, but as the <a href="/wiki/History_of_the_function_concept" title="History of the function concept">function concept developed</a> in the 17th–18th century, they began to be considered as functions of real-number-valued angle measures, and written with <a href="/wiki/Functional_notation" class="mw-redirect" title="Functional notation">functional notation</a>, for example sin(x). Parentheses are still often omitted to reduce clutter, but are sometimes necessary; for example the expression <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin x+y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin x+y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d73f903d656f0ab08059c51c063a3fc9a8016d71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.568ex; height:2.509ex;" alt="{\displaystyle \sin x+y}"> would typically be interpreted to mean <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(x)+y,}"> <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>y</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(x)+y,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a66b9aa22b7cfd52b3183027cfcb81ff63cbf2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.637ex; height:2.843ex;" alt="{\displaystyle \sin(x)+y,}"> so parentheses are required to express <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(x+y).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(x+y).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37e1b42175ec4a35aa84f2698e1adcab77ecca92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.637ex; height:2.843ex;" alt="{\displaystyle \sin(x+y).}"> A <a href="/wiki/Positive_integer" class="mw-redirect" title="Positive integer">positive integer</a> appearing as a superscript after the symbol of the function denotes <a href="/wiki/Exponentiation" title="Exponentiation">exponentiation</a>, not <a href="/wiki/Function_composition#Functional_powers" title="Function composition">function composition</a>. For example <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> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin ^{2}x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc0d6ba6bb181219b776ab25be991303f9e07d0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.627ex; height:2.676ex;" alt="{\displaystyle \sin ^{2}x}"> and <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><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)}"> denote <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> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(x)\cdot \sin(x),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5fc58e1d95d9b8382d1aa0d0ac3802a38aad8aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.315ex; height:2.843ex;" alt="{\displaystyle \sin(x)\cdot \sin(x),}"> not <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(\sin x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(\sin x).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfecbaeabb9ef965e58a0f96e8a17034ba247bd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.884ex; height:2.843ex;" alt="{\displaystyle \sin(\sin x).}"> This differs from the (historically later) general functional notation in which <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{2}(x)=(f\circ f)(x)=f(f(x)).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∘</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{2}(x)=(f\circ f)(x)=f(f(x)).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51b0fc08fd43fa5b7b1155839cfe92b83db7b547" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.563ex; height:3.176ex;" alt="{\displaystyle f^{2}(x)=(f\circ f)(x)=f(f(x)).}"> However, the exponent <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76a7a8e06f9f953ac8acb92698ce7ff6fa523bd7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle {-1}}"> is commonly used to denote the <a href="/wiki/Inverse_function" title="Inverse function">inverse function</a>, not the <a href="/wiki/Multiplicative_inverse" title="Multiplicative inverse">reciprocal</a>. For example <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin ^{-1}x}"> <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> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin ^{-1}x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43621a8bcd117b4e136e8c0b5d8a73f0bf3609a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.905ex; height:2.676ex;" alt="{\displaystyle \sin ^{-1}x}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin ^{-1}(x)}"> <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> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin ^{-1}(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5e27a0364b1563514c7ad91156162cda561dec3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.327ex; height:3.176ex;" alt="{\displaystyle \sin ^{-1}(x)}"> denote the <a href="/wiki/Inverse_trigonometric_function" class="mw-redirect" title="Inverse trigonometric function">inverse trigonometric function</a> alternatively written <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arcsin x\colon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arcsin</mi> <mo>⁡</mo> <mi>x</mi> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arcsin x\colon }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c54c791ff52b5762a30b04a06c96056928eb4c93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.326ex; height:2.176ex;" alt="{\displaystyle \arcsin x\colon }"> The equation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =\sin ^{-1}x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =\sin ^{-1}x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a2e14c13b4704c89bddd12ba7b8e85b0cf6c74c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.094ex; height:2.676ex;" alt="{\displaystyle \theta =\sin ^{-1}x}"> implies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \theta =x,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \theta =x,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1744aa589740474e82c26d514a0bc92be7dacf57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.408ex; height:2.509ex;" alt="{\displaystyle \sin \theta =x,}"> not <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta \cdot \sin x=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta \cdot \sin x=1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82cf2363cad1f47fa25fac58d0a3b0d358e91370" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.25ex; height:2.176ex;" alt="{\displaystyle \theta \cdot \sin x=1.}"> In this case, the superscript could be considered as denoting a composed or <a href="/wiki/Iterated_function" title="Iterated function">iterated function</a>, but negative superscripts other than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76a7a8e06f9f953ac8acb92698ce7ff6fa523bd7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle {-1}}"> are not in common use. <div class="mw-heading mw-heading2"><h2 id="Right-angled_triangle_definitions">Right-angled triangle definitions</h2>[<a href="/w/index.php?title=Trigonometric_functions&action=edit&section=2" title="Edit section: Right-angled triangle definitions">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:TrigonometryTriangle.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/TrigonometryTriangle.svg/220px-TrigonometryTriangle.svg.png" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/TrigonometryTriangle.svg/330px-TrigonometryTriangle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4f/TrigonometryTriangle.svg/440px-TrigonometryTriangle.svg.png 2x" data-file-width="400" data-file-height="300" /></a><figcaption>In this right triangle, denoting the measure of angle BAC as A: sin A = <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>⁠a/c⁠; cos A = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠b/c⁠; tan A = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠a/b⁠.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:TrigFunctionDiagram.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/TrigFunctionDiagram.svg/220px-TrigFunctionDiagram.svg.png" decoding="async" width="220" height="224" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/TrigFunctionDiagram.svg/330px-TrigFunctionDiagram.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/TrigFunctionDiagram.svg/440px-TrigFunctionDiagram.svg.png 2x" data-file-width="644" data-file-height="655" /></a><figcaption>Plot of the six trigonometric functions, the unit circle, and a line for the angle θ = 0.7 radians. The points labeled <style data-mw-deduplicate="TemplateStyles:r1239334494">@media screen{html.skin-theme-clientpref-night .mw-parser-output div:not(.notheme)>.tmp-color,html.skin-theme-clientpref-night .mw-parser-output p>.tmp-color,html.skin-theme-clientpref-night .mw-parser-output table:not(.notheme) .tmp-color{color:inherit!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output div:not(.notheme)>.tmp-color,html.skin-theme-clientpref-os .mw-parser-output p>.tmp-color,html.skin-theme-clientpref-os .mw-parser-output table:not(.notheme) .tmp-color{color:inherit!important}}</style>1, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">Sec(θ), <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">Csc(θ) represent the length of the line segment from the origin to that point. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">Sin(θ), <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">Tan(θ), and <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">1 are the heights to the line starting from the x-axis, while <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">Cos(θ), <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">1, and <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">Cot(θ) are lengths along the x-axis starting from the origin.</figcaption></figure> If the acute angle θ is given, then any right triangles that have an angle of θ are <a href="/wiki/Similarity_(geometry)" title="Similarity (geometry)">similar</a> to each other. This means that the ratio of any two side lengths depends only on θ. Thus these six ratios define six functions of θ, which are the trigonometric functions. In the following definitions, the <a href="/wiki/Hypotenuse" title="Hypotenuse">hypotenuse</a> is the length of the side opposite the right angle, opposite represents the side opposite the given angle θ, and adjacent represents the side between the angle θ and the right angle.<a href="#cite_note-2">[2]</a><a href="#cite_note-3">[3]</a> <table> <tbody><tr> <td style="padding-left: 2em; padding-right: 2em;"> <dl><dt>sine</dt> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \theta ={\frac {\mathrm {opposite} }{\mathrm {hypotenuse} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">h</mi> <mi mathvariant="normal">y</mi> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">e</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \theta ={\frac {\mathrm {opposite} }{\mathrm {hypotenuse} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e84fb0952ea5af6073c22a9709c8a2f88300542f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.713ex; height:5.843ex;" alt="{\displaystyle \sin \theta ={\frac {\mathrm {opposite} }{\mathrm {hypotenuse} }}}"></dd></dl> </td> <td style="padding-left: 2em; padding-right: 2em;"> <dl><dt>cosecant</dt> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \csc \theta ={\frac {\mathrm {hypotenuse} }{\mathrm {opposite} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>csc</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">h</mi> <mi mathvariant="normal">y</mi> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">e</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \csc \theta ={\frac {\mathrm {hypotenuse} }{\mathrm {opposite} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d566c8830461f28f09cfebb3aedd23697105c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.839ex; height:5.843ex;" alt="{\displaystyle \csc \theta ={\frac {\mathrm {hypotenuse} }{\mathrm {opposite} }}}"></dd></dl> </td></tr> <tr> <td style="padding-left: 2em; padding-right: 2em;"> <dl><dt>cosine</dt> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \theta ={\frac {\mathrm {adjacent} }{\mathrm {hypotenuse} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">j</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">h</mi> <mi mathvariant="normal">y</mi> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">e</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \theta ={\frac {\mathrm {adjacent} }{\mathrm {hypotenuse} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51de249e60290d55032eb1d31489ac350218f32f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.969ex; height:5.843ex;" alt="{\displaystyle \cos \theta ={\frac {\mathrm {adjacent} }{\mathrm {hypotenuse} }}}"></dd></dl> </td> <td style="padding-left: 2em; padding-right: 2em;"> <dl><dt>secant</dt> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sec \theta ={\frac {\mathrm {hypotenuse} }{\mathrm {adjacent} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sec</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">h</mi> <mi mathvariant="normal">y</mi> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">j</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sec \theta ={\frac {\mathrm {hypotenuse} }{\mathrm {adjacent} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6453a2306a2ad7934de638ac33bb07e86e7084a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.839ex; height:5.843ex;" alt="{\displaystyle \sec \theta ={\frac {\mathrm {hypotenuse} }{\mathrm {adjacent} }}}"></dd></dl> </td></tr> <tr> <td style="padding-left: 2em; padding-right: 2em;"> <dl><dt>tangent</dt> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan \theta ={\frac {\mathrm {opposite} }{\mathrm {adjacent} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">j</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan \theta ={\frac {\mathrm {opposite} }{\mathrm {adjacent} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6498813e920a97efef60c16baa75488fb2af3f07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.363ex; height:5.843ex;" alt="{\displaystyle \tan \theta ={\frac {\mathrm {opposite} }{\mathrm {adjacent} }}}"></dd></dl> </td> <td style="padding-left: 2em; padding-right: 2em;"> <dl><dt>cotangent</dt> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cot \theta ={\frac {\mathrm {adjacent} }{\mathrm {opposite} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cot</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">j</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">e</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cot \theta ={\frac {\mathrm {adjacent} }{\mathrm {opposite} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d3b0f28fb38886e66df1e89300a96500592754f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.103ex; height:5.843ex;" alt="{\displaystyle \cot \theta ={\frac {\mathrm {adjacent} }{\mathrm {opposite} }}}"></dd></dl> </td></tr></tbody></table> <a href="/wiki/Mnemonics_in_trigonometry" title="Mnemonics in trigonometry">Various mnemonics</a> can be used to remember these definitions. In a right-angled triangle, the sum of the two acute angles is a right angle, that is, 90° or <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠π/2⁠ <a href="/wiki/Radian" title="Radian">radians</a>. Therefore <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><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 )}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(90^{\circ }-\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>−</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(90^{\circ }-\theta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fb1d42d3e8d7d46582aef0da29a76e8801c6f7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.23ex; height:2.843ex;" alt="{\displaystyle \cos(90^{\circ }-\theta )}"> represent the same ratio, and thus are equal. This identity and analogous relationships between the other trigonometric functions are summarized in the following table. <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>Top: Trigonometric function sin θ for selected angles θ, π − θ, π + θ, and 2π − θ in the four quadrants. Bottom: Graph of sine versus angle. Angles from the top panel are identified.</figcaption></figure> <table class="wikitable sortable"> <caption>Summary of relationships between trigonometric functions<a href="#cite_note-4">[4]</a> </caption> <tbody><tr> <th rowspan="2">Function </th> <th rowspan="2">Description </th> <th colspan="2"><a href="/wiki/List_of_trigonometric_identities" title="List of trigonometric identities">Relationship</a> </th></tr> <tr> <th>using <a href="/wiki/Radian" title="Radian">radians</a> </th> <th>using <a href="/wiki/Degree_(angle)" title="Degree (angle)">degrees</a> </th></tr> <tr> <th>sine </th> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠opposite/hypotenuse⁠ </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \theta =\cos \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\csc \theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>θ</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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>csc</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \theta =\cos \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\csc \theta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1387e819274bb5fc07fc963d4e45746dc02d87e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:27.81ex; height:5.343ex;" alt="{\displaystyle \sin \theta =\cos \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\csc \theta }}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin x=\cos \left(90^{\circ }-x\right)={\frac {1}{\csc x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>−</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>csc</mi> <mo>⁡</mo> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin x=\cos \left(90^{\circ }-x\right)={\frac {1}{\csc x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/629e58780175213e7ea59f44e32a40c1b907985c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.773ex; height:5.176ex;" alt="{\displaystyle \sin x=\cos \left(90^{\circ }-x\right)={\frac {1}{\csc x}}}"> </td></tr> <tr> <th>cosine </th> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠adjacent/hypotenuse⁠ </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \theta =\sin \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\sec \theta }}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>θ</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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>sec</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \theta =\sin \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\sec \theta }}\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddc300f239616de2582d1df26c0814284b5e3771" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:28.198ex; height:5.343ex;" alt="{\displaystyle \cos \theta =\sin \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\sec \theta }}\,}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos x=\sin \left(90^{\circ }-x\right)={\frac {1}{\sec x}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>−</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>sec</mi> <mo>⁡</mo> <mi>x</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos x=\sin \left(90^{\circ }-x\right)={\frac {1}{\sec x}}\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bcd8e15be99ef87c951b9500dd493daf416d576" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:29.16ex; height:5.176ex;" alt="{\displaystyle \cos x=\sin \left(90^{\circ }-x\right)={\frac {1}{\sec x}}\,}"> </td></tr> <tr> <th>tangent </th> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠opposite/adjacent⁠ </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan \theta ={\frac {\sin \theta }{\cos \theta }}=\cot \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\cot \theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>cot</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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>cot</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan \theta ={\frac {\sin \theta }{\cos \theta }}=\cot \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\cot \theta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cea9aa84f9ffeb4b497fdd2d7b1a3b1177091c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:36.945ex; height:5.509ex;" alt="{\displaystyle \tan \theta ={\frac {\sin \theta }{\cos \theta }}=\cot \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\cot \theta }}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan x={\frac {\sin x}{\cos x}}=\cot \left(90^{\circ }-x\right)={\frac {1}{\cot x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> </mrow> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>cot</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>−</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>cot</mi> <mo>⁡</mo> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan x={\frac {\sin x}{\cos x}}=\cot \left(90^{\circ }-x\right)={\frac {1}{\cot x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44c279988419731d40c7a1a785f9c764a398022d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:38.146ex; height:5.176ex;" alt="{\displaystyle \tan x={\frac {\sin x}{\cos x}}=\cot \left(90^{\circ }-x\right)={\frac {1}{\cot x}}}"> </td></tr> <tr> <th>cotangent </th> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠adjacent/opposite⁠ </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cot \theta ={\frac {\cos \theta }{\sin \theta }}=\tan \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\tan \theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cot</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>tan</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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cot \theta ={\frac {\cos \theta }{\sin \theta }}=\tan \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\tan \theta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eda775f09115f4699b704a1d1b4d80e210f42514" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:37.205ex; height:5.509ex;" alt="{\displaystyle \cot \theta ={\frac {\cos \theta }{\sin \theta }}=\tan \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\tan \theta }}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cot x={\frac {\cos x}{\sin x}}=\tan \left(90^{\circ }-x\right)={\frac {1}{\tan x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cot</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> </mrow> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>tan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>−</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>tan</mi> <mo>⁡</mo> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cot x={\frac {\cos x}{\sin x}}=\tan \left(90^{\circ }-x\right)={\frac {1}{\tan x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffdbf89f9a8463e6ce9940eae1478c7d16a7b3f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:38.406ex; height:5.176ex;" alt="{\displaystyle \cot x={\frac {\cos x}{\sin x}}=\tan \left(90^{\circ }-x\right)={\frac {1}{\tan x}}}"> </td></tr> <tr> <th>secant </th> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠hypotenuse/adjacent⁠ </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sec \theta =\csc \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\cos \theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sec</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>csc</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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sec \theta =\csc \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\cos \theta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c213d292ff4a23a79a9d6df5b58e7ffe01f90c0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:27.936ex; height:5.343ex;" alt="{\displaystyle \sec \theta =\csc \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\cos \theta }}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sec x=\csc \left(90^{\circ }-x\right)={\frac {1}{\cos x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sec</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mi>csc</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>−</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sec x=\csc \left(90^{\circ }-x\right)={\frac {1}{\cos x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71a8156261cbc13a17b6d07cdd2aeb2c55e3fec6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.898ex; height:5.176ex;" alt="{\displaystyle \sec x=\csc \left(90^{\circ }-x\right)={\frac {1}{\cos x}}}"> </td></tr> <tr> <th>cosecant </th> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠hypotenuse/opposite⁠ </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \csc \theta =\sec \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\sin \theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>csc</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>sec</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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \csc \theta =\sec \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\sin \theta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/643a1a5f18ae1ccdb01853d35e05b83efc82766b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:27.68ex; height:5.343ex;" alt="{\displaystyle \csc \theta =\sec \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\sin \theta }}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \csc x=\sec \left(90^{\circ }-x\right)={\frac {1}{\sin x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>csc</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mi>sec</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>−</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \csc x=\sec \left(90^{\circ }-x\right)={\frac {1}{\sin x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd500f87c54117ffe3d01e8e1c61f74ef9ceedfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.643ex; height:5.176ex;" alt="{\displaystyle \csc x=\sec \left(90^{\circ }-x\right)={\frac {1}{\sin x}}}"> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Radians_versus_degrees">Radians versus degrees</h2>[<a href="/w/index.php?title=Trigonometric_functions&action=edit&section=3" title="Edit section: Radians versus degrees">edit</a>]</div> In geometric applications, the argument of a trigonometric function is generally the measure of an <a href="/wiki/Angle" title="Angle">angle</a>. For this purpose, any <a href="/wiki/Angular_unit" class="mw-redirect" title="Angular unit">angular unit</a> is convenient. One common unit is <a href="/wiki/Degree_(angle)" title="Degree (angle)">degrees</a>, in which a right angle is 90° and a complete turn is 360° (particularly in <a href="/wiki/Elementary_mathematics" title="Elementary mathematics">elementary mathematics</a>). However, in <a href="/wiki/Calculus" title="Calculus">calculus</a> and <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a>, the trigonometric functions are generally regarded more abstractly as functions of <a href="/wiki/Real_number" title="Real number">real</a> or <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>, rather than angles. In fact, the functions sin and cos can be defined for all complex numbers in terms of the <a href="/wiki/Exponential_function" title="Exponential function">exponential function</a>, via power series,<a href="#cite_note-:0-5">[5]</a> or as solutions to <a href="/wiki/Differential_equation" title="Differential equation">differential equations</a> given particular initial values<a href="#cite_note-6">[6]</a> (see below), without reference to any geometric notions. The other four trigonometric functions (tan, cot, sec, csc) can be defined as quotients and reciprocals of sin and cos, except where zero occurs in the denominator. It can be proved, for real arguments, that these definitions coincide with elementary geometric definitions if the argument is regarded as an angle in radians.<a href="#cite_note-:0-5">[5]</a> Moreover, these definitions result in simple expressions for the <a href="/wiki/Derivative" title="Derivative">derivatives</a> and <a href="/wiki/Antiderivative" title="Antiderivative">indefinite integrals</a> for the trigonometric functions.<a href="#cite_note-:1-7">[7]</a> Thus, in settings beyond elementary geometry, radians are regarded as the mathematically natural unit for describing angle measures. When <a href="/wiki/Radian" title="Radian">radians</a> (rad) are employed, the angle is given as the length of the <a href="/wiki/Arc_(geometry)" class="mw-redirect" title="Arc (geometry)">arc</a> of the <a href="/wiki/Unit_circle" title="Unit circle">unit circle</a> subtended by it: the angle that subtends an arc of length 1 on the unit circle is 1 rad (≈ 57.3°), and a complete <a href="/wiki/Turn_(angle)" title="Turn (angle)">turn</a> (360°) is an angle of 2π (≈ 6.28) rad. For real number x, the notation sin x, cos x, etc. refers to the value of the trigonometric functions evaluated at an angle of x rad. If units of degrees are intended, the degree sign must be explicitly shown (sin x°, cos x°, etc.). Using this standard notation, the argument x for the trigonometric functions satisfies the relationship x = (180x/π)°, so that, for example, sin π = sin 180° when we take x = π. In this way, the degree symbol can be regarded as a mathematical constant such that 1° = π/180 ≈ 0.0175. <div class="mw-heading mw-heading2"><h2 id="Unit-circle_definitions">Unit-circle definitions</h2>[<a href="/w/index.php?title=Trigonometric_functions&action=edit&section=4" title="Edit section: Unit-circle definitions">edit</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Circle-trig6.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Circle-trig6.svg/300px-Circle-trig6.svg.png" decoding="async" width="300" height="192" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Circle-trig6.svg/450px-Circle-trig6.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Circle-trig6.svg/600px-Circle-trig6.svg.png 2x" data-file-width="1250" data-file-height="800" /></a><figcaption>All of the trigonometric functions of the angle θ (theta) can be constructed geometrically in terms of a unit circle centered at O.</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> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Unit_Circle_Definitions_of_Six_Trigonometric_Functions.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/46/Unit_Circle_Definitions_of_Six_Trigonometric_Functions.svg/260px-Unit_Circle_Definitions_of_Six_Trigonometric_Functions.svg.png" decoding="async" width="260" height="250" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/46/Unit_Circle_Definitions_of_Six_Trigonometric_Functions.svg/390px-Unit_Circle_Definitions_of_Six_Trigonometric_Functions.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/46/Unit_Circle_Definitions_of_Six_Trigonometric_Functions.svg/520px-Unit_Circle_Definitions_of_Six_Trigonometric_Functions.svg.png 2x" data-file-width="665" data-file-height="640" /></a><figcaption>In this illustration, the six trigonometric functions of an arbitrary angle θ are represented as <a href="/wiki/Cartesian_coordinates" class="mw-redirect" title="Cartesian coordinates">Cartesian coordinates</a> of points related to the <a href="/wiki/Unit_circle" title="Unit circle">unit circle</a>. The ordinates of A, B and D are sin θ, tan θ and csc θ, respectively, while the abscissas of A, C and E are cos θ, cot θ and sec θ, respectively.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Trigonometric_function_quadrant_sign.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Trigonometric_function_quadrant_sign.svg/220px-Trigonometric_function_quadrant_sign.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Trigonometric_function_quadrant_sign.svg/330px-Trigonometric_function_quadrant_sign.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Trigonometric_function_quadrant_sign.svg/440px-Trigonometric_function_quadrant_sign.svg.png 2x" data-file-width="512" data-file-height="512" /></a><figcaption>Signs of trigonometric functions in each quadrant. <a href="/wiki/Mnemonics_in_trigonometry" title="Mnemonics in trigonometry">Mnemonics</a> like "all students take calculus" indicates when sine, cosine, and tangent are positive from quadrants I to IV.<a href="#cite_note-steuben-8">[8]</a></figcaption></figure> The six trigonometric functions can be defined as <a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">coordinate values</a> of points on the <a href="/wiki/Euclidean_plane" title="Euclidean plane">Euclidean plane</a> that are related to the <a href="/wiki/Unit_circle" title="Unit circle">unit circle</a>, which is the <a href="/wiki/Circle" title="Circle">circle</a> of radius one centered at the origin O of this coordinate system. While <a href="#Right-angled_triangle_definitions">right-angled triangle definitions</a> allow for the definition of the trigonometric functions for angles between 0 and <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><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}}}"> <a href="/wiki/Radian" title="Radian">radians</a> (90°), the unit circle definitions allow the domain of trigonometric functions to be extended to all positive and negative real numbers. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9027196ecb178d598958555ea01c43157d83597c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.604ex; height:2.176ex;" alt="{\displaystyle {\mathcal {L}}}"> be the <a href="/wiki/Ray_(geometry)" class="mw-redirect" title="Ray (geometry)">ray</a> obtained by rotating by an angle θ the positive half of the x-axis (<a href="/wiki/Counterclockwise" class="mw-redirect" title="Counterclockwise">counterclockwise</a> rotation for <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> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta >0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd8ceb635457435c97aa31c5ade3477fb5911256" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.998ex; height:2.509ex;" alt="{\displaystyle \theta >0,}"> and clockwise rotation for <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3da147239edfa6daac2c10f5258d18b41576d214" 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}">). This ray intersects the unit circle at the point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {A} =(x_{\mathrm {A} },y_{\mathrm {A} }).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {A} =(x_{\mathrm {A} },y_{\mathrm {A} }).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3230b96f1af7dd83efbb039d0aa3d4d76c3207aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.73ex; height:2.843ex;" alt="{\displaystyle \mathrm {A} =(x_{\mathrm {A} },y_{\mathrm {A} }).}"> The ray <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc122330f7ba9a19c076e1bc58b9686ea836a8e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.251ex; height:2.509ex;" alt="{\displaystyle {\mathcal {L}},}"> extended to a <a href="/wiki/Line_(geometry)" title="Line (geometry)">line</a> if necessary, intersects the line of equation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee42176e76ae6b56d68c42ced807e08b962a2b54" 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=1}"> at point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {B} =(1,y_{\mathrm {B} }),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {B} =(1,y_{\mathrm {B} }),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81d600e4ae0edd5a87c49dc30c4dd974d4ec94dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.932ex; height:2.843ex;" alt="{\displaystyle \mathrm {B} =(1,y_{\mathrm {B} }),}"> and the line of equation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10f53b404b1fdd041a589f1f2425e45a2edba110" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.416ex; height:2.509ex;" alt="{\displaystyle y=1}"> at point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {C} =(x_{\mathrm {C} },1).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">C</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">C</mi> </mrow> </mrow> </msub> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {C} =(x_{\mathrm {C} },1).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103c138dbef94f6720a6365c39f499edb6f9a307" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.178ex; height:2.843ex;" alt="{\displaystyle \mathrm {C} =(x_{\mathrm {C} },1).}"> The <a href="/wiki/Tangent_line" class="mw-redirect" title="Tangent line">tangent line</a> to the unit circle at the point A, is <a href="/wiki/Perpendicular" title="Perpendicular">perpendicular</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc122330f7ba9a19c076e1bc58b9686ea836a8e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.251ex; height:2.509ex;" alt="{\displaystyle {\mathcal {L}},}"> and intersects the y- and x-axes at points <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {D} =(0,y_{\mathrm {D} })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {D} =(0,y_{\mathrm {D} })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/284329c70a7619909a5a62aca6891b23e2bef156" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.507ex; height:2.843ex;" alt="{\displaystyle \mathrm {D} =(0,y_{\mathrm {D} })}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {E} =(x_{\mathrm {E} },0).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> </mrow> </msub> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {E} =(x_{\mathrm {E} },0).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09184d3521883c942d9bd0b8627033e7d963d606" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.015ex; height:2.843ex;" alt="{\displaystyle \mathrm {E} =(x_{\mathrm {E} },0).}"> The <a href="/wiki/Cartesian_coordinates" class="mw-redirect" title="Cartesian coordinates">coordinates</a> of these points give the values of all trigonometric functions for any arbitrary real value of θ in the following manner. The trigonometric functions cos and sin are defined, respectively, as the x- and y-coordinate values of point A. That is, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \theta =x_{\mathrm {A} }\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \theta =x_{\mathrm {A} }\quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14dc3543d05a1807bada67dbe5aacc7e2ae1a3be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.804ex; height:2.509ex;" alt="{\displaystyle \cos \theta =x_{\mathrm {A} }\quad }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \sin \theta =y_{\mathrm {A} }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \sin \theta =y_{\mathrm {A} }.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6b5ab79d78729cc1f5b6a31c303337f7c8b4e3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.005ex; height:2.509ex;" alt="{\displaystyle \quad \sin \theta =y_{\mathrm {A} }.}"><a href="#cite_note-9">[9]</a></dd></dl> In the range <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq \theta \leq \pi /2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mi>θ</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 0\leq \theta \leq \pi /2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aae201dc8951a9c20cb3c95f883614127224075" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.107ex; height:2.843ex;" alt="{\displaystyle 0\leq \theta \leq \pi /2}">, this definition coincides with the right-angled triangle definition, by taking the right-angled triangle to have the unit radius OA as <a href="/wiki/Hypotenuse" title="Hypotenuse">hypotenuse</a>. And since the equation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+y^{2}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+y^{2}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec84b90236512e8d27ff1a8f7707b60b63327de1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.7ex; height:3.009ex;" alt="{\displaystyle x^{2}+y^{2}=1}"> holds for all points <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {P} =(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {P} =(x,y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/405605ca539cd9d1bf962026e243a603b2b3e49b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.01ex; height:2.843ex;" alt="{\displaystyle \mathrm {P} =(x,y)}"> on the unit circle, this definition of cosine and sine also satisfies the <a href="/wiki/Pythagorean_identity" class="mw-redirect" title="Pythagorean identity">Pythagorean identity</a>. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos ^{2}\theta +\sin ^{2}\theta =1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e79d40676a67b3463099809cd6bd9e046d5781f" class="mwe-math-fallback-image-inline 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.}"></dd></dl> The other trigonometric functions can be found along the unit circle as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan \theta =y_{\mathrm {B} }\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan \theta =y_{\mathrm {B} }\quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/069d72909834b9e6e4b73cf581625b0a05815c88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.793ex; height:2.509ex;" alt="{\displaystyle \tan \theta =y_{\mathrm {B} }\quad }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \cot \theta =x_{\mathrm {C} },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mi>cot</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">C</mi> </mrow> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \cot \theta =x_{\mathrm {C} },}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9aef22b347c603c69d74109345f484bd456698a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.393ex; height:2.509ex;" alt="{\displaystyle \quad \cot \theta =x_{\mathrm {C} },}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \csc \theta \ =y_{\mathrm {D} }\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>csc</mi> <mo>⁡</mo> <mi>θ</mi> <mtext> </mtext> <mo>=</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \csc \theta \ =y_{\mathrm {D} }\quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ac8075d4c0ac1fa55840f4a9778368a37f7e94a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.087ex; height:2.509ex;" alt="{\displaystyle \csc \theta \ =y_{\mathrm {D} }\quad }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \sec \theta =x_{\mathrm {E} }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mi>sec</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \sec \theta =x_{\mathrm {E} }.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78b4d223146254ad0888ea1d29d86fb81f080132" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.208ex; height:2.509ex;" alt="{\displaystyle \quad \sec \theta =x_{\mathrm {E} }.}"></dd></dl> By applying the Pythagorean identity and geometric proof methods, these definitions can readily be shown to coincide with the definitions of tangent, cotangent, secant and cosecant in terms of sine and cosine, that is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan \theta ={\frac {\sin \theta }{\cos \theta }},\quad \cot \theta ={\frac {\cos \theta }{\sin \theta }},\quad \sec \theta ={\frac {1}{\cos \theta }},\quad \csc \theta ={\frac {1}{\sin \theta }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <mi>cot</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <mi>sec</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <mi>csc</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan \theta ={\frac {\sin \theta }{\cos \theta }},\quad \cot \theta ={\frac {\cos \theta }{\sin \theta }},\quad \sec \theta ={\frac {1}{\cos \theta }},\quad \csc \theta ={\frac {1}{\sin \theta }}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db831a3c3722d7263131957181b6a2779bfce1d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:62.885ex; height:5.509ex;" alt="{\displaystyle \tan \theta ={\frac {\sin \theta }{\cos \theta }},\quad \cot \theta ={\frac {\cos \theta }{\sin \theta }},\quad \sec \theta ={\frac {1}{\cos \theta }},\quad \csc \theta ={\frac {1}{\sin \theta }}.}"></dd></dl> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="//upload.wikimedia.org/wikipedia/commons/2/27/Trigonometric_functions_derivation_animation.svg"><img resource="/wiki/File:Trigonometric_functions.svg" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Trigonometric_functions.svg/300px-Trigonometric_functions.svg.png" decoding="async" width="300" height="196" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Trigonometric_functions.svg/450px-Trigonometric_functions.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/19/Trigonometric_functions.svg/600px-Trigonometric_functions.svg.png 2x" data-file-width="744" data-file-height="485" /></a><figcaption>Trigonometric functions: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">Sine, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">Cosine, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">Tangent, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">Cosecant (dotted), <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">Secant (dotted), <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">Cotangent (dotted) – <a class="external text" href="https://upload.wikimedia.org/wikipedia/commons/2/27/Trigonometric_functions_derivation_animation.svg">animation</a> </figcaption></figure> Since a rotation of an angle of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm 2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm 2\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5caec4734db21ac049a4e019e555af5bd2a37c2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.303ex; height:2.176ex;" alt="{\displaystyle \pm 2\pi }"> does not change the position or size of a shape, the points A, B, C, D, and E are the same for two angles whose difference is an integer multiple of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73efd1f6493490b058097060a572606d2c550a06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }">. Thus trigonometric functions are <a href="/wiki/Periodic_function" title="Periodic function">periodic functions</a> with period <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73efd1f6493490b058097060a572606d2c550a06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }">. That is, the equalities <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \theta =\sin \left(\theta +2k\pi \right)\quad }"> <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> <mrow> <mo>(</mo> <mrow> <mi>θ</mi> <mo>+</mo> <mn>2</mn> <mi>k</mi> <mi>π</mi> </mrow> <mo>)</mo> </mrow> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \theta =\sin \left(\theta +2k\pi \right)\quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd15247de5a268cee5cbbc8abaa834f995156ce1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.056ex; height:2.843ex;" alt="{\displaystyle \sin \theta =\sin \left(\theta +2k\pi \right)\quad }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \cos \theta =\cos \left(\theta +2k\pi \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>θ</mi> <mo>+</mo> <mn>2</mn> <mi>k</mi> <mi>π</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \cos \theta =\cos \left(\theta +2k\pi \right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74aa803d6bab98c385e1234988e86009946c99ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.567ex; height:2.843ex;" alt="{\displaystyle \quad \cos \theta =\cos \left(\theta +2k\pi \right)}"></dd></dl> hold for any angle θ and any <a href="/wiki/Integer" title="Integer">integer</a> k. The same is true for the four other trigonometric functions. By observing the sign and the monotonicity of the functions sine, cosine, cosecant, and secant in the four quadrants, one can show that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73efd1f6493490b058097060a572606d2c550a06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }"> is the smallest value for which they are periodic (i.e., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73efd1f6493490b058097060a572606d2c550a06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }"> is the <a href="/wiki/Periodic_function" title="Periodic function">fundamental period</a> of these functions). However, after a rotation by an angle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }">, the points B and C already return to their original position, so that the tangent function and the cotangent function have a fundamental period of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }">. That is, the equalities <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan \theta =\tan(\theta +k\pi )\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>+</mo> <mi>k</mi> <mi>π</mi> <mo stretchy="false">)</mo> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan \theta =\tan(\theta +k\pi )\quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2724a6a0811238927b570778e71751e401658a00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.901ex; height:2.843ex;" alt="{\displaystyle \tan \theta =\tan(\theta +k\pi )\quad }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \cot \theta =\cot(\theta +k\pi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mi>cot</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>cot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>+</mo> <mi>k</mi> <mi>π</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \cot \theta =\cot(\theta +k\pi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a49f207143efc425eea3bbf0aa50a11eb71d8c5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.381ex; height:2.843ex;" alt="{\displaystyle \quad \cot \theta =\cot(\theta +k\pi )}"></dd></dl> hold for any angle θ and any integer k. <div class="mw-heading mw-heading2"><h2 id="Algebraic_values">Algebraic values</h2>[<a href="/w/index.php?title=Trigonometric_functions&action=edit&section=5" title="Edit section: Algebraic values">edit</a>]</div> <figure class="mw-default-size mw-halign-right" 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/220px-Unit_circle_angles_color.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Unit_circle_angles_color.svg/330px-Unit_circle_angles_color.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Unit_circle_angles_color.svg/440px-Unit_circle_angles_color.svg.png 2x" data-file-width="720" data-file-height="720" /></a><figcaption>The <a href="/wiki/Unit_circle" title="Unit circle">unit circle</a>, with some points labeled with their cosine and sine (in this order), and the corresponding angles in radians and degrees.</figcaption></figure> The <a href="/wiki/Algebraic_expression" title="Algebraic expression">algebraic expressions</a> for the most important angles are as follows: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin 0=\sin 0^{\circ }\quad ={\frac {\sqrt {0}}{2}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mn>0</mn> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mspace width="1em" /> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>0</mn> </msqrt> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin 0=\sin 0^{\circ }\quad ={\frac {\sqrt {0}}{2}}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28fe011fbd11517be14b39b3cdf295acfc5b50a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:26.579ex; height:5.843ex;" alt="{\displaystyle \sin 0=\sin 0^{\circ }\quad ={\frac {\sqrt {0}}{2}}=0}"> (<a href="/wiki/Angle#Types_of_angles" title="Angle">zero angle</a>)</dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin {\frac {\pi }{6}}=\sin 30^{\circ }={\frac {\sqrt {1}}{2}}={\frac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> </mrow> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <msup> <mn>30</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>1</mn> </msqrt> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin {\frac {\pi }{6}}=\sin 30^{\circ }={\frac {\sqrt {1}}{2}}={\frac {1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46f2856ea32e09e872fd5371414e0bae796e69bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.261ex; height:5.843ex;" alt="{\displaystyle \sin {\frac {\pi }{6}}=\sin 30^{\circ }={\frac {\sqrt {1}}{2}}={\frac {1}{2}}}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin {\frac {\pi }{4}}=\sin 45^{\circ }={\frac {\sqrt {2}}{2}}={\frac {1}{\sqrt {2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mo>=</mo> <mi>sin</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> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin {\frac {\pi }{4}}=\sin 45^{\circ }={\frac {\sqrt {2}}{2}}={\frac {1}{\sqrt {2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/214fd46811398b0c3ce94e8c2a14ac479cce73a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:29.197ex; height:6.843ex;" alt="{\displaystyle \sin {\frac {\pi }{4}}=\sin 45^{\circ }={\frac {\sqrt {2}}{2}}={\frac {1}{\sqrt {2}}}}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin {\frac {\pi }{3}}=\sin 60^{\circ }={\frac {\sqrt {3}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>3</mn> </mfrac> </mrow> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <msup> <mn>60</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>3</mn> </msqrt> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin {\frac {\pi }{3}}=\sin 60^{\circ }={\frac {\sqrt {3}}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/976dfa7d5a0311d37f2eb0c73f82d4e01b01ab72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.164ex; height:5.843ex;" alt="{\displaystyle \sin {\frac {\pi }{3}}=\sin 60^{\circ }={\frac {\sqrt {3}}{2}}}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin {\frac {\pi }{2}}=\sin 90^{\circ }={\frac {\sqrt {4}}{2}}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>4</mn> </msqrt> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin {\frac {\pi }{2}}=\sin 90^{\circ }={\frac {\sqrt {4}}{2}}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/086c77876c678749a43890929359e536300f401d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:26.425ex; height:5.843ex;" alt="{\displaystyle \sin {\frac {\pi }{2}}=\sin 90^{\circ }={\frac {\sqrt {4}}{2}}=1}"> (<a href="/wiki/Right_angle" title="Right angle">right angle</a>)</dd></dl> Writing the numerators as <a href="/wiki/Square_roots" class="mw-redirect" title="Square roots">square roots</a> of consecutive non-negative integers, with a denominator of 2, provides an easy way to remember the values.<a href="#cite_note-Larson_2013-10">[10]</a> Such simple expressions generally do not exist for other angles which are rational multiples of a right angle. <ul><li>For an angle which, measured in degrees, is a multiple of three, the <a href="/wiki/Exact_trigonometric_values" title="Exact trigonometric values">exact trigonometric values</a> of the sine and the cosine may be expressed in terms of square roots. These values of the sine and the cosine may thus be constructed by <a href="/wiki/Compass-and-straightedge_construction" class="mw-redirect" title="Compass-and-straightedge construction">ruler and compass</a>.</li> <li>For an angle of an integer number of degrees, the sine and the cosine may be expressed in terms of square roots and the <a href="/wiki/Cube_root" title="Cube root">cube root</a> of a non-real <a href="/wiki/Complex_number" title="Complex number">complex number</a>. <a href="/wiki/Galois_theory" title="Galois theory">Galois theory</a> allows a proof that, if the angle is not a multiple of 3°, non-real cube roots are unavoidable.</li> <li>For an angle which, expressed in degrees, is a <a href="/wiki/Rational_number" title="Rational number">rational number</a>, the sine and the cosine are <a href="/wiki/Algebraic_number" title="Algebraic number">algebraic numbers</a>, which may be expressed in terms of <a href="/wiki/Nth_root" title="Nth root">nth roots</a>. This results from the fact that the <a href="/wiki/Galois_group" title="Galois group">Galois groups</a> of the <a href="/wiki/Cyclotomic_polynomial" title="Cyclotomic polynomial">cyclotomic polynomials</a> are <a href="/wiki/Cyclic_group" title="Cyclic group">cyclic</a>.</li> <li>For an angle which, expressed in degrees, is not a rational number, then either the angle or both the sine and the cosine are <a href="/wiki/Transcendental_number" title="Transcendental number">transcendental numbers</a>. This is a corollary of <a href="/wiki/Baker%27s_theorem" title="Baker's theorem">Baker's theorem</a>, proved in 1966.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Simple_algebraic_values">Simple algebraic values</h3>[<a href="/w/index.php?title=Trigonometric_functions&action=edit&section=6" title="Edit section: Simple algebraic values">edit</a>]</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/Exact_trigonometric_values#Common_angles" title="Exact trigonometric values">Exact trigonometric values § Common angles</a></div> The following table lists the sines, cosines, and tangents of multiples of 15 degrees from 0 to 90 degrees. <table class="wikitable" style="text-align:center;"> <tbody><tr> <th colspan="2">Angle, θ, in </th> <th rowspan="2"><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><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 )}"> </th> <th rowspan="2"><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><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 )}"> </th> <th rowspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan(\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan(\theta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2dbdbf63a30c035f2e8232b47930e745680b6e31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.259ex; height:2.843ex;" alt="{\displaystyle \tan(\theta )}"> </th></tr> <tr> <th>radians </th> <th>degrees </th></tr> <tr> <td><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><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}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd0e1e92cf5770c2bfbb1de8b4b7bf904c9deef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.217ex; height:2.343ex;" alt="{\displaystyle 0^{\circ }}"> </td> <td><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><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}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"> </td> <td><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><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}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\pi }{12}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>12</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\pi }{12}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b1272b2462bfc7c89c80c80be3e3712829c8d6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.161ex; height:4.676ex;" alt="{\displaystyle {\frac {\pi }{12}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 15^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>15</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 15^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6b2ba390041accdbd09de910f720bd6e873d137" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.343ex;" alt="{\displaystyle 15^{\circ }}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> <mn>4</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78c627f0ebe511a980e4f73e69210a111a36bda0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.873ex; height:5.843ex;" alt="{\displaystyle {\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> <mn>4</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3b490d0765455c4d35af5b84445852edea3f2a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.873ex; height:5.843ex;" alt="{\displaystyle {\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2-{\sqrt {3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2-{\sqrt {3}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df3c5bc3d7b3e1148963e3279582cc8f07a4b789" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.101ex; height:2.843ex;" alt="{\displaystyle 2-{\sqrt {3}}}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\pi }{6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\pi }{6}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da430867901fd359c000b52f2bd70b36cf5e2182" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.168ex; height:4.676ex;" alt="{\displaystyle {\frac {\pi }{6}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 30^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>30</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 30^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42f29df9c101d2a8dae3f1552342cfe4c3adb76c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.343ex;" alt="{\displaystyle 30^{\circ }}"> </td> <td><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><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}}}"> </td> <td><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><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}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\sqrt {3}}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>3</mn> </msqrt> <mn>3</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\sqrt {3}}{3}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d772a07a833dc01b7dec1aaea164b6accc2b68e" 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}}{3}}}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\pi }{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\pi }{4}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f89d7c88c1c93dce69a46052a8e276e231063de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.168ex; height:4.676ex;" alt="{\displaystyle {\frac {\pi }{4}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 45^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>45</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 45^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c28223ddedeb94a84bb15474cc64b5ce436cbe50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.343ex;" alt="{\displaystyle 45^{\circ }}"> </td> <td><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><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}}}"> </td> <td><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><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}}}"><a href="#cite_note-11">[a]</a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\pi }{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>3</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\pi }{3}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c83c684a603005cda4feb8eea0254143ffb0e16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.168ex; height:4.676ex;" alt="{\displaystyle {\frac {\pi }{3}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 60^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>60</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 60^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08c42292485b447b7f627a7accd90d5b439c11d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.343ex;" alt="{\displaystyle 60^{\circ }}"> </td> <td><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><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}}}"> </td> <td><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><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}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {3}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b19c09494138b5082459afac7f9a8d99c546fcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:2.843ex;" alt="{\displaystyle {\sqrt {3}}}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {5\pi }{12}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mn>12</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {5\pi }{12}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8e60a8a09f8e6c420879481acb6d8ead181e822" 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 {5\pi }{12}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 75^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>75</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 75^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/565c6cb6b28eeab72c86cdb73940a1ad0d2d1d26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.343ex;" alt="{\displaystyle 75^{\circ }}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> <mn>4</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3b490d0765455c4d35af5b84445852edea3f2a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.873ex; height:5.843ex;" alt="{\displaystyle {\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> <mn>4</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78c627f0ebe511a980e4f73e69210a111a36bda0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.873ex; height:5.843ex;" alt="{\displaystyle {\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2+{\sqrt {3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2+{\sqrt {3}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/442e3ac1b93fc34af6ec2955a62d030e4f818df5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.101ex; height:2.843ex;" alt="{\displaystyle 2+{\sqrt {3}}}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\pi }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\pi }{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98f98bef5d4981ff6e2aa827d4699e347fb30db2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.168ex; height:4.676ex;" alt="{\displaystyle {\frac {\pi }{2}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 90^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 90^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c326d317eddef3ad3e6625e018a708e290a039f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.343ex;" alt="{\displaystyle 90^{\circ }}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"> </td> <td><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><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}"> </td> <td data-sort-value="" style="background: var(--background-color-interactive, #ececec); color: var(--color-base, inherit); vertical-align: middle; text-align: center;" class="table-na">Undefined </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Definitions_in_analysis">Definitions in analysis</h2>[<a href="/w/index.php?title=Trigonometric_functions&action=edit&section=7" title="Edit section: Definitions in analysis">edit</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Trigonometrija-graf.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Trigonometrija-graf.svg/220px-Trigonometrija-graf.svg.png" decoding="async" width="220" height="131" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Trigonometrija-graf.svg/330px-Trigonometrija-graf.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Trigonometrija-graf.svg/440px-Trigonometrija-graf.svg.png 2x" data-file-width="1532" data-file-height="915" /></a><figcaption><a href="/wiki/Graph_of_a_function" title="Graph of a function">Graphs</a> of sine, cosine and tangent</figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Taylorsine.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Taylorsine.svg/220px-Taylorsine.svg.png" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Taylorsine.svg/330px-Taylorsine.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Taylorsine.svg/440px-Taylorsine.svg.png 2x" data-file-width="1600" data-file-height="1200" /></a><figcaption>The sine function (blue) is closely approximated by its <a href="/wiki/Taylor%27s_theorem" title="Taylor's theorem">Taylor polynomial</a> of degree 7 (pink) for a full cycle centered on the origin.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Taylor_cos.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Taylor_cos.gif/220px-Taylor_cos.gif" decoding="async" width="220" height="145" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Taylor_cos.gif/330px-Taylor_cos.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/51/Taylor_cos.gif/440px-Taylor_cos.gif 2x" data-file-width="1381" data-file-height="913" /></a><figcaption>Animation for the approximation of cosine via Taylor polynomials.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Taylorreihenentwicklung_des_Kosinus.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Taylorreihenentwicklung_des_Kosinus.svg/220px-Taylorreihenentwicklung_des_Kosinus.svg.png" decoding="async" width="220" height="176" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Taylorreihenentwicklung_des_Kosinus.svg/330px-Taylorreihenentwicklung_des_Kosinus.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Taylorreihenentwicklung_des_Kosinus.svg/440px-Taylorreihenentwicklung_des_Kosinus.svg.png 2x" data-file-width="600" data-file-height="480" /></a><figcaption><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb9af7ed6f44822021b74bb82b431022c7fd66b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.25ex; height:2.843ex;" alt="{\displaystyle \cos(x)}"> together with the first Taylor polynomials <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{n}(x)=\sum _{k=0}^{n}(-1)^{k}{\frac {x^{2k}}{(2k)!}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{n}(x)=\sum _{k=0}^{n}(-1)^{k}{\frac {x^{2k}}{(2k)!}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc9f8e57daa3b8730a0a6e8a0387feb5b7d93605" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; margin-left: -0.089ex; width:23.604ex; height:7.009ex;" alt="{\displaystyle p_{n}(x)=\sum _{k=0}^{n}(-1)^{k}{\frac {x^{2k}}{(2k)!}}}"></figcaption></figure> <a href="/wiki/G._H._Hardy" title="G. H. Hardy">G. H. Hardy</a> noted in his 1908 work <a href="/wiki/A_Course_of_Pure_Mathematics" title="A Course of Pure Mathematics">A Course of Pure Mathematics</a> that the definition of the trigonometric functions in terms of the unit circle is not satisfactory, because it depends implicitly on a notion of angle that can be measured by a real number.<a href="#cite_note-Hardy-12">[11]</a> Thus in modern analysis, trigonometric functions are usually constructed without reference to geometry. Various ways exist in the literature for defining the trigonometric functions in a manner suitable for analysis; they include: <ul><li>Using the "geometry" of the unit circle, which requires formulating the arc length of a circle (or area of a sector) analytically.<a href="#cite_note-Hardy-12">[11]</a></li> <li>By a power series, which is particularly well-suited to complex variables.<a href="#cite_note-Hardy-12">[11]</a><a href="#cite_note-WW-13">[12]</a></li> <li>By using an infinite product expansion.<a href="#cite_note-Hardy-12">[11]</a></li> <li>By inverting the inverse trigonometric functions, which can be defined as integrals of algebraic or rational functions.<a href="#cite_note-Hardy-12">[11]</a></li> <li>As solutions of a differential equation.<a href="#cite_note-BS-14">[13]</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Definition_by_differential_equations">Definition by differential equations</h3>[<a href="/w/index.php?title=Trigonometric_functions&action=edit&section=8" title="Edit section: Definition by differential equations">edit</a>]</div> Sine and cosine can be defined as the unique solution to the <a href="/wiki/Initial_value_problem" title="Initial value problem">initial value problem</a>:<a href="#cite_note-FOOTNOTEBartleSherbert1999247-15">[14]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{dx}}\sin x=\cos x,\ {\frac {d}{dx}}\cos x=-\sin x,\ \sin(0)=0,\ \cos(0)=1.}"> <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> <mi>x</mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>,</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo>,</mo> <mtext> </mtext> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dx}}\sin x=\cos x,\ {\frac {d}{dx}}\cos x=-\sin x,\ \sin(0)=0,\ \cos(0)=1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3745ae123ee19601b18a6aadffdc5221a4fcd58f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:61.427ex; height:5.509ex;" alt="{\displaystyle {\frac {d}{dx}}\sin x=\cos x,\ {\frac {d}{dx}}\cos x=-\sin x,\ \sin(0)=0,\ \cos(0)=1.}"></dd></dl> Differentiating again, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {d^{2}}{dx^{2}}}\sin x={\frac {d}{dx}}\cos x=-\sin x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {d^{2}}{dx^{2}}}\sin x={\frac {d}{dx}}\cos x=-\sin x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6aa176d98502db389885f452e3e50aae92a1d09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:29.243ex; height:4.343ex;" alt="{\textstyle {\frac {d^{2}}{dx^{2}}}\sin x={\frac {d}{dx}}\cos x=-\sin x}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {d^{2}}{dx^{2}}}\cos x=-{\frac {d}{dx}}\sin x=-\cos x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {d^{2}}{dx^{2}}}\cos x=-{\frac {d}{dx}}\sin x=-\cos x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1afdbe2076b1758ca7d7a31dd1bf66b42c23a97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:31.306ex; height:4.343ex;" alt="{\textstyle {\frac {d^{2}}{dx^{2}}}\cos x=-{\frac {d}{dx}}\sin x=-\cos x}">, so both sine and cosine are solutions of the same <a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">ordinary differential equation</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y''+y=0\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mo>″</mo> </msup> <mo>+</mo> <mi>y</mi> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y''+y=0\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97942cf21391288ac44faf2a2463b15f656b9ebb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.588ex; height:2.843ex;" alt="{\displaystyle y''+y=0\,.}"></dd></dl> Sine is the unique solution with y(0) = 0 and y′(0) = 1; cosine is the unique solution with y(0) = 1 and y′(0) = 0. One can then prove, as a theorem, that solutions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos ,\sin }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>,</mo> <mi>sin</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos ,\sin }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb709f682d4c55a7ce4af4cbef8aa5da94f4e72b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.001ex; height:2.509ex;" alt="{\displaystyle \cos ,\sin }"> are periodic, having the same period. Writing this period as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73efd1f6493490b058097060a572606d2c550a06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }"> is then a definition of the real number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"> which is independent of geometry. Applying the <a href="/wiki/Quotient_rule" title="Quotient rule">quotient rule</a> to the tangent <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan x=\sin x/\cos x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan x=\sin x/\cos x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ec4622b47b0e15e523b31c4cff77abec0ae8323" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.125ex; height:2.843ex;" alt="{\displaystyle \tan x=\sin x/\cos x}">, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{dx}}\tan x={\frac {\cos ^{2}x+\sin ^{2}x}{\cos ^{2}x}}=1+\tan ^{2}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>tan</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>x</mi> </mrow> <mrow> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>x</mi> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dx}}\tan x={\frac {\cos ^{2}x+\sin ^{2}x}{\cos ^{2}x}}=1+\tan ^{2}x\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c99ccb58bba30b061c734a4668dd9973deacc12c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:41.395ex; height:6.009ex;" alt="{\displaystyle {\frac {d}{dx}}\tan x={\frac {\cos ^{2}x+\sin ^{2}x}{\cos ^{2}x}}=1+\tan ^{2}x\,,}"></dd></dl> so the tangent function satisfies the ordinary differential equation <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y'=1+y^{2}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo>=</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y'=1+y^{2}\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00156229736a17c06756aa0a949e4ed8a375a2c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.195ex; height:3.009ex;" alt="{\displaystyle y'=1+y^{2}\,.}"></dd></dl> It is the unique solution with y(0) = 0. <div class="mw-heading mw-heading3"><h3 id="Power_series_expansion">Power series expansion</h3>[<a href="/w/index.php?title=Trigonometric_functions&action=edit&section=9" title="Edit section: Power series expansion">edit</a>]</div> The basic trigonometric functions can be defined by the following power series expansions<a href="#cite_note-16">[15]</a>. These series are also known as the <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a> or <a href="/wiki/Maclaurin_series" class="mw-redirect" title="Maclaurin series">Maclaurin series</a> of these trigonometric functions: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\sin x&=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots \\[6mu]&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}x^{2n+1}\\[8pt]\cos x&=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots \\[6mu]&=\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="0.633em 1.1em 0.633em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>x</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow> <mn>3</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mrow> <mn>5</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mrow> <mn>7</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mrow> <mn>4</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mrow> <mn>6</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mo>.</mo> </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 \\[6mu]&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}x^{2n+1}\\[8pt]\cos x&=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots \\[6mu]&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}x^{2n}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/396790dc41b52c5381ef1683a279d05ba5d64f79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -14.171ex; width:33.752ex; height:29.509ex;" alt="{\displaystyle {\begin{aligned}\sin x&=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots \\[6mu]&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}x^{2n+1}\\[8pt]\cos x&=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots \\[6mu]&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}x^{2n}.\end{aligned}}}"></dd></dl> The <a href="/wiki/Radius_of_convergence" title="Radius of convergence">radius of convergence</a> of these series is infinite. Therefore, the sine and the cosine can be extended to <a href="/wiki/Entire_function" title="Entire function">entire functions</a> (also called "sine" and "cosine"), which are (by definition) <a href="/wiki/Complex-valued_function" class="mw-redirect" title="Complex-valued function">complex-valued functions</a> that are defined and <a href="/wiki/Holomorphic" class="mw-redirect" title="Holomorphic">holomorphic</a> on the whole <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a>. Term-by-term differentiation shows that the sine and cosine defined by the series obey the differential equation discussed previously, and conversely one can obtain these series from elementary recursion relations derived from the differential equation. Being defined as fractions of entire functions, the other trigonometric functions may be extended to <a href="/wiki/Meromorphic_function" title="Meromorphic function">meromorphic functions</a>, that is functions that are holomorphic in the whole complex plane, except some isolated points called <a href="/wiki/Zeros_and_poles" title="Zeros and poles">poles</a>. Here, the poles are the numbers of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle (2k+1){\frac {\pi }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle (2k+1){\frac {\pi }{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2de46d790d680297f2bf80f8a8f3baeba3bd2771" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:9.964ex; height:3.176ex;" alt="{\textstyle (2k+1){\frac {\pi }{2}}}"> for the tangent and the secant, or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf859397db5c3d7bddebe20b20a69d8191f2448f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.543ex; height:2.176ex;" alt="{\displaystyle k\pi }"> for the cotangent and the cosecant, where k is an arbitrary integer. Recurrences relations may also be computed for the coefficients of the <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a> of the other trigonometric functions. These series have a finite <a href="/wiki/Radius_of_convergence" title="Radius of convergence">radius of convergence</a>. Their coefficients have a <a href="/wiki/Combinatorics" title="Combinatorics">combinatorial</a> interpretation: they enumerate <a href="/wiki/Alternating_permutation" title="Alternating permutation">alternating permutations</a> of finite sets.<a href="#cite_note-17">[16]</a> More precisely, defining <dl><dd>Un, the nth <a href="/wiki/Up/down_number" class="mw-redirect" title="Up/down number">up/down number</a>,</dd> <dd>Bn, the nth <a href="/wiki/Bernoulli_number" title="Bernoulli number">Bernoulli number</a>, and</dd> <dd>En, is the nth <a href="/wiki/Euler_number" class="mw-redirect" title="Euler number">Euler number</a>,</dd></dl> one has the following series expansions:<a href="#cite_note-18">[17]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\tan x&{}=\sum _{n=0}^{\infty }{\frac {U_{2n+1}}{(2n+1)!}}x^{2n+1}\\[8mu]&{}=\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}2^{2n}\left(2^{2n}-1\right)B_{2n}}{(2n)!}}x^{2n-1}\\[5mu]&{}=x+{\frac {1}{3}}x^{3}+{\frac {2}{15}}x^{5}+{\frac {17}{315}}x^{7}+\cdots ,\qquad {\text{for }}|x|<{\frac {\pi }{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="0.744em 0.578em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>tan</mi> <mo>⁡</mo> <mi>x</mi> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow 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> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> </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> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>15</mn> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>17</mn> <mn>315</mn> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>,</mo> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>for </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\tan x&{}=\sum _{n=0}^{\infty }{\frac {U_{2n+1}}{(2n+1)!}}x^{2n+1}\\[8mu]&{}=\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}2^{2n}\left(2^{2n}-1\right)B_{2n}}{(2n)!}}x^{2n-1}\\[5mu]&{}=x+{\frac {1}{3}}x^{3}+{\frac {2}{15}}x^{5}+{\frac {17}{315}}x^{7}+\cdots ,\qquad {\text{for }}|x|<{\frac {\pi }{2}}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0de1a2399f8c3b723b71c5e24e8f0136fd4bb18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.171ex; width:58.945ex; height:21.509ex;" alt="{\displaystyle {\begin{aligned}\tan x&{}=\sum _{n=0}^{\infty }{\frac {U_{2n+1}}{(2n+1)!}}x^{2n+1}\\[8mu]&{}=\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}2^{2n}\left(2^{2n}-1\right)B_{2n}}{(2n)!}}x^{2n-1}\\[5mu]&{}=x+{\frac {1}{3}}x^{3}+{\frac {2}{15}}x^{5}+{\frac {17}{315}}x^{7}+\cdots ,\qquad {\text{for }}|x|<{\frac {\pi }{2}}.\end{aligned}}}"></dd></dl> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\csc x&=\sum _{n=0}^{\infty }{\frac {(-1)^{n+1}2\left(2^{2n-1}-1\right)B_{2n}}{(2n)!}}x^{2n-1}\\[5mu]&=x^{-1}+{\frac {1}{6}}x+{\frac {7}{360}}x^{3}+{\frac {31}{15120}}x^{5}+\cdots ,\qquad {\text{for }}0<|x|<\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="0.578em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>csc</mi> <mo>⁡</mo> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>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> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> </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> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>7</mn> <mn>360</mn> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>31</mn> <mn>15120</mn> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>,</mo> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>for </mtext> </mrow> <mn>0</mn> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>π</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\csc x&=\sum _{n=0}^{\infty }{\frac {(-1)^{n+1}2\left(2^{2n-1}-1\right)B_{2n}}{(2n)!}}x^{2n-1}\\[5mu]&=x^{-1}+{\frac {1}{6}}x+{\frac {7}{360}}x^{3}+{\frac {31}{15120}}x^{5}+\cdots ,\qquad {\text{for }}0<|x|<\pi .\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd4b74fe732906b09b6205ecfa81326222ae0320" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.171ex; width:66.757ex; height:13.509ex;" alt="{\displaystyle {\begin{aligned}\csc x&=\sum _{n=0}^{\infty }{\frac {(-1)^{n+1}2\left(2^{2n-1}-1\right)B_{2n}}{(2n)!}}x^{2n-1}\\[5mu]&=x^{-1}+{\frac {1}{6}}x+{\frac {7}{360}}x^{3}+{\frac {31}{15120}}x^{5}+\cdots ,\qquad {\text{for }}0<|x|<\pi .\end{aligned}}}"></dd></dl> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\sec x&=\sum _{n=0}^{\infty }{\frac {U_{2n}}{(2n)!}}x^{2n}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}E_{2n}}{(2n)!}}x^{2n}\\[5mu]&=1+{\frac {1}{2}}x^{2}+{\frac {5}{24}}x^{4}+{\frac {61}{720}}x^{6}+\cdots ,\qquad {\text{for }}|x|<{\frac {\pi }{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="0.578em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>sec</mi> <mo>⁡</mo> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> <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> <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> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> </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> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>24</mn> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>61</mn> <mn>720</mn> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>,</mo> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>for </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sec x&=\sum _{n=0}^{\infty }{\frac {U_{2n}}{(2n)!}}x^{2n}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}E_{2n}}{(2n)!}}x^{2n}\\[5mu]&=1+{\frac {1}{2}}x^{2}+{\frac {5}{24}}x^{4}+{\frac {61}{720}}x^{6}+\cdots ,\qquad {\text{for }}|x|<{\frac {\pi }{2}}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e8d1437d98b6b4d05d4da50fe2e18bd39a7ff9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:58.4ex; height:13.176ex;" alt="{\displaystyle {\begin{aligned}\sec x&=\sum _{n=0}^{\infty }{\frac {U_{2n}}{(2n)!}}x^{2n}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}E_{2n}}{(2n)!}}x^{2n}\\[5mu]&=1+{\frac {1}{2}}x^{2}+{\frac {5}{24}}x^{4}+{\frac {61}{720}}x^{6}+\cdots ,\qquad {\text{for }}|x|<{\frac {\pi }{2}}.\end{aligned}}}"></dd></dl> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\cot x&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}2^{2n}B_{2n}}{(2n)!}}x^{2n-1}\\[5mu]&=x^{-1}-{\frac {1}{3}}x-{\frac {1}{45}}x^{3}-{\frac {2}{945}}x^{5}-\cdots ,\qquad {\text{for }}0<|x|<\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="0.578em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>cot</mi> <mo>⁡</mo> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> </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> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mi>x</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>45</mn> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>945</mn> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>−</mo> <mo>⋯</mo> <mo>,</mo> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>for </mtext> </mrow> <mn>0</mn> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>π</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\cot x&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}2^{2n}B_{2n}}{(2n)!}}x^{2n-1}\\[5mu]&=x^{-1}-{\frac {1}{3}}x-{\frac {1}{45}}x^{3}-{\frac {2}{945}}x^{5}-\cdots ,\qquad {\text{for }}0<|x|<\pi .\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c6645779d6fe606694a0ac157fcdee271c7e795" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:63.389ex; height:13.176ex;" alt="{\displaystyle {\begin{aligned}\cot x&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}2^{2n}B_{2n}}{(2n)!}}x^{2n-1}\\[5mu]&=x^{-1}-{\frac {1}{3}}x-{\frac {1}{45}}x^{3}-{\frac {2}{945}}x^{5}-\cdots ,\qquad {\text{for }}0<|x|<\pi .\end{aligned}}}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Continued_fraction_expansion">Continued fraction expansion</h3>[<a href="/w/index.php?title=Trigonometric_functions&action=edit&section=10" title="Edit section: Continued fraction expansion">edit</a>]</div> The following <a href="/wiki/Continued_fraction" title="Continued fraction">continued fractions</a> are valid in the whole complex plane: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin x={\cfrac {x}{1+{\cfrac {x^{2}}{2\cdot 3-x^{2}+{\cfrac {2\cdot 3x^{2}}{4\cdot 5-x^{2}+{\cfrac {4\cdot 5x^{2}}{6\cdot 7-x^{2}+\ddots }}}}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>⋅</mo> <mn>3</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>⋅</mo> <mn>3</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mo>⋅</mo> <mn>5</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mo>⋅</mo> <mn>5</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> <mo>⋅</mo> <mn>7</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>⋱</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin x={\cfrac {x}{1+{\cfrac {x^{2}}{2\cdot 3-x^{2}+{\cfrac {2\cdot 3x^{2}}{4\cdot 5-x^{2}+{\cfrac {4\cdot 5x^{2}}{6\cdot 7-x^{2}+\ddots }}}}}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2dd5fcbfe285b52286c6cbbac551a43a948c8c43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -15.338ex; width:54.203ex; height:19.509ex;" alt="{\displaystyle \sin x={\cfrac {x}{1+{\cfrac {x^{2}}{2\cdot 3-x^{2}+{\cfrac {2\cdot 3x^{2}}{4\cdot 5-x^{2}+{\cfrac {4\cdot 5x^{2}}{6\cdot 7-x^{2}+\ddots }}}}}}}}}"></dd></dl> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos x={\cfrac {1}{1+{\cfrac {x^{2}}{1\cdot 2-x^{2}+{\cfrac {1\cdot 2x^{2}}{3\cdot 4-x^{2}+{\cfrac {3\cdot 4x^{2}}{5\cdot 6-x^{2}+\ddots }}}}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>⋅</mo> <mn>2</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>⋅</mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>⋅</mo> <mn>4</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>⋅</mo> <mn>4</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> <mo>⋅</mo> <mn>6</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>⋱</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos x={\cfrac {1}{1+{\cfrac {x^{2}}{1\cdot 2-x^{2}+{\cfrac {1\cdot 2x^{2}}{3\cdot 4-x^{2}+{\cfrac {3\cdot 4x^{2}}{5\cdot 6-x^{2}+\ddots }}}}}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/434dc975c06321a5d4b158c8307bbab4785453e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -15.338ex; width:54.459ex; height:19.509ex;" alt="{\displaystyle \cos x={\cfrac {1}{1+{\cfrac {x^{2}}{1\cdot 2-x^{2}+{\cfrac {1\cdot 2x^{2}}{3\cdot 4-x^{2}+{\cfrac {3\cdot 4x^{2}}{5\cdot 6-x^{2}+\ddots }}}}}}}}}"></dd></dl> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan x={\cfrac {x}{1-{\cfrac {x^{2}}{3-{\cfrac {x^{2}}{5-{\cfrac {x^{2}}{7-\ddots }}}}}}}}={\cfrac {1}{{\cfrac {1}{x}}-{\cfrac {1}{{\cfrac {3}{x}}-{\cfrac {1}{{\cfrac {5}{x}}-{\cfrac {1}{{\cfrac {7}{x}}-\ddots }}}}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> <mo>−</mo> <mo>⋱</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>−</mo> <mo>⋱</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan x={\cfrac {x}{1-{\cfrac {x^{2}}{3-{\cfrac {x^{2}}{5-{\cfrac {x^{2}}{7-\ddots }}}}}}}}={\cfrac {1}{{\cfrac {1}{x}}-{\cfrac {1}{{\cfrac {3}{x}}-{\cfrac {1}{{\cfrac {5}{x}}-{\cfrac {1}{{\cfrac {7}{x}}-\ddots }}}}}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/904f76221938b08b8b0840e6bd8bb7a164e73f68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -17.338ex; width:59.956ex; height:21.509ex;" alt="{\displaystyle \tan x={\cfrac {x}{1-{\cfrac {x^{2}}{3-{\cfrac {x^{2}}{5-{\cfrac {x^{2}}{7-\ddots }}}}}}}}={\cfrac {1}{{\cfrac {1}{x}}-{\cfrac {1}{{\cfrac {3}{x}}-{\cfrac {1}{{\cfrac {5}{x}}-{\cfrac {1}{{\cfrac {7}{x}}-\ddots }}}}}}}}}"></dd></dl> The last one was used in the historically first <a href="/wiki/Proof_that_%CF%80_is_irrational" title="Proof that π is irrational">proof that π is irrational</a>.<a href="#cite_note-19">[18]</a> <div class="mw-heading mw-heading3"><h3 id="Partial_fraction_expansion">Partial fraction expansion</h3>[<a href="/w/index.php?title=Trigonometric_functions&action=edit&section=11" title="Edit section: Partial fraction expansion">edit</a>]</div> There is a series representation as <a href="/wiki/Partial_fraction_expansion" class="mw-redirect" title="Partial fraction expansion">partial fraction expansion</a> where just translated <a href="/wiki/Multiplicative_inverse" title="Multiplicative inverse">reciprocal functions</a> are summed up, such that the <a href="/wiki/Pole_(complex_analysis)" class="mw-redirect" title="Pole (complex analysis)">poles</a> of the cotangent function and the reciprocal functions match:<a href="#cite_note-Aigner_2000-20">[19]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \cot \pi x=\lim _{N\to \infty }\sum _{n=-N}^{N}{\frac {1}{x+n}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mi>cot</mi> <mo>⁡</mo> <mi>π</mi> <mi>x</mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi>N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>x</mi> <mo>+</mo> <mi>n</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi \cot \pi x=\lim _{N\to \infty }\sum _{n=-N}^{N}{\frac {1}{x+n}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ac00ba0889abc421eea10ceb59869fd45c62b04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:28.536ex; height:7.509ex;" alt="{\displaystyle \pi \cot \pi x=\lim _{N\to \infty }\sum _{n=-N}^{N}{\frac {1}{x+n}}.}"></dd></dl> This identity can be proved with the <a href="/wiki/Gustav_Herglotz" title="Gustav Herglotz">Herglotz</a> trick.<a href="#cite_note-Remmert_1991-21">[20]</a> Combining the (–n)th with the nth term lead to <a href="/wiki/Absolute_convergence" title="Absolute convergence">absolutely convergent</a> series: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \cot \pi x={\frac {1}{x}}+2x\sum _{n=1}^{\infty }{\frac {1}{x^{2}-n^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mi>cot</mi> <mo>⁡</mo> <mi>π</mi> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mo>+</mo> <mn>2</mn> <mi>x</mi> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi \cot \pi x={\frac {1}{x}}+2x\sum _{n=1}^{\infty }{\frac {1}{x^{2}-n^{2}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d659f067ce3fc400d3e268c009344a4d80825d31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:31.75ex; height:6.843ex;" alt="{\displaystyle \pi \cot \pi x={\frac {1}{x}}+2x\sum _{n=1}^{\infty }{\frac {1}{x^{2}-n^{2}}}.}"></dd></dl> Similarly, one can find a partial fraction expansion for the secant, cosecant and tangent functions: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \csc \pi x=\sum _{n=-\infty }^{\infty }{\frac {(-1)^{n}}{x+n}}={\frac {1}{x}}+2x\sum _{n=1}^{\infty }{\frac {(-1)^{n}}{x^{2}-n^{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mi>csc</mi> <mo>⁡</mo> <mi>π</mi> <mi>x</mi> <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> <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>x</mi> <mo>+</mo> <mi>n</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mo>+</mo> <mn>2</mn> <mi>x</mi> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow 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>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi \csc \pi x=\sum _{n=-\infty }^{\infty }{\frac {(-1)^{n}}{x+n}}={\frac {1}{x}}+2x\sum _{n=1}^{\infty }{\frac {(-1)^{n}}{x^{2}-n^{2}}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3829cac6320e9ab5354b78d45d001fa4bb4cd4d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:47.138ex; height:6.843ex;" alt="{\displaystyle \pi \csc \pi x=\sum _{n=-\infty }^{\infty }{\frac {(-1)^{n}}{x+n}}={\frac {1}{x}}+2x\sum _{n=1}^{\infty }{\frac {(-1)^{n}}{x^{2}-n^{2}}},}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi ^{2}\csc ^{2}\pi x=\sum _{n=-\infty }^{\infty }{\frac {1}{(x+n)^{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>csc</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>π</mi> <mi>x</mi> <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>x</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 \pi ^{2}\csc ^{2}\pi x=\sum _{n=-\infty }^{\infty }{\frac {1}{(x+n)^{2}}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83a202257921c286c1ef4115874599ca4561127f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:28.443ex; height:6.843ex;" alt="{\displaystyle \pi ^{2}\csc ^{2}\pi x=\sum _{n=-\infty }^{\infty }{\frac {1}{(x+n)^{2}}},}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \sec \pi x=\sum _{n=0}^{\infty }(-1)^{n}{\frac {(2n+1)}{(n+{\tfrac {1}{2}})^{2}-x^{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mi>sec</mi> <mo>⁡</mo> <mi>π</mi> <mi>x</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> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi \sec \pi x=\sum _{n=0}^{\infty }(-1)^{n}{\frac {(2n+1)}{(n+{\tfrac {1}{2}})^{2}-x^{2}}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fa62cf520135be3ad83191a7603e7636ca85908" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:35.665ex; height:7.176ex;" alt="{\displaystyle \pi \sec \pi x=\sum _{n=0}^{\infty }(-1)^{n}{\frac {(2n+1)}{(n+{\tfrac {1}{2}})^{2}-x^{2}}},}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \tan \pi x=2x\sum _{n=0}^{\infty }{\frac {1}{(n+{\tfrac {1}{2}})^{2}-x^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mi>tan</mi> <mo>⁡</mo> <mi>π</mi> <mi>x</mi> <mo>=</mo> <mn>2</mn> <mi>x</mi> <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> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi \tan \pi x=2x\sum _{n=0}^{\infty }{\frac {1}{(n+{\tfrac {1}{2}})^{2}-x^{2}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76de92ad9a05c77a7f620b472165a00f7978e0fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:33.311ex; height:7.176ex;" alt="{\displaystyle \pi \tan \pi x=2x\sum _{n=0}^{\infty }{\frac {1}{(n+{\tfrac {1}{2}})^{2}-x^{2}}}.}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Infinite_product_expansion">Infinite product expansion</h3>[<a href="/w/index.php?title=Trigonometric_functions&action=edit&section=12" title="Edit section: Infinite product expansion">edit</a>]</div> The following infinite product for the sine is due to <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a>, and is of great importance in complex analysis:<a href="#cite_note-22">[21]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin z=z\prod _{n=1}^{\infty }\left(1-{\frac {z^{2}}{n^{2}\pi ^{2}}}\right),\quad z\in \mathbb {C} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>z</mi> <mo>=</mo> <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> <mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="1em" /> <mi>z</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin z=z\prod _{n=1}^{\infty }\left(1-{\frac {z^{2}}{n^{2}\pi ^{2}}}\right),\quad z\in \mathbb {C} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32a6ecb11ce2f07f3b234a80773c49216892be72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:35.473ex; height:6.843ex;" alt="{\displaystyle \sin z=z\prod _{n=1}^{\infty }\left(1-{\frac {z^{2}}{n^{2}\pi ^{2}}}\right),\quad z\in \mathbb {C} .}"></dd></dl> This may be obtained from the partial fraction decomposition of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cot z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cot</mi> <mo>⁡</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cot z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/798b34c844311c1ca3a1b630e5e9187fa289a275" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.575ex; height:2.009ex;" alt="{\displaystyle \cot z}"> given above, which is the logarithmic derivative of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/840154d23e7487c8c0d9bef213611822d5b09463" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.331ex; height:2.176ex;" alt="{\displaystyle \sin z}">.<a href="#cite_note-23">[22]</a> From this, it can be deduced also that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos z=\prod _{n=1}^{\infty }\left(1-{\frac {z^{2}}{(n-1/2)^{2}\pi ^{2}}}\right),\quad z\in \mathbb {C} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>z</mi> <mo>=</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="1em" /> <mi>z</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos z=\prod _{n=1}^{\infty }\left(1-{\frac {z^{2}}{(n-1/2)^{2}\pi ^{2}}}\right),\quad z\in \mathbb {C} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f72c1f05c17f9f8a165f6837eb0bb5dc3e905d76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:42.391ex; height:6.843ex;" alt="{\displaystyle \cos z=\prod _{n=1}^{\infty }\left(1-{\frac {z^{2}}{(n-1/2)^{2}\pi ^{2}}}\right),\quad z\in \mathbb {C} .}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Euler's_formula_and_the_exponential_function">Euler's formula and the exponential function</h3>[<a href="/w/index.php?title=Trigonometric_functions&action=edit&section=13" title="Edit section: Euler's formula and the exponential function">edit</a>]</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/220px-Sinus_und_Kosinus_am_Einheitskreis_3.svg.png" decoding="async" width="220" height="215" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Sinus_und_Kosinus_am_Einheitskreis_3.svg/330px-Sinus_und_Kosinus_am_Einheitskreis_3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Sinus_und_Kosinus_am_Einheitskreis_3.svg/440px-Sinus_und_Kosinus_am_Einheitskreis_3.svg.png 2x" data-file-width="418" data-file-height="409" /></a><figcaption><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><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 )}"> and <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><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 )}"> are the real and imaginary part of <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><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 }}"> respectively.</figcaption></figure> <a href="/wiki/Euler%27s_formula" title="Euler's formula">Euler's formula</a> relates sine and cosine to the <a href="/wiki/Exponential_function" title="Exponential function">exponential function</a>: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{ix}=\cos x+i\sin x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>x</mi> </mrow> </msup> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{ix}=\cos x+i\sin x.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a0baf9cdc69dd49332b5a170ebd62f9636c17d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:19.999ex; height:2.843ex;" alt="{\displaystyle e^{ix}=\cos x+i\sin x.}"></dd></dl> This formula is commonly considered for real values of x, but it remains true for all complex values. Proof: Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{1}(x)=\cos x+i\sin x,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{1}(x)=\cos x+i\sin x,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/806434ff9eee1c64863cc2f13239cf662fe3ef18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.508ex; height:2.843ex;" alt="{\displaystyle f_{1}(x)=\cos x+i\sin x,}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{2}(x)=e^{ix}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>x</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{2}(x)=e^{ix}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f98d3c61e540e50b3149ab0f1f31f8773ef9ff3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.901ex; height:3.176ex;" alt="{\displaystyle f_{2}(x)=e^{ix}.}"> One has <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle df_{j}(x)/dx=if_{j}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mi>i</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle df_{j}(x)/dx=if_{j}(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0f32cbc3e855cb51f8b05126608528d77377c9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.201ex; height:3.009ex;" alt="{\displaystyle df_{j}(x)/dx=if_{j}(x)}"> for j = 1, 2. The <a href="/wiki/Quotient_rule" title="Quotient rule">quotient rule</a> implies thus that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d/dx\,(f_{1}(x)/f_{2}(x))=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>d</mi> <mi>x</mi> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d/dx\,(f_{1}(x)/f_{2}(x))=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd2386dc852ca2a6faad4d4395c2c344eaf6fc1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.209ex; height:2.843ex;" alt="{\displaystyle d/dx\,(f_{1}(x)/f_{2}(x))=0}">. Therefore, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{1}(x)/f_{2}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{1}(x)/f_{2}(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83e89dd002f58b2cfe96d65103325c0824bce0a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.827ex; height:2.843ex;" alt="{\displaystyle f_{1}(x)/f_{2}(x)}"> is a constant function, which equals 1, as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{1}(0)=f_{2}(0)=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{1}(0)=f_{2}(0)=1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/baf20bf5158ce5e47bea507742385a24ba68ca62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.337ex; height:2.843ex;" alt="{\displaystyle f_{1}(0)=f_{2}(0)=1.}"> This proves the formula. One has <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}e^{ix}&=\cos x+i\sin x\\[5pt]e^{-ix}&=\cos x-i\sin x.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.8em 0.3em" 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>x</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>x</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>−</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}e^{ix}&=\cos x+i\sin x\\[5pt]e^{-ix}&=\cos x-i\sin x.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d374fafbe34908c7766b67e4c51797589906940" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:22.029ex; height:7.509ex;" alt="{\displaystyle {\begin{aligned}e^{ix}&=\cos x+i\sin x\\[5pt]e^{-ix}&=\cos x-i\sin x.\end{aligned}}}"></dd></dl> Solving this <a href="/wiki/Linear_system" title="Linear system">linear system</a> in sine and cosine, one can express them in terms of the exponential function: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\sin x&={\frac {e^{ix}-e^{-ix}}{2i}}\\[5pt]\cos x&={\frac {e^{ix}+e^{-ix}}{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="0.8em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>x</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>x</mi> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>i</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> </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>x</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>x</mi> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sin x&={\frac {e^{ix}-e^{-ix}}{2i}}\\[5pt]\cos x&={\frac {e^{ix}+e^{-ix}}{2}}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/590e4a1bbe3ccdb7521fe06a6e5b56e538d4e729" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:19.927ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}\sin x&={\frac {e^{ix}-e^{-ix}}{2i}}\\[5pt]\cos x&={\frac {e^{ix}+e^{-ix}}{2}}.\end{aligned}}}"></dd></dl> When x is real, this may be rewritten as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos x=\operatorname {Re} \left(e^{ix}\right),\qquad \sin x=\operatorname {Im} \left(e^{ix}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mi>Re</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>x</mi> </mrow> </msup> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="2em" /> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mi>Im</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>x</mi> </mrow> </msup> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos x=\operatorname {Re} \left(e^{ix}\right),\qquad \sin x=\operatorname {Im} \left(e^{ix}\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ef70603cc5b854703e6bc8fb192f21a883bfe0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:37.348ex; height:3.343ex;" alt="{\displaystyle \cos x=\operatorname {Re} \left(e^{ix}\right),\qquad \sin x=\operatorname {Im} \left(e^{ix}\right).}"></dd></dl> Most <a href="/wiki/List_of_trigonometric_identities" title="List of trigonometric identities">trigonometric identities</a> can be proved by expressing trigonometric functions in terms of the complex exponential function by using above formulas, and then using the identity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{a+b}=e^{a}e^{b}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{a+b}=e^{a}e^{b}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b99c3f7d5d874674c43bcaf77044e546a79ac09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.474ex; height:2.676ex;" alt="{\displaystyle e^{a+b}=e^{a}e^{b}}"> for simplifying the result. Euler's formula can also be used to define the basic trigonometric function directly, as follows, using the language of <a href="/wiki/Topological_group" title="Topological group">topological groups</a>.<a href="#cite_note-24">[23]</a> The set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"> of complex numbers of unit modulus is a compact and connected topological group, which has a neighborhood of the identity that is homeomorphic to the real line. Therefore, it is isomorphic as a topological group to the one-dimensional torus group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} /\mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} /\mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f62160dc28538c41d74ffcfbca4a7b3c68693880" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.391ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} /\mathbb {Z} }">, via an isomorphism <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e:\mathbb {R} /\mathbb {Z} \to U.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">→</mo> <mi>U</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e:\mathbb {R} /\mathbb {Z} \to U.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c53a1a384ed8a7ea3b1e6642fc0892270e1321b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.455ex; height:2.843ex;" alt="{\displaystyle e:\mathbb {R} /\mathbb {Z} \to U.}"> In pedestrian terms <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e(t)=\exp(2\pi it)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>i</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e(t)=\exp(2\pi it)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12b2b39e93192773bca4a0e13c5f3d3b6a9a9d2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.329ex; height:2.843ex;" alt="{\displaystyle e(t)=\exp(2\pi it)}">, and this isomorphism is unique up to taking complex conjugates. For a nonzero real number <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><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}"> (the base), the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\mapsto e(t/a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo stretchy="false">↦</mo> <mi>e</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\mapsto e(t/a)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d7d5cbcb8e06365cad64575879b7abefbf86998" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.578ex; height:2.843ex;" alt="{\displaystyle t\mapsto e(t/a)}"> defines an isomorphism of the group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} /a\mathbb {Z} \to U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">→</mo> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} /a\mathbb {Z} \to U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0dd2e1d6caf8f89d54f1a6e25ababd8e97f483d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.017ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} /a\mathbb {Z} \to U}">. The real and imaginary parts of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e(t/a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e(t/a)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535855d6cd3853e2af10c5b7681032ecca664e50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.125ex; height:2.843ex;" alt="{\displaystyle e(t/a)}"> are the cosine and sine, where <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><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}"> is used as the base for measuring angles. For example, when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=2\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/213761e935138cf03ebb0f65eab478227702bc73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.823ex; height:2.176ex;" alt="{\displaystyle a=2\pi }">, we get the measure in radians, and the usual trigonometric functions. When <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=360}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mn>360</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=360}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f637a5ef55506e9a958663144d7c237b1d3bc153" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.816ex; height:2.176ex;" alt="{\displaystyle a=360}">, we get the sine and cosine of angles measured in degrees. Note that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=2\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/213761e935138cf03ebb0f65eab478227702bc73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.823ex; height:2.176ex;" alt="{\displaystyle a=2\pi }"> is the unique value at which the derivative <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{dt}}e(t/a)}"> <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>t</mi> </mrow> </mfrac> </mrow> <mi>e</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dt}}e(t/a)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41bb5f118bd1e7258557f8627d9d0a63ff8b170a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.016ex; height:5.509ex;" alt="{\displaystyle {\frac {d}{dt}}e(t/a)}"> becomes a <a href="/wiki/Unit_vector" title="Unit vector">unit vector</a> with positive imaginary part at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43469ec032d858feae5aa87029e22eaaf0109e9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.101ex; height:2.176ex;" alt="{\displaystyle t=0}">. This fact can, in turn, be used to define the constant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73efd1f6493490b058097060a572606d2c550a06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }">. <div class="mw-heading mw-heading3"><h3 id="Definition_via_integration">Definition via integration</h3>[<a href="/w/index.php?title=Trigonometric_functions&action=edit&section=14" title="Edit section: Definition via integration">edit</a>]</div> Another way to define the trigonometric functions in analysis is using integration.<a href="#cite_note-Hardy-12">[11]</a><a href="#cite_note-25">[24]</a> For a real number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}">, put <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta (t)=\int _{0}^{t}{\frac {d\tau }{1+\tau ^{2}}}=\arctan t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>τ</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta (t)=\int _{0}^{t}{\frac {d\tau }{1+\tau ^{2}}}=\arctan t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5446f05cb1a80b0650cad56dcc83d48792fc9e05" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:28.457ex; height:6.176ex;" alt="{\displaystyle \theta (t)=\int _{0}^{t}{\frac {d\tau }{1+\tau ^{2}}}=\arctan t}"> where this defines this inverse tangent function. Also, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"> is defined by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2}}\pi =\int _{0}^{\infty }{\frac {d\tau }{1+\tau ^{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> <mi>π</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>τ</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}\pi =\int _{0}^{\infty }{\frac {d\tau }{1+\tau ^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0795868f0c34777e87a400470a4e4efdce6c121d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.307ex; height:5.843ex;" alt="{\displaystyle {\frac {1}{2}}\pi =\int _{0}^{\infty }{\frac {d\tau }{1+\tau ^{2}}}}"> a definition that goes back to <a href="/wiki/Karl_Weierstrass" title="Karl Weierstrass">Karl Weierstrass</a>.<a href="#cite_note-26">[25]</a> On the interval <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\pi /2<\theta <\pi /2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo><</mo> <mi>θ</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 -\pi /2<\theta <\pi /2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e3e288e37620b8dfa7c8ee54537bbfa7424f22b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.409ex; height:2.843ex;" alt="{\displaystyle -\pi /2<\theta <\pi /2}">, the trigonometric functions are defined by inverting the relation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =\arctan t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =\arctan t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7d7bcc089626308d299719fe69b4312720ca6fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.882ex; height:2.176ex;" alt="{\displaystyle \theta =\arctan t}">. Thus we define the trigonometric functions by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan \theta =t,\quad \cos \theta =(1+t^{2})^{-1/2},\quad \sin \theta =t(1+t^{2})^{-1/2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>t</mi> <mo>,</mo> <mspace width="1em" /> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mspace width="1em" /> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>t</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan \theta =t,\quad \cos \theta =(1+t^{2})^{-1/2},\quad \sin \theta =t(1+t^{2})^{-1/2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26ece962d0beaf964c06d83ce78ad55185bb2953" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:54.812ex; height:3.343ex;" alt="{\displaystyle \tan \theta =t,\quad \cos \theta =(1+t^{2})^{-1/2},\quad \sin \theta =t(1+t^{2})^{-1/2}}"> where the point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (t,\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (t,\theta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/918d28e5a2f2d5652fe1f9dbfbc427c3a0928fba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.773ex; height:2.843ex;" alt="{\displaystyle (t,\theta )}"> is on the graph of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =\arctan t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =\arctan t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7d7bcc089626308d299719fe69b4312720ca6fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.882ex; height:2.176ex;" alt="{\displaystyle \theta =\arctan t}"> and the positive square root is taken. This defines the trigonometric functions on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-\pi /2,\pi /2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>,</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-\pi /2,\pi /2)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449f2d8ba994d85245c644d5584d7f49cf9699dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.965ex; height:2.843ex;" alt="{\displaystyle (-\pi /2,\pi /2)}">. The definition can be extended to all real numbers by first observing that, as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta \to \pi /2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo stretchy="false">→</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta \to \pi /2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a9d7c4d4350b7583273b48c66525cb856cfbac0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.361ex; height:2.843ex;" alt="{\displaystyle \theta \to \pi /2}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\to \infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a34d7a61899d577d950881b4a44888d43f3fa93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.777ex; height:2.009ex;" alt="{\displaystyle t\to \infty }">, and so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \theta =(1+t^{2})^{-1/2}\to 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mo stretchy="false">→</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \theta =(1+t^{2})^{-1/2}\to 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99debcb0dae28be096d176f71598a0657ec88639" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.146ex; height:3.343ex;" alt="{\displaystyle \cos \theta =(1+t^{2})^{-1/2}\to 0}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \theta =t(1+t^{2})^{-1/2}\to 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>t</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mo stretchy="false">→</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \theta =t(1+t^{2})^{-1/2}\to 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8eee2398e392f600847a9335d7e15d59c437bd97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.731ex; height:3.343ex;" alt="{\displaystyle \sin \theta =t(1+t^{2})^{-1/2}\to 1}">. Thus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/611e5c70de1d1cf4ebc3b70d2b5467f45d17a483" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.589ex; height:2.176ex;" alt="{\displaystyle \cos \theta }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efa733f6703578b0c3af870a3170b4ab0dd99c00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.333ex; height:2.176ex;" alt="{\displaystyle \sin \theta }"> are extended continuously so that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(\pi /2)=0,\sin(\pi /2)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(\pi /2)=0,\sin(\pi /2)=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/296446088b217fddabe4cc1e1bcb3d5dc250f291" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.455ex; height:2.843ex;" alt="{\displaystyle \cos(\pi /2)=0,\sin(\pi /2)=1}">. Now the conditions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(\theta +\pi )=-\cos(\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>+</mo> <mi>π</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(\theta +\pi )=-\cos(\theta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b046d35169f8e4cf8d9f4d4f0cc3ce8060de1d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.488ex; height:2.843ex;" alt="{\displaystyle \cos(\theta +\pi )=-\cos(\theta )}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(\theta +\pi )=-\sin(\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>+</mo> <mi>π</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(\theta +\pi )=-\sin(\theta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/661f015f38413d85bb8656ac7acd5ee461887cad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.977ex; height:2.843ex;" alt="{\displaystyle \sin(\theta +\pi )=-\sin(\theta )}"> define the sine and cosine as periodic functions with period <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73efd1f6493490b058097060a572606d2c550a06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }">, for all real numbers. Proving the basic properties of sine and cosine, including the fact that sine and cosine are analytic, one may first establish the addition formulae. First, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arctan s+\arctan t=\arctan {\frac {s+t}{1-st}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arctan</mi> <mo>⁡</mo> <mi>s</mi> <mo>+</mo> <mi>arctan</mi> <mo>⁡</mo> <mi>t</mi> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>s</mi> <mo>+</mo> <mi>t</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>s</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arctan s+\arctan t=\arctan {\frac {s+t}{1-st}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fc1259ba06342205c91a0ba57b5dda9c5b4c8db" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:35.198ex; height:5.343ex;" alt="{\displaystyle \arctan s+\arctan t=\arctan {\frac {s+t}{1-st}}}"> holds, provided <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arctan s+\arctan t\in (-\pi /2,\pi /2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arctan</mi> <mo>⁡</mo> <mi>s</mi> <mo>+</mo> <mi>arctan</mi> <mo>⁡</mo> <mi>t</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>,</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arctan s+\arctan t\in (-\pi /2,\pi /2)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b08b04c9630afd8704bb77fbac3fda07be1c1357" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.283ex; height:2.843ex;" alt="{\displaystyle \arctan s+\arctan t\in (-\pi /2,\pi /2)}">, since <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arctan s+\arctan t=\int _{-s}^{t}{\frac {d\tau }{1+\tau ^{2}}}=\int _{0}^{\frac {s+t}{1-st}}{\frac {d\tau }{1+\tau ^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arctan</mi> <mo>⁡</mo> <mi>s</mi> <mo>+</mo> <mi>arctan</mi> <mo>⁡</mo> <mi>t</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>τ</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>s</mi> <mo>+</mo> <mi>t</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>s</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>τ</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arctan s+\arctan t=\int _{-s}^{t}{\frac {d\tau }{1+\tau ^{2}}}=\int _{0}^{\frac {s+t}{1-st}}{\frac {d\tau }{1+\tau ^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d56067cebd2d043f8fe822649c0fa68d50896c4a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:49.67ex; height:7.176ex;" alt="{\displaystyle \arctan s+\arctan t=\int _{-s}^{t}{\frac {d\tau }{1+\tau ^{2}}}=\int _{0}^{\frac {s+t}{1-st}}{\frac {d\tau }{1+\tau ^{2}}}}"> after the substitution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau \to {\frac {s+\tau }{1-s\tau }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>s</mi> <mo>+</mo> <mi>τ</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>s</mi> <mi>τ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau \to {\frac {s+\tau }{1-s\tau }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1c07eb3e128fe13ce4e22a8fd7dd80d145bdd5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.947ex; height:5.176ex;" alt="{\displaystyle \tau \to {\frac {s+\tau }{1-s\tau }}}">. In particular, the limiting case as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s\to \infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7834a6c0df76be4b2d2901b6fff0b665a66f9625" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.028ex; height:1.843ex;" alt="{\displaystyle s\to \infty }"> gives <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arctan t+{\frac {\pi }{2}}=\arctan(-1/t),\quad t\in (-\infty ,0).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arctan</mi> <mo>⁡</mo> <mi>t</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <mi>t</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arctan t+{\frac {\pi }{2}}=\arctan(-1/t),\quad t\in (-\infty ,0).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b61f78899b2319e6ebb87d620ddf89d2cade4bc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:44.869ex; height:4.676ex;" alt="{\displaystyle \arctan t+{\frac {\pi }{2}}=\arctan(-1/t),\quad t\in (-\infty ,0).}"> Thus we have <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \left(\theta +{\frac {\pi }{2}}\right)={\frac {-1}{t{\sqrt {1+(-1/t)^{2}}}}}={\frac {-1}{\sqrt {1+t^{2}}}}=-\cos(\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>θ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \left(\theta +{\frac {\pi }{2}}\right)={\frac {-1}{t{\sqrt {1+(-1/t)^{2}}}}}={\frac {-1}{\sqrt {1+t^{2}}}}=-\cos(\theta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b85b46b69b48c8ae4f552bf6d5d881a979d6d132" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:54.127ex; height:6.509ex;" alt="{\displaystyle \sin \left(\theta +{\frac {\pi }{2}}\right)={\frac {-1}{t{\sqrt {1+(-1/t)^{2}}}}}={\frac {-1}{\sqrt {1+t^{2}}}}=-\cos(\theta )}"> and <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \left(\theta +{\frac {\pi }{2}}\right)={\frac {1}{\sqrt {1+(-1/t)^{2}}}}={\frac {t}{\sqrt {1+t^{2}}}}=\sin(\theta ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>θ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>t</mi> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \left(\theta +{\frac {\pi }{2}}\right)={\frac {1}{\sqrt {1+(-1/t)^{2}}}}={\frac {t}{\sqrt {1+t^{2}}}}=\sin(\theta ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e53e6e4227feb92609ed79e778d11336903fcda8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:51.739ex; height:6.509ex;" alt="{\displaystyle \cos \left(\theta +{\frac {\pi }{2}}\right)={\frac {1}{\sqrt {1+(-1/t)^{2}}}}={\frac {t}{\sqrt {1+t^{2}}}}=\sin(\theta ).}"> So the sine and cosine functions are related by translation over a quarter period <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi /2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi /2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b44e3d874a0b229fded7ffce67a0677dd5b8b67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.657ex; height:2.843ex;" alt="{\displaystyle \pi /2}">. <div class="mw-heading mw-heading3"><h3 id="Definitions_using_functional_equations">Definitions using functional equations</h3>[<a href="/w/index.php?title=Trigonometric_functions&action=edit&section=15" title="Edit section: Definitions using functional equations">edit</a>]</div> One can also define the trigonometric functions using various <a href="/wiki/Functional_equation" title="Functional equation">functional equations</a>. For example,<a href="#cite_note-Kannappan_2009-27">[26]</a> the sine and the cosine form the unique pair of <a href="/wiki/Continuous_function" title="Continuous function">continuous functions</a> that satisfy the difference formula <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(x-y)=\cos x\cos y+\sin x\sin y\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mi>cos</mi> <mo>⁡</mo> <mi>y</mi> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mi>sin</mi> <mo>⁡</mo> <mi>y</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(x-y)=\cos x\cos y+\sin x\sin y\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a70c9f04e8d46c409d6677070a30e7ba0eed0883" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.798ex; height:2.843ex;" alt="{\displaystyle \cos(x-y)=\cos x\cos y+\sin x\sin y\,}"></dd></dl> and the added condition <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<x\cos x<\sin x<x\quad {\text{ for }}\quad 0<x<1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>x</mi> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo><</mo> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo><</mo> <mi>x</mi> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> for </mtext> </mrow> <mspace width="1em" /> <mn>0</mn> <mo><</mo> <mi>x</mi> <mo><</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<x\cos x<\sin x<x\quad {\text{ for }}\quad 0<x<1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63db02f4b47b708e01b5392e6668ddb7efc1ce72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:41.995ex; height:2.176ex;" alt="{\displaystyle 0<x\cos x<\sin x<x\quad {\text{ for }}\quad 0<x<1.}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="In_the_complex_plane">In the complex plane</h3>[<a href="/w/index.php?title=Trigonometric_functions&action=edit&section=16" title="Edit section: In the complex plane">edit</a>]</div> The sine and cosine of a <a href="/wiki/Complex_number" title="Complex number">complex number</a> <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><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}"> 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 follows: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\sin z&=\sin x\cosh y+i\cos x\sinh y\\[5pt]\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="0.8em 0.3em" 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> </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> </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\\[5pt]\cos z&=\cos x\cosh y-i\sin x\sinh y\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1646655eab602e234f42df85cae241ffbb867cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:34.278ex; height:7.176ex;" alt="{\displaystyle {\begin{aligned}\sin z&=\sin x\cosh y+i\cos x\sinh y\\[5pt]\cos z&=\cos x\cosh y-i\sin x\sinh y\end{aligned}}}"></dd></dl> By taking advantage of <a href="/wiki/Domain_coloring" title="Domain coloring">domain coloring</a>, it is possible to graph the trigonometric functions as complex-valued functions. Various features unique to the complex functions can be seen from the graph; for example, the sine and cosine functions can be seen to be unbounded as the imaginary part of <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><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}"> becomes larger (since the color white represents infinity), and the fact that the functions contain simple <a href="/wiki/Zeros_and_poles" title="Zeros and poles">zeros or poles</a> is apparent from the fact that the hue cycles around each zero or pole exactly once. Comparing these graphs with those of the corresponding Hyperbolic functions highlights the relationships between the two. <table style="text-align:center"> <caption>Trigonometric functions in the complex plane </caption> <tbody><tr> <td><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Trig-sin.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Trig-sin.png/220px-Trig-sin.png" decoding="async" width="220" height="218" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Trig-sin.png/330px-Trig-sin.png 1.5x, //upload.wikimedia.org/wikipedia/commons/0/0f/Trig-sin.png 2x" data-file-width="419" data-file-height="416" /></a><figcaption></figcaption></figure> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin z\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>z</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin z\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b16de9fc6e761481f5aaa8396cdb582c8fb9d536" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.718ex; height:2.176ex;" alt="{\displaystyle \sin z\,}"> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Trig-cos.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Trig-cos.png/220px-Trig-cos.png" decoding="async" width="220" height="218" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Trig-cos.png/330px-Trig-cos.png 1.5x, //upload.wikimedia.org/wikipedia/commons/e/ee/Trig-cos.png 2x" data-file-width="419" data-file-height="416" /></a><figcaption></figcaption></figure> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos z\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>z</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos z\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9eca9983bcba495eed5b5bd6131dd275720bebda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.973ex; height:1.676ex;" alt="{\displaystyle \cos z\,}"> </td> <td><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Trig-tan.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/Trig-tan.png/220px-Trig-tan.png" decoding="async" width="220" height="218" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/Trig-tan.png/330px-Trig-tan.png 1.5x, //upload.wikimedia.org/wikipedia/commons/8/80/Trig-tan.png 2x" data-file-width="419" data-file-height="416" /></a><figcaption></figcaption></figure> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan z\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mi>z</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan z\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3490be6289a83b5c6e593bf3876b0044daf993e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.222ex; height:2.009ex;" alt="{\displaystyle \tan z\,}"> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Trig-cot.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/27/Trig-cot.png/220px-Trig-cot.png" decoding="async" width="220" height="218" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/27/Trig-cot.png/330px-Trig-cot.png 1.5x, //upload.wikimedia.org/wikipedia/commons/2/27/Trig-cot.png 2x" data-file-width="419" data-file-height="416" /></a><figcaption></figcaption></figure> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cot z\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cot</mi> <mo>⁡</mo> <mi>z</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cot z\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3588e798a21d2cff6da8dccd73f88058e5ef94a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.962ex; height:2.009ex;" alt="{\displaystyle \cot z\,}"> </td> <td><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Trig-sec.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/09/Trig-sec.png/220px-Trig-sec.png" decoding="async" width="220" height="218" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/09/Trig-sec.png/330px-Trig-sec.png 1.5x, //upload.wikimedia.org/wikipedia/commons/0/09/Trig-sec.png 2x" data-file-width="419" data-file-height="416" /></a><figcaption></figcaption></figure> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sec z\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sec</mi> <mo>⁡</mo> <mi>z</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sec z\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a516162cf1d7a696142eefc96eaf98267eb23ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.843ex; height:1.676ex;" alt="{\displaystyle \sec z\,}"> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Trig-csc.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Trig-csc.png/220px-Trig-csc.png" decoding="async" width="220" height="218" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Trig-csc.png/330px-Trig-csc.png 1.5x, //upload.wikimedia.org/wikipedia/commons/d/d4/Trig-csc.png 2x" data-file-width="419" data-file-height="416" /></a><figcaption></figcaption></figure> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \csc z\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>csc</mi> <mo>⁡</mo> <mi>z</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \csc z\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d8270cf9f0494bf63bb5077e1cfa13e424b41d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.843ex; height:1.676ex;" alt="{\displaystyle \csc z\,}"> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Periodicity_and_asymptotes">Periodicity and asymptotes</h2>[<a href="/w/index.php?title=Trigonometric_functions&action=edit&section=17" title="Edit section: Periodicity and asymptotes">edit</a>]</div> The cosine and sine functions are <a href="/wiki/Periodic_function" title="Periodic function">periodic</a>, with period <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73efd1f6493490b058097060a572606d2c550a06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }">, which is the smallest positive period: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(z+2\pi )=\cos(z),\quad \sin(z+2\pi )=\sin(z).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</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>z</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</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>z</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(z+2\pi )=\cos(z),\quad \sin(z+2\pi )=\sin(z).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9df9f8b591c2d89940ac0b6de5caeb9e6532eb1f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.393ex; height:2.843ex;" alt="{\displaystyle \cos(z+2\pi )=\cos(z),\quad \sin(z+2\pi )=\sin(z).}"> Consequently, the secant and cosecant also have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73efd1f6493490b058097060a572606d2c550a06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }"> as their period. The functions sine and cosine also have semiperiods <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }">, and <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(z+\pi )=-\cos(z),\quad \sin(z+\pi )=-\sin(z).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo>+</mo> <mi>π</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo>+</mo> <mi>π</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(z+\pi )=-\cos(z),\quad \sin(z+\pi )=-\sin(z).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddb06c4f67f2c6c125787922f1a9610ee3708a2f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.459ex; height:2.843ex;" alt="{\displaystyle \cos(z+\pi )=-\cos(z),\quad \sin(z+\pi )=-\sin(z).}"> It therefore follows that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan(z+\pi )=\tan(z),\quad \cot(z+\pi )=\cot(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo>+</mo> <mi>π</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <mi>cot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo>+</mo> <mi>π</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>cot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan(z+\pi )=\tan(z),\quad \cot(z+\pi )=\cot(z)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc5819c039fd3768bdaefc96b74b22d2c96b10c3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.406ex; height:2.843ex;" alt="{\displaystyle \tan(z+\pi )=\tan(z),\quad \cot(z+\pi )=\cot(z)}"> as well as other identities such as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos ^{2}(z+\pi )=\cos ^{2}(z),\quad \sin ^{2}(z+\pi )=\sin(z),\quad \cos(z+\pi )\sin(z+\pi )=\cos(z)\sin(z).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo>+</mo> <mi>π</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>z</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo>+</mo> <mi>π</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo>+</mo> <mi>π</mi> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo>+</mo> <mi>π</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos ^{2}(z+\pi )=\cos ^{2}(z),\quad \sin ^{2}(z+\pi )=\sin(z),\quad \cos(z+\pi )\sin(z+\pi )=\cos(z)\sin(z).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da54d0d66fc362858cdc850c5942b788244b7eb4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:84.328ex; height:3.176ex;" alt="{\displaystyle \cos ^{2}(z+\pi )=\cos ^{2}(z),\quad \sin ^{2}(z+\pi )=\sin(z),\quad \cos(z+\pi )\sin(z+\pi )=\cos(z)\sin(z).}"> We also have <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(x+\pi /2)=-\sin(x),\quad \sin(x+\pi /2)=\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>+</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <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> <mspace width="1em" /> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(x+\pi /2)=-\sin(x),\quad \sin(x+\pi /2)=\cos(x).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23fe48f4ac089d1be79d83ff07cf4499e42b224c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:49.88ex; height:2.843ex;" alt="{\displaystyle \cos(x+\pi /2)=-\sin(x),\quad \sin(x+\pi /2)=\cos(x).}"> The function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(z)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/292b02242178488542ee5c526562b6f1cad812c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.753ex; height:2.843ex;" alt="{\displaystyle \sin(z)}"> has a unique zero (at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b92bfc06485cc90286474b14a516a68d8bfdd7b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.349ex; height:2.176ex;" alt="{\displaystyle z=0}">) in the strip <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\pi <\Re (z)<\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>π</mi> <mo><</mo> <mi mathvariant="normal">ℜ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo><</mo> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\pi <\Re (z)<\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/649051d7d3898df7ce373c645f05aa2b13391da8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.491ex; height:2.843ex;" alt="{\displaystyle -\pi <\Re (z)<\pi }">. The function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(z)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48e7d26bd35f069db0efa8925a0f93befc6d0566" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.009ex; height:2.843ex;" alt="{\displaystyle \cos(z)}"> has the pair of zeros <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=\pm \pi /2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mo>±</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=\pm \pi /2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a9052258e688c7bb8b277c42f6aed573dc2c8cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.652ex; height:2.843ex;" alt="{\displaystyle z=\pm \pi /2}"> in the same domain. Because of the periodicity, the zeros of sine are <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \mathbb {Z} =\left\{\dots ,-2\pi ,-\pi ,0,\pi ,2\pi ,\dots \right\}\subset \mathbb {C} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mo>…</mo> <mo>,</mo> <mo>−</mo> <mn>2</mn> <mi>π</mi> <mo>,</mo> <mo>−</mo> <mi>π</mi> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mi>π</mi> <mo>,</mo> <mn>2</mn> <mi>π</mi> <mo>,</mo> <mo>…</mo> </mrow> <mo>}</mo> </mrow> <mo>⊂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi \mathbb {Z} =\left\{\dots ,-2\pi ,-\pi ,0,\pi ,2\pi ,\dots \right\}\subset \mathbb {C} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1aa6c7c5e28602e478bb381f7a7c0e441935f441" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.198ex; height:2.843ex;" alt="{\displaystyle \pi \mathbb {Z} =\left\{\dots ,-2\pi ,-\pi ,0,\pi ,2\pi ,\dots \right\}\subset \mathbb {C} .}"> There zeros of cosine are <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\pi }{2}}+\pi \mathbb {Z} =\left\{\dots ,-{\frac {3\pi }{2}},-{\frac {\pi }{2}},{\frac {\pi }{2}},{\frac {3\pi }{2}},\dots \right\}\subset \mathbb {C} .}"> <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>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mo>…</mo> <mo>,</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mo>…</mo> </mrow> <mo>}</mo> </mrow> <mo>⊂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\pi }{2}}+\pi \mathbb {Z} =\left\{\dots ,-{\frac {3\pi }{2}},-{\frac {\pi }{2}},{\frac {\pi }{2}},{\frac {3\pi }{2}},\dots \right\}\subset \mathbb {C} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6503bde57134cab7820c6d0cce7ba26f7f2a40f4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:45.516ex; height:6.176ex;" alt="{\displaystyle {\frac {\pi }{2}}+\pi \mathbb {Z} =\left\{\dots ,-{\frac {3\pi }{2}},-{\frac {\pi }{2}},{\frac {\pi }{2}},{\frac {3\pi }{2}},\dots \right\}\subset \mathbb {C} .}"> All of the zeros are simple zeros, and each function has derivative <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bfeaa85da53ad1947d8000926cfea33827ef1e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.971ex; height:2.176ex;" alt="{\displaystyle \pm 1}"> at each of the zeros. The tangent function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan(z)=\sin(z)/\cos(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan(z)=\sin(z)/\cos(z)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9bdb665245c757d79927924b4b2bc87608b6de1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.667ex; height:2.843ex;" alt="{\displaystyle \tan(z)=\sin(z)/\cos(z)}"> has a simple zero at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b92bfc06485cc90286474b14a516a68d8bfdd7b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.349ex; height:2.176ex;" alt="{\displaystyle z=0}"> and vertical asymptotes at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=\pm \pi /2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mo>±</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=\pm \pi /2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a9052258e688c7bb8b277c42f6aed573dc2c8cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.652ex; height:2.843ex;" alt="{\displaystyle z=\pm \pi /2}">, where it has a simple pole of residue <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/704fb0427140d054dd267925495e78164fee9aac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}">. Again, owing to the periodicity, the zeros are all the integer multiples of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"> and the poles are odd multiples of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi /2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi /2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b44e3d874a0b229fded7ffce67a0677dd5b8b67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.657ex; height:2.843ex;" alt="{\displaystyle \pi /2}">, all having the same residue. The poles correspond to vertical asymptotes <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{x\to \pi ^{-}}\tan(x)=+\infty ,\quad \lim _{x\to \pi ^{+}}\tan(x)=-\infty .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> </mrow> </munder> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mspace width="1em" /> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mrow> </munder> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to \pi ^{-}}\tan(x)=+\infty ,\quad \lim _{x\to \pi ^{+}}\tan(x)=-\infty .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d393c37e4052b20b5a0fcd11a295e6d92d29858" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:41.694ex; height:4.343ex;" alt="{\displaystyle \lim _{x\to \pi ^{-}}\tan(x)=+\infty ,\quad \lim _{x\to \pi ^{+}}\tan(x)=-\infty .}"> The cotangent function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cot(z)=\cos(z)/\sin(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cot(z)=\cos(z)/\sin(z)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a73a2a84cc2cd29a0e965a14a21e9a859db7b66b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.407ex; height:2.843ex;" alt="{\displaystyle \cot(z)=\cos(z)/\sin(z)}"> has a simple pole of residue 1 at the integer multiples of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"> and simple zeros at odd multiples of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi /2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi /2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b44e3d874a0b229fded7ffce67a0677dd5b8b67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.657ex; height:2.843ex;" alt="{\displaystyle \pi /2}">. The poles correspond to vertical asymptotes <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{x\to 0^{-}}\cot(x)=-\infty ,\quad \lim _{x\to 0^{+}}\cot(x)=+\infty .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> </mrow> </munder> <mi>cot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mspace width="1em" /> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mrow> </munder> <mi>cot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to 0^{-}}\cot(x)=-\infty ,\quad \lim _{x\to 0^{+}}\cot(x)=+\infty .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e44020ac6feda4d05ba8769f27c6250a87ac213" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:40.93ex; height:4.343ex;" alt="{\displaystyle \lim _{x\to 0^{-}}\cot(x)=-\infty ,\quad \lim _{x\to 0^{+}}\cot(x)=+\infty .}"> <div class="mw-heading mw-heading2"><h2 id="Basic_identities">Basic identities</h2>[<a href="/w/index.php?title=Trigonometric_functions&action=edit&section=18" title="Edit section: Basic identities">edit</a>]</div> Many <a href="/wiki/Identity_(mathematics)" title="Identity (mathematics)">identities</a> interrelate the trigonometric functions. This section contains the most basic ones; for more identities, see <a href="/wiki/List_of_trigonometric_identities" title="List of trigonometric identities">List of trigonometric identities</a>. These identities may be proved geometrically from the unit-circle definitions or the right-angled-triangle definitions (although, for the latter definitions, care must be taken for angles that are not in the interval [0, π/2], see <a href="/wiki/Proofs_of_trigonometric_identities" title="Proofs of trigonometric identities">Proofs of trigonometric identities</a>). For non-geometrical proofs using only tools of <a href="/wiki/Calculus" title="Calculus">calculus</a>, one may use directly the differential equations, in a way that is similar to that of the <a href="#Euler's_formula_and_the_exponential_function">above proof</a> of Euler's identity. One can also use Euler's identity for expressing all trigonometric functions in terms of complex exponentials and using properties of the exponential function. <div class="mw-heading mw-heading3"><h3 id="Parity">Parity</h3>[<a href="/w/index.php?title=Trigonometric_functions&action=edit&section=19" title="Edit section: Parity">edit</a>]</div> The cosine and the secant are <a href="/wiki/Even_function" class="mw-redirect" title="Even function">even functions</a>; the other trigonometric functions are <a href="/wiki/Odd_function" class="mw-redirect" title="Odd function">odd functions</a>. That is: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\sin(-x)&=-\sin x\\\cos(-x)&=\cos x\\\tan(-x)&=-\tan x\\\cot(-x)&=-\cot x\\\csc(-x)&=-\csc x\\\sec(-x)&=\sec x.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>tan</mi> <mo>⁡</mo> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mi>cot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>cot</mi> <mo>⁡</mo> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mi>csc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>csc</mi> <mo>⁡</mo> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mi>sec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>sec</mi> <mo>⁡</mo> <mi>x</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sin(-x)&=-\sin x\\\cos(-x)&=\cos x\\\tan(-x)&=-\tan x\\\cot(-x)&=-\cot x\\\csc(-x)&=-\csc x\\\sec(-x)&=\sec x.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cac6e94beccce00404c42433ff64d402094913a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.671ex; width:19.428ex; height:18.509ex;" alt="{\displaystyle {\begin{aligned}\sin(-x)&=-\sin x\\\cos(-x)&=\cos x\\\tan(-x)&=-\tan x\\\cot(-x)&=-\cot x\\\csc(-x)&=-\csc x\\\sec(-x)&=\sec x.\end{aligned}}}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Periods">Periods</h3>[<a href="/w/index.php?title=Trigonometric_functions&action=edit&section=20" title="Edit section: Periods">edit</a>]</div> All trigonometric functions are <a href="/wiki/Periodic_function" title="Periodic function">periodic functions</a> of period 2π. This is the smallest period, except for the tangent and the cotangent, which have π as smallest period. This means that, for every integer k, one has <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{lrl}\sin(x+&2k\pi )&=\sin x\\\cos(x+&2k\pi )&=\cos x\\\tan(x+&k\pi )&=\tan x\\\cot(x+&k\pi )&=\cot x\\\csc(x+&2k\pi )&=\csc x\\\sec(x+&2k\pi )&=\sec x.\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left right left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> </mtd> <mtd> <mn>2</mn> <mi>k</mi> <mi>π</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> </mtd> <mtd> <mn>2</mn> <mi>k</mi> <mi>π</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> </mtd> <mtd> <mi>k</mi> <mi>π</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>=</mo> <mi>tan</mi> <mo>⁡</mo> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mi>cot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> </mtd> <mtd> <mi>k</mi> <mi>π</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>=</mo> <mi>cot</mi> <mo>⁡</mo> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mi>csc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> </mtd> <mtd> <mn>2</mn> <mi>k</mi> <mi>π</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>=</mo> <mi>csc</mi> <mo>⁡</mo> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mi>sec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> </mtd> <mtd> <mn>2</mn> <mi>k</mi> <mi>π</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>=</mo> <mi>sec</mi> <mo>⁡</mo> <mi>x</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{lrl}\sin(x+&2k\pi )&=\sin x\\\cos(x+&2k\pi )&=\cos x\\\tan(x+&k\pi )&=\tan x\\\cot(x+&k\pi )&=\cot x\\\csc(x+&2k\pi )&=\csc x\\\sec(x+&2k\pi )&=\sec x.\end{array}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66e396d87b8ac1334a550695386ac4f249afd30c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.338ex; width:25.207ex; height:19.843ex;" alt="{\displaystyle {\begin{array}{lrl}\sin(x+&2k\pi )&=\sin x\\\cos(x+&2k\pi )&=\cos x\\\tan(x+&k\pi )&=\tan x\\\cot(x+&k\pi )&=\cot x\\\csc(x+&2k\pi )&=\csc x\\\sec(x+&2k\pi )&=\sec x.\end{array}}}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Pythagorean_identity">Pythagorean identity</h3>[<a href="/w/index.php?title=Trigonometric_functions&action=edit&section=21" title="Edit section: Pythagorean identity">edit</a>]</div> The Pythagorean identity, is the expression of the <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a> in terms of trigonometric functions. It is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin ^{2}x+\cos ^{2}x=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin ^{2}x+\cos ^{2}x=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d1938e6828e597c076248d3ec430e0a7e5f98c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.61ex; height:2.843ex;" alt="{\displaystyle \sin ^{2}x+\cos ^{2}x=1}">.</dd></dl> Dividing through by either <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos ^{2}x}"> <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>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos ^{2}x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc33e5f302f647fa86db284b6dd1b7b943cb80a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.882ex; height:2.676ex;" alt="{\displaystyle \cos ^{2}x}"> or <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> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin ^{2}x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc0d6ba6bb181219b776ab25be991303f9e07d0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.627ex; height:2.676ex;" alt="{\displaystyle \sin ^{2}x}"> gives <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan ^{2}x+1=\sec ^{2}x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo>=</mo> <msup> <mi>sec</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan ^{2}x+1=\sec ^{2}x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0e1f4cc39227f4ec9aef940fb7f7fd733ce72b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.984ex; height:2.843ex;" alt="{\displaystyle \tan ^{2}x+1=\sec ^{2}x}"></dd></dl> and <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+\cot ^{2}x=\csc ^{2}x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <msup> <mi>cot</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <msup> <mi>csc</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+\cot ^{2}x=\csc ^{2}x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da61d3108247a5b574a90589cbdeb2e4f65bfba0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.724ex; height:2.843ex;" alt="{\displaystyle 1+\cot ^{2}x=\csc ^{2}x}">.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Sum_and_difference_formulas">Sum and difference formulas</h3>[<a href="/w/index.php?title=Trigonometric_functions&action=edit&section=22" title="Edit section: Sum and difference formulas">edit</a>]</div> The sum and difference formulas allow expanding the sine, the cosine, and the tangent of a sum or a difference of two angles in terms of sines and cosines and tangents of the angles themselves. These can be derived geometrically, using arguments that date to <a href="/wiki/Ptolemy" title="Ptolemy">Ptolemy</a>. One can also produce them algebraically using <a href="/wiki/Euler%27s_formula" title="Euler's formula">Euler's formula</a>. <dl><dt>Sum</dt> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\sin \left(x+y\right)&=\sin x\cos y+\cos x\sin y,\\[5mu]\cos \left(x+y\right)&=\cos x\cos y-\sin x\sin y,\\[5mu]\tan(x+y)&={\frac {\tan x+\tan y}{1-\tan x\tan 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="0.578em 0.578em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>+</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mi>cos</mi> <mo>⁡</mo> <mi>y</mi> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mi>sin</mi> <mo>⁡</mo> <mi>y</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>+</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mi>cos</mi> <mo>⁡</mo> <mi>y</mi> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mi>sin</mi> <mo>⁡</mo> <mi>y</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>tan</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mi>tan</mi> <mo>⁡</mo> <mi>y</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>tan</mi> <mo>⁡</mo> <mi>x</mi> <mi>tan</mi> <mo>⁡</mo> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sin \left(x+y\right)&=\sin x\cos y+\cos x\sin y,\\[5mu]\cos \left(x+y\right)&=\cos x\cos y-\sin x\sin y,\\[5mu]\tan(x+y)&={\frac {\tan x+\tan y}{1-\tan x\tan y}}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a94648d4600a711a8851dfaea622a269be4eda5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:37.058ex; height:13.176ex;" alt="{\displaystyle {\begin{aligned}\sin \left(x+y\right)&=\sin x\cos y+\cos x\sin y,\\[5mu]\cos \left(x+y\right)&=\cos x\cos y-\sin x\sin y,\\[5mu]\tan(x+y)&={\frac {\tan x+\tan y}{1-\tan x\tan y}}.\end{aligned}}}"></dd> <dt>Difference</dt> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\sin \left(x-y\right)&=\sin x\cos y-\cos x\sin y,\\[5mu]\cos \left(x-y\right)&=\cos x\cos y+\sin x\sin y,\\[5mu]\tan(x-y)&={\frac {\tan x-\tan y}{1+\tan x\tan 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="0.578em 0.578em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mi>cos</mi> <mo>⁡</mo> <mi>y</mi> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mi>sin</mi> <mo>⁡</mo> <mi>y</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mi>cos</mi> <mo>⁡</mo> <mi>y</mi> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mi>sin</mi> <mo>⁡</mo> <mi>y</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>tan</mi> <mo>⁡</mo> <mi>x</mi> <mo>−</mo> <mi>tan</mi> <mo>⁡</mo> <mi>y</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>tan</mi> <mo>⁡</mo> <mi>x</mi> <mi>tan</mi> <mo>⁡</mo> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sin \left(x-y\right)&=\sin x\cos y-\cos x\sin y,\\[5mu]\cos \left(x-y\right)&=\cos x\cos y+\sin x\sin y,\\[5mu]\tan(x-y)&={\frac {\tan x-\tan y}{1+\tan x\tan y}}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a627a03bba700c34bee8de20cfa09d78b127716" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:37.058ex; height:13.176ex;" alt="{\displaystyle {\begin{aligned}\sin \left(x-y\right)&=\sin x\cos y-\cos x\sin y,\\[5mu]\cos \left(x-y\right)&=\cos x\cos y+\sin x\sin y,\\[5mu]\tan(x-y)&={\frac {\tan x-\tan y}{1+\tan x\tan y}}.\end{aligned}}}"></dd></dl> When the two angles are equal, the sum formulas reduce to simpler equations known as the <a href="/wiki/Double-angle_formulae" class="mw-redirect" title="Double-angle formulae">double-angle formulae</a>. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\sin 2x&=2\sin x\cos x={\frac {2\tan x}{1+\tan ^{2}x}},\\[5mu]\cos 2x&=\cos ^{2}x-\sin ^{2}x=2\cos ^{2}x-1=1-2\sin ^{2}x={\frac {1-\tan ^{2}x}{1+\tan ^{2}x}},\\[5mu]\tan 2x&={\frac {2\tan x}{1-\tan ^{2}x}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.578em 0.578em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mn>2</mn> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>tan</mi> <mo>⁡</mo> <mi>x</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mn>2</mn> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>x</mi> <mo>−</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mn>2</mn> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>x</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>tan</mi> <mo>⁡</mo> <mn>2</mn> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>tan</mi> <mo>⁡</mo> <mi>x</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sin 2x&=2\sin x\cos x={\frac {2\tan x}{1+\tan ^{2}x}},\\[5mu]\cos 2x&=\cos ^{2}x-\sin ^{2}x=2\cos ^{2}x-1=1-2\sin ^{2}x={\frac {1-\tan ^{2}x}{1+\tan ^{2}x}},\\[5mu]\tan 2x&={\frac {2\tan x}{1-\tan ^{2}x}}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d631768e76acdd703625fbcaf9cdf8b8e3e9200f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.005ex; width:67.963ex; height:19.176ex;" alt="{\displaystyle {\begin{aligned}\sin 2x&=2\sin x\cos x={\frac {2\tan x}{1+\tan ^{2}x}},\\[5mu]\cos 2x&=\cos ^{2}x-\sin ^{2}x=2\cos ^{2}x-1=1-2\sin ^{2}x={\frac {1-\tan ^{2}x}{1+\tan ^{2}x}},\\[5mu]\tan 2x&={\frac {2\tan x}{1-\tan ^{2}x}}.\end{aligned}}}"></dd></dl> These identities can be used to derive the <a href="/wiki/Product-to-sum_identities" class="mw-redirect" title="Product-to-sum identities">product-to-sum identities</a>. By setting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=\tan {\tfrac {1}{2}}\theta ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>θ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=\tan {\tfrac {1}{2}}\theta ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5443ee3becc0fed5c9756a11a95ec32abdcaa8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:11.08ex; height:3.509ex;" alt="{\displaystyle t=\tan {\tfrac {1}{2}}\theta ,}"> all trigonometric functions of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"> can be expressed as <a href="/wiki/Rational_fraction" class="mw-redirect" title="Rational fraction">rational fractions</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}">: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\sin \theta &={\frac {2t}{1+t^{2}}},\\[5mu]\cos \theta &={\frac {1-t^{2}}{1+t^{2}}},\\[5mu]\tan \theta &={\frac {2t}{1-t^{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="0.578em 0.578em 0.3em" 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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>t</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>t</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sin \theta &={\frac {2t}{1+t^{2}}},\\[5mu]\cos \theta &={\frac {1-t^{2}}{1+t^{2}}},\\[5mu]\tan \theta &={\frac {2t}{1-t^{2}}}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9eb5a515edf456acce4c943b43121632bef4d27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.838ex; width:16.067ex; height:18.843ex;" alt="{\displaystyle {\begin{aligned}\sin \theta &={\frac {2t}{1+t^{2}}},\\[5mu]\cos \theta &={\frac {1-t^{2}}{1+t^{2}}},\\[5mu]\tan \theta &={\frac {2t}{1-t^{2}}}.\end{aligned}}}"></dd></dl> Together with <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\theta ={\frac {2}{1+t^{2}}}\,dt,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\theta ={\frac {2}{1+t^{2}}}\,dt,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73140a7504e0e72b1c0b3c85dbc1465ac65aa5b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.227ex; height:5.676ex;" alt="{\displaystyle d\theta ={\frac {2}{1+t^{2}}}\,dt,}"></dd></dl> this is the <a href="/wiki/Tangent_half-angle_substitution" title="Tangent half-angle substitution">tangent half-angle substitution</a>, which reduces the computation of <a href="/wiki/Integral" title="Integral">integrals</a> and <a href="/wiki/Antiderivative" title="Antiderivative">antiderivatives</a> of trigonometric functions to that of rational fractions. <div class="mw-heading mw-heading3"><h3 id="Derivatives_and_antiderivatives">Derivatives and antiderivatives</h3>[<a href="/w/index.php?title=Trigonometric_functions&action=edit&section=23" title="Edit section: Derivatives and antiderivatives">edit</a>]</div> The <a href="/wiki/Derivative" title="Derivative">derivatives</a> of trigonometric functions result from those of sine and cosine by applying the <a href="/wiki/Quotient_rule" title="Quotient rule">quotient rule</a>. The values given for the <a href="/wiki/Antiderivative" title="Antiderivative">antiderivatives</a> in the following table can be verified by differentiating them. The number C is a <a href="/wiki/Constant_of_integration" title="Constant of integration">constant of integration</a>. <table class="wikitable" style="text-align: center;"> <tbody><tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></th> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f'(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f'(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0cd7d7c75340e779d82658e19d1720ce84ab127" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.144ex; height:3.009ex;" alt="{\displaystyle f'(x)}"></th> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \int f(x)\,dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>∫</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \int f(x)\,dx}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfc65d54f82712b31932316909fb9ed3aed0e56c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.155ex; height:3.176ex;" alt="{\textstyle \int f(x)\,dx}"> </th></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09b4b55580d6a821a07ad9fe35be88976917b10b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.572ex; height:2.176ex;" alt="{\displaystyle \sin x}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/184ba70c3a71df25a25c09f34cd7f8175a9b5280" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.828ex; height:1.676ex;" alt="{\displaystyle \cos x}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\cos x+C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\cos x+C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0404cc47799b9e733668119b4eb64eca96effa8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.63ex; height:2.343ex;" alt="{\displaystyle -\cos x+C}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/184ba70c3a71df25a25c09f34cd7f8175a9b5280" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.828ex; height:1.676ex;" alt="{\displaystyle \cos x}"></td> <td><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> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\sin x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/506006811ed806ee55fb1db8a7c1060c012fa313" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.768ex; height:2.343ex;" alt="{\displaystyle -\sin x}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin x+C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin x+C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85ed3e6c28cc2694301b1f6e37fc67b6150c2894" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.179ex; height:2.343ex;" alt="{\displaystyle \sin x+C}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3983a722002d77dd3d0babab871c50488aef9f4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.076ex; height:2.009ex;" alt="{\displaystyle \tan x}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sec ^{2}x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>sec</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sec ^{2}x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00967000e7758a9ca4dbca1caf30ab3d1ee23b9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.752ex; height:2.676ex;" alt="{\displaystyle \sec ^{2}x}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln \left|\sec x\right|+C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>|</mo> <mrow> <mi>sec</mi> <mo>⁡</mo> <mi>x</mi> </mrow> <mo>|</mo> </mrow> <mo>+</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln \left|\sec x\right|+C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83451b94bea8c22c437b184e33aca0f6b229cc1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.538ex; height:2.843ex;" alt="{\displaystyle \ln \left|\sec x\right|+C}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \csc x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>csc</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \csc x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0e0fc784f0e9c8c3346f7891568f4ab216c1155" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.698ex; height:1.676ex;" alt="{\displaystyle \csc x}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\csc x\cot x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>csc</mi> <mo>⁡</mo> <mi>x</mi> <mi>cot</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\csc x\cot x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c3d54c37b1e9075f6abe5818d4c708880737a36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.096ex; height:2.176ex;" alt="{\displaystyle -\csc x\cot x}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln \left|\csc x-\cot x\right|+C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>|</mo> <mrow> <mi>csc</mi> <mo>⁡</mo> <mi>x</mi> <mo>−</mo> <mi>cot</mi> <mo>⁡</mo> <mi>x</mi> </mrow> <mo>|</mo> </mrow> <mo>+</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln \left|\csc x-\cot x\right|+C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9ffaf6f1abf8a76102ed43a7469c9b007160e2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.194ex; height:2.843ex;" alt="{\displaystyle \ln \left|\csc x-\cot x\right|+C}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sec x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sec</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sec x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32c8904bcf644b6f3ab015742047994329f741bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.698ex; height:1.676ex;" alt="{\displaystyle \sec x}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sec x\tan x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sec</mi> <mo>⁡</mo> <mi>x</mi> <mi>tan</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sec x\tan x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80c5bfca686cdb169d8de98a75c7454be33428f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.161ex; height:2.009ex;" alt="{\displaystyle \sec x\tan x}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln \left|\sec x+\tan x\right|+C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>|</mo> <mrow> <mi>sec</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mi>tan</mi> <mo>⁡</mo> <mi>x</mi> </mrow> <mo>|</mo> </mrow> <mo>+</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln \left|\sec x+\tan x\right|+C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abb06f267a6fdd61b10d81a1336d7835cdd6d752" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.454ex; height:2.843ex;" alt="{\displaystyle \ln \left|\sec x+\tan x\right|+C}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cot x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cot</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cot x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/905c0a51e91def9119e96c665363c5853f030a96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.816ex; height:2.009ex;" alt="{\displaystyle \cot x}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\csc ^{2}x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msup> <mi>csc</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\csc ^{2}x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a1f99390e05a4690b44a4a328886f11ce1e38f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.947ex; height:2.843ex;" alt="{\displaystyle -\csc ^{2}x}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\ln \left|\csc x\right|+C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>|</mo> <mrow> <mi>csc</mi> <mo>⁡</mo> <mi>x</mi> </mrow> <mo>|</mo> </mrow> <mo>+</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\ln \left|\csc x\right|+C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25b5d7fe0a2b1772f27ad540c1511158201ce78a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.733ex; height:2.843ex;" alt="{\displaystyle -\ln \left|\csc x\right|+C}"> </td></tr></tbody></table> Note: For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<x<\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>x</mi> <mo><</mo> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<x<\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5072e331d80251147f44dc1bca9faa5d19c96a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.021ex; height:2.176ex;" alt="{\displaystyle 0<x<\pi }"> the integral of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \csc x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>csc</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \csc x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0e0fc784f0e9c8c3346f7891568f4ab216c1155" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.698ex; height:1.676ex;" alt="{\displaystyle \csc x}"> can also be written as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\operatorname {arsinh} (\cot x),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>arsinh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>cot</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\operatorname {arsinh} (\cot x),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97183217052b1052ddfd680e19851bec00951e2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.69ex; height:2.843ex;" alt="{\displaystyle -\operatorname {arsinh} (\cot x),}"> and for the integral of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sec x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sec</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sec x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32c8904bcf644b6f3ab015742047994329f741bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.698ex; height:1.676ex;" alt="{\displaystyle \sec x}"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\pi /2<x<\pi /2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo><</mo> <mi>x</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 -\pi /2<x<\pi /2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/244b2a56dbd0cdd7c5f1b8a3b8edbf35a65014de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.649ex; height:2.843ex;" alt="{\displaystyle -\pi /2<x<\pi /2}"> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {arsinh} (\tan x),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arsinh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>tan</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {arsinh} (\tan x),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac0082bd8717d7b9b733948f3f1ae516c53a1875" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.755ex; height:2.843ex;" alt="{\displaystyle \operatorname {arsinh} (\tan x),}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {arsinh} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arsinh</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {arsinh} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78e4506c4572caf5a4700de12122e9690f1f9d3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.222ex; height:2.176ex;" alt="{\displaystyle \operatorname {arsinh} }"> is the <a href="/wiki/Inverse_hyperbolic_sine" class="mw-redirect" title="Inverse hyperbolic sine">inverse hyperbolic sine</a>. Alternatively, the derivatives of the 'co-functions' can be obtained using trigonometric identities and the chain rule: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {d\cos x}{dx}}&={\frac {d}{dx}}\sin(\pi /2-x)=-\cos(\pi /2-x)=-\sin x\,,\\{\frac {d\csc x}{dx}}&={\frac {d}{dx}}\sec(\pi /2-x)=-\sec(\pi /2-x)\tan(\pi /2-x)=-\csc x\cot x\,,\\{\frac {d\cot x}{dx}}&={\frac {d}{dx}}\tan(\pi /2-x)=-\sec ^{2}(\pi /2-x)=-\csc ^{2}x\,.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <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>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mspace width="thinmathspace" /> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>csc</mi> <mo>⁡</mo> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>sec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>sec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>csc</mi> <mo>⁡</mo> <mi>x</mi> <mi>cot</mi> <mo>⁡</mo> <mi>x</mi> <mspace width="thinmathspace" /> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>cot</mi> <mo>⁡</mo> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <msup> <mi>sec</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <msup> <mi>csc</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>x</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {d\cos x}{dx}}&={\frac {d}{dx}}\sin(\pi /2-x)=-\cos(\pi /2-x)=-\sin x\,,\\{\frac {d\csc x}{dx}}&={\frac {d}{dx}}\sec(\pi /2-x)=-\sec(\pi /2-x)\tan(\pi /2-x)=-\csc x\cot x\,,\\{\frac {d\cot x}{dx}}&={\frac {d}{dx}}\tan(\pi /2-x)=-\sec ^{2}(\pi /2-x)=-\csc ^{2}x\,.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41f6bcda342982f9a6c122ab0f59e6bb45855159" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.425ex; margin-bottom: -0.246ex; width:75.026ex; height:16.509ex;" alt="{\displaystyle {\begin{aligned}{\frac {d\cos x}{dx}}&={\frac {d}{dx}}\sin(\pi /2-x)=-\cos(\pi /2-x)=-\sin x\,,\\{\frac {d\csc x}{dx}}&={\frac {d}{dx}}\sec(\pi /2-x)=-\sec(\pi /2-x)\tan(\pi /2-x)=-\csc x\cot x\,,\\{\frac {d\cot x}{dx}}&={\frac {d}{dx}}\tan(\pi /2-x)=-\sec ^{2}(\pi /2-x)=-\csc ^{2}x\,.\end{aligned}}}"></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Inverse_functions">Inverse functions</h2>[<a href="/w/index.php?title=Trigonometric_functions&action=edit&section=24" title="Edit section: Inverse functions">edit</a>]</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/Inverse_trigonometric_functions" title="Inverse trigonometric functions">Inverse trigonometric functions</a></div> The trigonometric functions are periodic, and hence not <a href="/wiki/Injective_function" title="Injective function">injective</a>, so strictly speaking, they do not have an <a href="/wiki/Inverse_function" title="Inverse function">inverse function</a>. However, on each interval on which a trigonometric function is <a href="/wiki/Monotonic" class="mw-redirect" title="Monotonic">monotonic</a>, one can define an inverse function, and this defines inverse trigonometric functions as <a href="/wiki/Multivalued_function" title="Multivalued function">multivalued functions</a>. To define a true inverse function, one must restrict the domain to an interval where the function is monotonic, and is thus <a href="/wiki/Bijection" title="Bijection">bijective</a> from this interval to its image by the function. The common choice for this interval, called the set of <a href="/wiki/Principal_value" title="Principal value">principal values</a>, is given in the following table. As usual, the inverse trigonometric functions are denoted with the prefix "arc" before the name or its abbreviation of the function. <table class="wikitable" style="text-align: center;"> <tbody><tr> <th>Function</th> <th>Definition</th> <th>Domain</th> <th>Set of principal values </th></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=\arcsin x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>arcsin</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=\arcsin x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7eb83fc5e328a407b9bdb3a6a5a9aaed8044b95c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.933ex; height:2.509ex;" alt="{\displaystyle y=\arcsin x}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin y=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>y</mi> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin y=x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cabdeb55e583822d40ffa632af0d3080d063eb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.826ex; height:2.509ex;" alt="{\displaystyle \sin y=x}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1\leq x\leq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> <mo>≤</mo> <mi>x</mi> <mo>≤</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1\leq x\leq 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c07d30937eb8028698054698012f2d76caad7fe4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.66ex; height:2.343ex;" alt="{\displaystyle -1\leq x\leq 1}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle -{\frac {\pi }{2}}\leq y\leq {\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> <mo>≤</mo> <mi>y</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 -{\frac {\pi }{2}}\leq y\leq {\frac {\pi }{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bc00eba7339cda98c1f8847bc6c0b9a3f17e6c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:12.717ex; height:3.176ex;" alt="{\textstyle -{\frac {\pi }{2}}\leq y\leq {\frac {\pi }{2}}}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=\arccos x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>arccos</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=\arccos x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41bf0f13cd55dd0ff3c5c7cc6258bffa5a654f0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.188ex; height:2.009ex;" alt="{\displaystyle y=\arccos x}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos y=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>y</mi> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos y=x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e7846086fe6da9edcdc3a94d23aea280cb44035" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.082ex; height:2.009ex;" alt="{\displaystyle \cos y=x}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1\leq x\leq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> <mo>≤</mo> <mi>x</mi> <mo>≤</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1\leq x\leq 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c07d30937eb8028698054698012f2d76caad7fe4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.66ex; height:2.343ex;" alt="{\displaystyle -1\leq x\leq 1}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 0\leq y\leq \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mi>y</mi> <mo>≤</mo> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle 0\leq y\leq \pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/251f11b548502319154d5139c0fe18009edb7041" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.847ex; height:2.509ex;" alt="{\textstyle 0\leq y\leq \pi }"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=\arctan x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=\arctan x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53b245f79e8ccc0051e708c62e797559da23d75b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.437ex; height:2.343ex;" alt="{\displaystyle y=\arctan x}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan y=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mi>y</mi> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan y=x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2436900ff8234effa467e9970b14bb8b6222c271" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.33ex; height:2.343ex;" alt="{\displaystyle \tan y=x}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\infty <x<\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo><</mo> <mi>x</mi> <mo><</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\infty <x<\infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bfcfd777298df7ef46c75fab394e644d5fbafe1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.982ex; height:2.176ex;" alt="{\displaystyle -\infty <x<\infty }"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle -{\frac {\pi }{2}}<y<{\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> <mo><</mo> <mi>y</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 -{\frac {\pi }{2}}<y<{\frac {\pi }{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44a69d057ac0c7427266ea3ebc317b72242ccc6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:12.717ex; height:3.176ex;" alt="{\textstyle -{\frac {\pi }{2}}<y<{\frac {\pi }{2}}}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=\operatorname {arccot} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>arccot</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=\operatorname {arccot} x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99ca0d666abb65de24db033220beada025c8c768" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.177ex; height:2.343ex;" alt="{\displaystyle y=\operatorname {arccot} x}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cot y=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cot</mi> <mo>⁡</mo> <mi>y</mi> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cot y=x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43b80a1f772b33bbad296f4375179874e0e0a1c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.07ex; height:2.343ex;" alt="{\displaystyle \cot y=x}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\infty <x<\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo><</mo> <mi>x</mi> <mo><</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\infty <x<\infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bfcfd777298df7ef46c75fab394e644d5fbafe1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.982ex; height:2.176ex;" alt="{\displaystyle -\infty <x<\infty }"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 0<y<\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>y</mi> <mo><</mo> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle 0<y<\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad4b57ec4e172279f55d56530a19e647bebdfa98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.847ex; height:2.509ex;" alt="{\textstyle 0<y<\pi }"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=\operatorname {arcsec} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>arcsec</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=\operatorname {arcsec} x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4276eae9a683b0a9b871b0335af6e8eda2f32c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.058ex; height:2.009ex;" alt="{\displaystyle y=\operatorname {arcsec} x}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sec y=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sec</mi> <mo>⁡</mo> <mi>y</mi> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sec y=x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8763f7a37657a22f0a9e636ecb0d1e336a94f233" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.952ex; height:2.009ex;" alt="{\displaystyle \sec y=x}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x<-1{\text{ or }}x>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo><</mo> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> or </mtext> </mrow> <mi>x</mi> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x<-1{\text{ or }}x>1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6aa060b113400fe7ba220b2a240bcd2fefb9f5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.225ex; height:2.343ex;" alt="{\displaystyle x<-1{\text{ or }}x>1}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 0\leq y\leq \pi ,\;y\neq {\frac {\pi }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mi>y</mi> <mo>≤</mo> <mi>π</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mi>y</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\leq y\leq \pi ,\;y\neq {\frac {\pi }{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2084d9fbbd7ac296fea44a876edfa83809bcba42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:17.558ex; height:3.176ex;" alt="{\textstyle 0\leq y\leq \pi ,\;y\neq {\frac {\pi }{2}}}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=\operatorname {arccsc} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>arccsc</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=\operatorname {arccsc} x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/792670892a9afb014fe67c308402dc41028f88a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.058ex; height:2.009ex;" alt="{\displaystyle y=\operatorname {arccsc} x}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \csc y=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>csc</mi> <mo>⁡</mo> <mi>y</mi> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \csc y=x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/109975c3ebcdc28b85fbeef3dd3c98bf0cc6a26b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.952ex; height:2.009ex;" alt="{\displaystyle \csc y=x}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x<-1{\text{ or }}x>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo><</mo> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> or </mtext> </mrow> <mi>x</mi> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x<-1{\text{ or }}x>1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6aa060b113400fe7ba220b2a240bcd2fefb9f5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.225ex; height:2.343ex;" alt="{\displaystyle x<-1{\text{ or }}x>1}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle -{\frac {\pi }{2}}\leq y\leq {\frac {\pi }{2}},\;y\neq 0}"> <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> <mo>≤</mo> <mi>y</mi> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mi>y</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle -{\frac {\pi }{2}}\leq y\leq {\frac {\pi }{2}},\;y\neq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f18a776c0223a9970a57a2e33b2354616c280f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:19.812ex; height:3.176ex;" alt="{\textstyle -{\frac {\pi }{2}}\leq y\leq {\frac {\pi }{2}},\;y\neq 0}"> </td></tr></tbody></table> The notations sin−1, cos−1, etc. are often used for arcsin and arccos, etc. When this notation is used, inverse functions could be confused with multiplicative inverses. The notation with the "arc" prefix avoids such a confusion, though "arcsec" for arcsecant can be confused with "<a href="/wiki/Arcsecond" class="mw-redirect" title="Arcsecond">arcsecond</a>". Just like the sine and cosine, the inverse trigonometric functions can also be expressed in terms of infinite series. They can also be expressed in terms of <a href="/wiki/Complex_logarithm" title="Complex logarithm">complex logarithms</a>. <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2>[<a href="/w/index.php?title=Trigonometric_functions&action=edit&section=25" title="Edit section: Applications">edit</a>]</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/Uses_of_trigonometry" title="Uses of trigonometry">Uses of trigonometry</a></div> <div class="mw-heading mw-heading3"><h3 id="Angles_and_sides_of_a_triangle">Angles and sides of a triangle</h3>[<a href="/w/index.php?title=Trigonometric_functions&action=edit&section=26" title="Edit section: Angles and sides of a triangle">edit</a>]</div> In this section A, B, C denote the three (interior) angles of a triangle, and a, b, c denote the lengths of the respective opposite edges. They are related by various formulas, which are named by the trigonometric functions they involve. <div class="mw-heading mw-heading4"><h4 id="Law_of_sines">Law of sines</h4>[<a href="/w/index.php?title=Trigonometric_functions&action=edit&section=27" title="Edit section: Law of sines">edit</a>]</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/Law_of_sines" title="Law of sines">Law of sines</a></div> The law of sines states that for an arbitrary triangle with sides a, b, and c and angles opposite those sides A, B and C: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\sin A}{a}}={\frac {\sin B}{b}}={\frac {\sin C}{c}}={\frac {2\Delta }{abc}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>A</mi> </mrow> <mi>a</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>B</mi> </mrow> <mi>b</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>C</mi> </mrow> <mi>c</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi mathvariant="normal">Δ</mi> </mrow> <mrow> <mi>a</mi> <mi>b</mi> <mi>c</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\sin A}{a}}={\frac {\sin B}{b}}={\frac {\sin C}{c}}={\frac {2\Delta }{abc}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7076f44b9aa649f5c21eeb030eeb9c9f2bf0d162" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:31.523ex; height:5.509ex;" alt="{\displaystyle {\frac {\sin A}{a}}={\frac {\sin B}{b}}={\frac {\sin C}{c}}={\frac {2\Delta }{abc}},}"> where Δ is the area of the triangle, or, equivalently, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {a}{\sin A}}={\frac {b}{\sin B}}={\frac {c}{\sin C}}=2R,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>A</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>B</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>C</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> <mi>R</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a}{\sin A}}={\frac {b}{\sin B}}={\frac {c}{\sin C}}=2R,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1419ccbac53bfe833611e68a722eef65eee1245d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:30.379ex; height:5.509ex;" alt="{\displaystyle {\frac {a}{\sin A}}={\frac {b}{\sin B}}={\frac {c}{\sin C}}=2R,}"> where R is the triangle's <a href="/wiki/Circumscribed_circle" title="Circumscribed circle">circumradius</a>. It can be proved by dividing the triangle into two right ones and using the above definition of sine. The law of sines is useful for computing the lengths of the unknown sides in a triangle if two angles and one side are known. This is a common situation occurring in <a href="/wiki/Triangulation" title="Triangulation">triangulation</a>, a technique to determine unknown distances by measuring two angles and an accessible enclosed distance. <div class="mw-heading mw-heading4"><h4 id="Law_of_cosines">Law of cosines</h4>[<a href="/w/index.php?title=Trigonometric_functions&action=edit&section=28" title="Edit section: Law of cosines">edit</a>]</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/Law_of_cosines" title="Law of cosines">Law of cosines</a></div> The law of cosines (also known as the cosine formula or cosine rule) is an extension of the <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos C,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>a</mi> <mi>b</mi> <mi>cos</mi> <mo>⁡</mo> <mi>C</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos C,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e17c2539aed06efeb90b3d60d1e2ccd0da210fd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.864ex; height:3.009ex;" alt="{\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos C,}"> or equivalently, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos C={\frac {a^{2}+b^{2}-c^{2}}{2ab}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>C</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>a</mi> <mi>b</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos C={\frac {a^{2}+b^{2}-c^{2}}{2ab}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afddf4b33705cd7055834106f391883272a37988" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:21.924ex; height:5.843ex;" alt="{\displaystyle \cos C={\frac {a^{2}+b^{2}-c^{2}}{2ab}}.}"> In this formula the angle at C is opposite to the side c. This theorem can be proved by dividing the triangle into two right ones and using the <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a>. The law of cosines can be used to determine a side of a triangle if two sides and the angle between them are known. It can also be used to find the cosines of an angle (and consequently the angles themselves) if the lengths of all the sides are known. <div class="mw-heading mw-heading4"><h4 id="Law_of_tangents">Law of tangents</h4>[<a href="/w/index.php?title=Trigonometric_functions&action=edit&section=29" title="Edit section: Law of tangents">edit</a>]</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/Law_of_tangents" title="Law of tangents">Law of tangents</a></div> The law of tangents says that: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\tan {\frac {A-B}{2}}}{\tan {\frac {A+B}{2}}}}={\frac {a-b}{a+b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>A</mi> <mo>−</mo> <mi>B</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> <mrow> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>A</mi> <mo>+</mo> <mi>B</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>−</mo> <mi>b</mi> </mrow> <mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\tan {\frac {A-B}{2}}}{\tan {\frac {A+B}{2}}}}={\frac {a-b}{a+b}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fefe8217e3c6576f40c7702257a3253e2afa5c67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:18.18ex; height:8.509ex;" alt="{\displaystyle {\frac {\tan {\frac {A-B}{2}}}{\tan {\frac {A+B}{2}}}}={\frac {a-b}{a+b}}}">.</dd></dl> <div class="mw-heading mw-heading4"><h4 id="Law_of_cotangents">Law of cotangents</h4>[<a href="/w/index.php?title=Trigonometric_functions&action=edit&section=30" title="Edit section: Law of cotangents">edit</a>]</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/Law_of_cotangents" title="Law of cotangents">Law of cotangents</a></div> If s is the triangle's semiperimeter, (a + b + c)/2, and r is the radius of the triangle's <a href="/wiki/Incircle" class="mw-redirect" title="Incircle">incircle</a>, then rs is the triangle's area. Therefore <a href="/wiki/Heron%27s_formula" title="Heron's formula">Heron's formula</a> implies that: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r={\sqrt {{\frac {1}{s}}(s-a)(s-b)(s-c)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>s</mi> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>c</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r={\sqrt {{\frac {1}{s}}(s-a)(s-b)(s-c)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f218bf20a5847c859002590e4da0f173e19d37f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:28.924ex; height:6.176ex;" alt="{\displaystyle r={\sqrt {{\frac {1}{s}}(s-a)(s-b)(s-c)}}}">.</dd></dl> The law of cotangents says that:<a href="#cite_note-Allen_1976-28">[27]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cot {\frac {A}{2}}={\frac {s-a}{r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cot</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>A</mi> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>s</mi> <mo>−</mo> <mi>a</mi> </mrow> <mi>r</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cot {\frac {A}{2}}={\frac {s-a}{r}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0dfa86b8f1a97ed733e1518d70a9c94a2ed5638" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.161ex; height:5.343ex;" alt="{\displaystyle \cot {\frac {A}{2}}={\frac {s-a}{r}}}"></dd></dl> It follows that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\cot {\dfrac {A}{2}}}{s-a}}={\frac {\cot {\dfrac {B}{2}}}{s-b}}={\frac {\cot {\dfrac {C}{2}}}{s-c}}={\frac {1}{r}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>cot</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mi>A</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> </mrow> <mrow> <mi>s</mi> <mo>−</mo> <mi>a</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>cot</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mi>B</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> </mrow> <mrow> <mi>s</mi> <mo>−</mo> <mi>b</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>cot</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mi>C</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> </mrow> <mrow> <mi>s</mi> <mo>−</mo> <mi>c</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\cot {\dfrac {A}{2}}}{s-a}}={\frac {\cot {\dfrac {B}{2}}}{s-b}}={\frac {\cot {\dfrac {C}{2}}}{s-c}}={\frac {1}{r}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ae32a93073b58b031eff5666b2165b26902b57f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:32.691ex; height:8.509ex;" alt="{\displaystyle {\frac {\cot {\dfrac {A}{2}}}{s-a}}={\frac {\cot {\dfrac {B}{2}}}{s-b}}={\frac {\cot {\dfrac {C}{2}}}{s-c}}={\frac {1}{r}}.}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Periodic_functions">Periodic functions</h3>[<a href="/w/index.php?title=Trigonometric_functions&action=edit&section=31" title="Edit section: Periodic functions">edit</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Lissajous_curve_5by4.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Lissajous_curve_5by4.svg/220px-Lissajous_curve_5by4.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Lissajous_curve_5by4.svg/330px-Lissajous_curve_5by4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Lissajous_curve_5by4.svg/440px-Lissajous_curve_5by4.svg.png 2x" data-file-width="600" data-file-height="600" /></a><figcaption>A <a href="/wiki/Lissajous_curve" title="Lissajous curve">Lissajous curve</a>, a figure formed with a trigonometry-based function.</figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Synthesis_square.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0a/Synthesis_square.gif/330px-Synthesis_square.gif" decoding="async" width="330" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0a/Synthesis_square.gif/495px-Synthesis_square.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/0/0a/Synthesis_square.gif 2x" data-file-width="500" data-file-height="250" /></a><figcaption>An animation of the <a href="/wiki/Additive_synthesis" title="Additive synthesis">additive synthesis</a> of a <a href="/wiki/Square_wave" title="Square wave">square wave</a> with an increasing number of harmonics</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Sawtooth_Fourier_Animation.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Sawtooth_Fourier_Animation.gif/290px-Sawtooth_Fourier_Animation.gif" decoding="async" width="290" height="375" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/b/b7/Sawtooth_Fourier_Animation.gif 1.5x" data-file-width="380" data-file-height="492" /></a><figcaption>Sinusoidal basis functions (bottom) can form a sawtooth wave (top) when added. All the basis functions have nodes at the nodes of the sawtooth, and all but the fundamental (k = 1) have additional nodes. The oscillation seen about the sawtooth when k is large is called the <a href="/wiki/Gibbs_phenomenon" title="Gibbs phenomenon">Gibbs phenomenon</a>.</figcaption></figure> The trigonometric functions are also important in physics. The sine and the cosine functions, for example, are used to describe <a href="/wiki/Simple_harmonic_motion" title="Simple harmonic motion">simple harmonic motion</a>, which models many natural phenomena, such as the movement of a mass attached to a spring and, for small angles, the pendular motion of a mass hanging by a string. The sine and cosine functions are one-dimensional projections of <a href="/wiki/Uniform_circular_motion" class="mw-redirect" title="Uniform circular motion">uniform circular motion</a>. Trigonometric functions also prove to be useful in the study of general <a href="/wiki/Periodic_function" title="Periodic function">periodic functions</a>. The characteristic wave patterns of periodic functions are useful for modeling recurring phenomena such as sound or light <a href="/wiki/Wave" title="Wave">waves</a>.<a href="#cite_note-Farlow_1993-29">[28]</a> Under rather general conditions, a periodic function f (x) can be expressed as a sum of sine waves or cosine waves in a <a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a>.<a href="#cite_note-Folland_1992-30">[29]</a> Denoting the sine or cosine <a href="/wiki/Basis_functions" class="mw-redirect" title="Basis functions">basis functions</a> by φk, the expansion of the periodic function f (t) takes the form: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)=\sum _{k=1}^{\infty }c_{k}\varphi _{k}(t).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t)=\sum _{k=1}^{\infty }c_{k}\varphi _{k}(t).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2217234c9d189ecbd7f08116a49d0bbefd7633fd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.768ex; height:6.843ex;" alt="{\displaystyle f(t)=\sum _{k=1}^{\infty }c_{k}\varphi _{k}(t).}"> For example, the <a href="/wiki/Square_wave" title="Square wave">square wave</a> can be written as the <a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\text{square}}(t)={\frac {4}{\pi }}\sum _{k=1}^{\infty }{\sin {\big (}(2k-1)t{\big )} \over 2k-1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>square</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mi>π</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <mrow> <mn>2</mn> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\text{square}}(t)={\frac {4}{\pi }}\sum _{k=1}^{\infty }{\sin {\big (}(2k-1)t{\big )} \over 2k-1}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0c74941497dde03bd6cf2caeb3e0a9551e11e73" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:33.924ex; height:7.176ex;" alt="{\displaystyle f_{\text{square}}(t)={\frac {4}{\pi }}\sum _{k=1}^{\infty }{\sin {\big (}(2k-1)t{\big )} \over 2k-1}.}"> In the animation of a square wave at top right it can be seen that just a few terms already produce a fairly good approximation. The superposition of several terms in the expansion of a <a href="/wiki/Sawtooth_wave" title="Sawtooth wave">sawtooth wave</a> are shown underneath. <div class="mw-heading mw-heading2"><h2 id="History">History</h2>[<a href="/w/index.php?title=Trigonometric_functions&action=edit&section=32" title="Edit section: History">edit</a>]</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> While the early study of trigonometry can be traced to antiquity, the trigonometric functions 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). The functions of sine and <a href="/wiki/Versine" title="Versine">versine</a> (1 – cosine) can be traced back to the <a href="/wiki/Jy%C4%81,_koti-jy%C4%81_and_utkrama-jy%C4%81" title="Jyā, koti-jyā and utkrama-jyā">jyā and koti-jyā</a> functions used in <a href="/wiki/Gupta_period" class="mw-redirect" title="Gupta period">Gupta period</a> <a href="/wiki/Indian_astronomy" title="Indian astronomy">Indian astronomy</a> (<a href="/wiki/Aryabhatiya" title="Aryabhatiya">Aryabhatiya</a>, <a href="/wiki/Surya_Siddhanta" title="Surya Siddhanta">Surya Siddhanta</a>), via translation from Sanskrit to Arabic and then from Arabic to Latin.<a href="#cite_note-Boyer_1991-31">[30]</a> (See <a href="/wiki/Aryabhata%27s_sine_table" class="mw-redirect" title="Aryabhata's sine table">Aryabhata's sine table</a>.) 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>.<a href="#cite_note-Gingerich_1986-32">[31]</a> With the exception of the sine (which was adopted from Indian mathematics), the other five modern trigonometric functions were discovered by Persian and Arab mathematicians, including the cosine, tangent, cotangent, secant and cosecant.<a href="#cite_note-Gingerich_1986-32">[31]</a> <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. Circa 830, <a href="/wiki/Habash_al-Hasib_al-Marwazi" class="mw-redirect" title="Habash al-Hasib al-Marwazi">Habash al-Hasib al-Marwazi</a> discovered the cotangent, and produced tables of tangents and cotangents.<a href="#cite_note-Sesiano-33">[32]</a><a href="#cite_note-Britannica-34">[33]</a> <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°.<a href="#cite_note-Britannica-34">[33]</a> The trigonometric functions were later studied by mathematicians including <a href="/wiki/Omar_Khayy%C3%A1m" class="mw-redirect" title="Omar Khayyám">Omar Khayyám</a>, <a href="/wiki/Bh%C4%81skara_II" title="Bhāskara II">Bhāskara II</a>, <a href="/wiki/Nasir_al-Din_al-Tusi" title="Nasir al-Din al-Tusi">Nasir al-Din al-Tusi</a>, <a href="/wiki/Jamsh%C4%ABd_al-K%C4%81sh%C4%AB" class="mw-redirect" title="Jamshīd al-Kāshī">Jamshīd al-Kāshī</a> (14th century), <a href="/wiki/Ulugh_Beg" title="Ulugh Beg">Ulugh Beg</a> (14th century), <a href="/wiki/Regiomontanus" title="Regiomontanus">Regiomontanus</a> (1464), <a href="/wiki/Georg_Joachim_Rheticus" title="Georg Joachim Rheticus">Rheticus</a>, and Rheticus' student <a href="/wiki/Valentinus_Otho" title="Valentinus Otho">Valentinus Otho</a>. <a href="/wiki/Madhava_of_Sangamagrama" title="Madhava of Sangamagrama">Madhava of Sangamagrama</a> (c. 1400) made early strides in the <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">analysis</a> of trigonometric functions in terms of <a href="/wiki/Series_(mathematics)" title="Series (mathematics)">infinite series</a>.<a href="#cite_note-mact-biog-35">[34]</a> (See <a href="/wiki/Madhava_series" title="Madhava series">Madhava series</a> and <a href="/wiki/Madhava%27s_sine_table" title="Madhava's sine table">Madhava's sine table</a>.) The tangent function was brought to Europe by <a href="/wiki/Giovanni_Bianchini" title="Giovanni Bianchini">Giovanni Bianchini</a> in 1467 in trigonometry tables he created to support the calculation of stellar coordinates.<a href="#cite_note-36">[35]</a> The terms tangent and secant were first introduced by the Danish mathematician <a href="/wiki/Thomas_Fincke" title="Thomas Fincke">Thomas Fincke</a> in his book Geometria rotundi (1583).<a href="#cite_note-Fincke-37">[36]</a> The 17th century French mathematician <a href="/wiki/Albert_Girard" title="Albert Girard">Albert Girard</a> made the first published use of the abbreviations sin, cos, and tan in his book Trigonométrie.<a href="#cite_note-MacTutor-38">[37]</a> In a paper published in 1682, <a href="/wiki/Gottfried_Leibniz" class="mw-redirect" title="Gottfried Leibniz">Gottfried Leibniz</a> proved that sin x is not an <a href="/wiki/Algebraic_function" title="Algebraic function">algebraic function</a> of x.<a href="#cite_note-Bourbaki_1994-39">[38]</a> Though introduced as ratios of sides of a <a href="/wiki/Right_triangle" title="Right triangle">right triangle</a>, and thus appearing to be <a href="/wiki/Rational_function" title="Rational function">rational functions</a>, Leibnitz result established that they are actually <a href="/wiki/Transcendental_function" title="Transcendental function">transcendental functions</a> of their argument. The task of assimilating circular functions into algebraic expressions was accomplished by Euler in his <a href="/wiki/Introduction_to_the_Analysis_of_the_Infinite" class="mw-redirect" title="Introduction to the Analysis of the Infinite">Introduction to the Analysis of the Infinite</a> (1748). His method was to show that the sine and cosine functions are <a href="/wiki/Alternating_series" title="Alternating series">alternating series</a> formed from the even and odd terms respectively of the <a href="/wiki/Exponential_function" title="Exponential function">exponential series</a>. He presented "<a href="/wiki/Euler%27s_formula" title="Euler's formula">Euler's formula</a>", as well as near-modern abbreviations (sin., cos., tang., cot., sec., and cosec.).<a href="#cite_note-Boyer_1991-31">[30]</a> A few functions were common historically, but are now seldom used, such as the <a href="/wiki/Chord_(geometry)" title="Chord (geometry)">chord</a>, the <a href="/wiki/Versine" title="Versine">versine</a> (which appeared in the earliest tables<a href="#cite_note-Boyer_1991-31">[30]</a>), the <a href="/wiki/Coversine" class="mw-redirect" title="Coversine">coversine</a>, the <a href="/wiki/Haversine" class="mw-redirect" title="Haversine">haversine</a>,<a href="#cite_note-40">[39]</a> the <a href="/wiki/Exsecant" title="Exsecant">exsecant</a> and the <a href="/wiki/Excosecant" class="mw-redirect" title="Excosecant">excosecant</a>. The <a href="/wiki/List_of_trigonometric_identities" title="List of trigonometric identities">list of trigonometric identities</a> shows more relations between these functions. <ul><li>crd(θ) = 2 sin(<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠θ/2⁠)</li> <li>versin(θ) = 1 − cos(θ) = 2 sin2(<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠θ/2⁠)</li> <li>coversin(θ) = 1 − sin(θ) = versin(<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠π/2⁠ − θ)</li> <li>haversin(θ) = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠versin(θ) = sin2(<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠θ/2⁠)</li> <li>exsec(θ) = sec(θ) − 1</li> <li>excsc(θ) = exsec(<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠π/2⁠ − θ) = csc(θ) − 1</li></ul> Historically, trigonometric functions were often combined with <a href="/wiki/Logarithm" title="Logarithm">logarithms</a> in compound functions like the logarithmic sine, logarithmic cosine, logarithmic secant, logarithmic cosecant, logarithmic tangent and logarithmic cotangent.<a href="#cite_note-Hammer_1897-41">[40]</a><a href="#cite_note-Heß_1916-42">[41]</a><a href="#cite_note-Lötzbeyer_1950-43">[42]</a><a href="#cite_note-Roegel_2016-44">[43]</a> <div class="mw-heading mw-heading2"><h2 id="Etymology">Etymology</h2>[<a href="/w/index.php?title=Trigonometric_functions&action=edit&section=33" title="Edit section: Etymology">edit</a>]</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> The word sine derives<a href="#cite_note-45">[44]</a> from <a href="/wiki/Latin" title="Latin">Latin</a> <a href="https://en.wiktionary.org/wiki/sinus" class="extiw" title="wikt:sinus">sinus</a>, meaning "bend; bay", and more specifically "the hanging fold of the upper part of a <a href="/wiki/Toga" title="Toga">toga</a>", "the bosom of a garment", which was chosen as the translation of what was interpreted as the Arabic word jaib, meaning "pocket" or "fold" in the twelfth-century translations of works by <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> into <a href="/wiki/Medieval_Latin" title="Medieval Latin">Medieval Latin</a>.<a href="#cite_note-46">[45]</a> The choice was based on a misreading of the Arabic written form j-y-b (<a href="https://en.wiktionary.org/wiki/%D8%AC%D9%8A%D8%A8" class="extiw" title="wikt:جيب">جيب</a>), which itself originated as a <a href="/wiki/Transliteration" title="Transliteration">transliteration</a> from Sanskrit jīvā, which along with its synonym jyā (the standard Sanskrit term for the sine) translates to "bowstring", being in turn adopted from <a href="/wiki/Ancient_Greek_language" class="mw-redirect" title="Ancient Greek language">Ancient Greek</a> <a href="/wiki/Chord_(geometry)" title="Chord (geometry)">χορδή</a> "string".<a href="#cite_note-Plofker_2009-47">[46]</a> The word tangent comes from Latin tangens meaning "touching", since the line touches the circle of unit radius, whereas secant stems from Latin secans—"cutting"—since the line cuts the circle.<a href="#cite_note-48">[47]</a> The prefix "<a href="/wiki/Co_(function_prefix)" class="mw-redirect" title="Co (function prefix)">co-</a>" (in "cosine", "cotangent", "cosecant") is found in <a href="/wiki/Edmund_Gunter" title="Edmund Gunter">Edmund Gunter</a>'s Canon triangulorum (1620), which defines the cosinus as an abbreviation for the sinus complementi (sine of the <a href="/wiki/Complementary_angle" class="mw-redirect" title="Complementary angle">complementary angle</a>) and proceeds to define the cotangens similarly.<a href="#cite_note-Gunter_1620-49">[48]</a><a href="#cite_note-Roegel_2010-50">[49]</a> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Trigonometric_functions&action=edit&section=34" title="Edit section: See also">edit</a>]</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: 25em;"> <ul><li><a href="/wiki/Bh%C4%81skara_I%27s_sine_approximation_formula" title="Bhāskara I's sine approximation formula">Bhāskara I's sine approximation formula</a></li> <li><a href="/wiki/Small-angle_approximation" title="Small-angle approximation">Small-angle approximation</a></li> <li><a href="/wiki/Differentiation_of_trigonometric_functions" title="Differentiation of trigonometric functions">Differentiation of trigonometric functions</a></li> <li><a href="/wiki/Generalized_trigonometry" title="Generalized trigonometry">Generalized trigonometry</a></li> <li><a href="/wiki/Generating_trigonometric_tables" class="mw-redirect" title="Generating trigonometric tables">Generating trigonometric tables</a></li> <li><a href="/wiki/List_of_integrals_of_trigonometric_functions" title="List of integrals of trigonometric functions">List of integrals of trigonometric functions</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/Polar_sine" title="Polar sine">Polar sine</a> – a generalization to vertex angles</li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Trigonometric_functions&action=edit&section=35" title="Edit section: Notes">edit</a>]</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-11"><a href="#cite_ref-11">^</a> Also equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\sqrt {2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sqrt {2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db5f73af5805a28d23fa6782b4199ff31aa108ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:3.934ex; height:6.176ex;" alt="{\displaystyle {\frac {1}{\sqrt {2}}}}"> </li> </ol></div></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-klein-1"><a href="#cite_ref-klein_1-0">^</a> <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="CITEREFKlein1924" class="citation book cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Felix_Klein" title="Felix Klein">Klein, Felix</a> (1924) [1902]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=5t8fAAAAIAAJ&pg=PA175">"Die goniometrischen Funktionen"</a>. Elementarmathematik vom höheren Standpunkt aus: Arithmetik, Algebra, Analysis (in German). Vol. 1 (3rd ed.). Berlin: J. Springer. Ch. 3.2, p. 175 ff.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Die+goniometrischen+Funktionen&rft.btitle=Elementarmathematik+vom+h%C3%B6heren+Standpunkt+aus%3A+Arithmetik%2C+Algebra%2C+Analysis&rft.place=Berlin&rft.pages=%3Cspan+class%3D%22nowrap%22%3ECh.+3.2%3C%2Fspan%3E%2C+p.-175+ff.&rft.edition=3rd&rft.pub=J.+Springer&rft.date=1924&rft.aulast=Klein&rft.aufirst=Felix&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D5t8fAAAAIAAJ%26pg%3DPA175&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrigonometric+functions" class="Z3988"> Translated as <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKlein1932" class="citation book cs1"><a rel="nofollow" class="external text" href="https://archive.org/details/geometryelementa0000feli/page/162/?q=%22ii.+the+goniometric+functions%22">"The Goniometric Functions"</a>. Elementary Mathematics from an Advanced Standpoint: Arithmetic, Algebra, Analysis. Translated by Hedrick, E. R.; Noble, C. A. Macmillan. 1932. Ch. 3.2, p. 162 ff.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+Goniometric+Functions&rft.btitle=Elementary+Mathematics+from+an+Advanced+Standpoint%3A+Arithmetic%2C+Algebra%2C+Analysis&rft.pages=Ch.+3.2%2C+p.-162+ff.&rft.pub=Macmillan&rft.date=1932&rft.aulast=Klein&rft.aufirst=Felix&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fgeometryelementa0000feli%2Fpage%2F162%2F%3Fq%3D%2522ii.%2Bthe%2Bgoniometric%2Bfunctions%2522&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrigonometric+functions" class="Z3988"> </li> <li id="cite_note-2"><a href="#cite_ref-2">^</a> <a href="#CITEREFProtterMorrey1970">Protter & Morrey (1970</a>, pp. APP-2, APP-3) </li> <li id="cite_note-3"><a href="#cite_ref-3">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.mathsisfun.com/sine-cosine-tangent.html">"Sine, Cosine, Tangent"</a>. www.mathsisfun.com. Retrieved 2020-08-29.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.mathsisfun.com&rft.atitle=Sine%2C+Cosine%2C+Tangent&rft_id=https%3A%2F%2Fwww.mathsisfun.com%2Fsine-cosine-tangent.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrigonometric+functions" class="Z3988"> </li> <li id="cite_note-4"><a href="#cite_ref-4">^</a> <a href="#CITEREFProtterMorrey1970">Protter & Morrey (1970</a>, p. APP-7) </li> <li id="cite_note-:0-5">^ <a href="#cite_ref-:0_5-0">a</a> <a href="#cite_ref-:0_5-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRudin,_Walter,_1921–2010" class="citation book cs1">Rudin, Walter, 1921–2010. <a rel="nofollow" class="external text" href="https://www.worldcat.org/oclc/1502474">Principles of mathematical analysis</a> (Third ed.). New York. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-07-054235-X" title="Special:BookSources/0-07-054235-X"><bdi>0-07-054235-X</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/1502474">1502474</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Principles+of+mathematical+analysis&rft.place=New+York&rft.edition=Third&rft_id=info%3Aoclcnum%2F1502474&rft.isbn=0-07-054235-X&rft.au=Rudin%2C+Walter%2C+1921%E2%80%932010&rft_id=https%3A%2F%2Fwww.worldcat.org%2Foclc%2F1502474&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrigonometric+functions" class="Z3988"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: multiple names: authors list (<a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">link</a>) CS1 maint: numeric names: authors list (<a href="/wiki/Category:CS1_maint:_numeric_names:_authors_list" title="Category:CS1 maint: numeric names: authors list">link</a>) </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDiamond2014" class="citation journal cs1">Diamond, Harvey (2014). <a rel="nofollow" class="external text" href="https://www.tandfonline.com/doi/full/10.4169/math.mag.87.1.37">"Defining Exponential and Trigonometric Functions Using Differential Equations"</a>. Mathematics Magazine. 87 (1): 37–42. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4169%2Fmath.mag.87.1.37">10.4169/math.mag.87.1.37</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0025-570X">0025-570X</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:126217060">126217060</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+Magazine&rft.atitle=Defining+Exponential+and+Trigonometric+Functions+Using+Differential+Equations&rft.volume=87&rft.issue=1&rft.pages=37-42&rft.date=2014&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A126217060%23id-name%3DS2CID&rft.issn=0025-570X&rft_id=info%3Adoi%2F10.4169%2Fmath.mag.87.1.37&rft.aulast=Diamond&rft.aufirst=Harvey&rft_id=https%3A%2F%2Fwww.tandfonline.com%2Fdoi%2Ffull%2F10.4169%2Fmath.mag.87.1.37&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrigonometric+functions" class="Z3988"> </li> <li id="cite_note-:1-7"><a href="#cite_ref-:1_7-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSpivak1967" class="citation book cs1">Spivak, Michael (1967). 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Washington, DC: Mathematical Association of America. p. 119. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-88385-525-6" title="Special:BookSources/978-0-88385-525-6"><bdi>978-0-88385-525-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Twenty+years+before+the+blackboard%3A+the+lessons+and+humor+of+a+mathematics+teacher&rft.place=Washington%2C+DC&rft.series=Spectrum+series&rft.pages=119&rft.pub=Mathematical+Association+of+America&rft.date=1998&rft.isbn=978-0-88385-525-6&rft.aulast=Stueben&rft.aufirst=Michael&rft.au=Sandford%2C+Diane&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dqnd0P-Ja-O8C%26dq%3D%2522All%2BStudents%2BTake%2BCalculus%2522%26pg%3DPA119&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrigonometric+functions" class="Z3988"> </li> <li id="cite_note-9"><a href="#cite_ref-9">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBityutskov2011" class="citation web cs1">Bityutskov, V.I. (2011-02-07). <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php/Trigonometric_functions">"Trigonometric Functions"</a>. Encyclopedia of Mathematics. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20171229231821/https://www.encyclopediaofmath.org/index.php/Trigonometric_functions">Archived</a> from the original on 2017-12-29. Retrieved 2017-12-29.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Encyclopedia+of+Mathematics&rft.atitle=Trigonometric+Functions&rft.date=2011-02-07&rft.aulast=Bityutskov&rft.aufirst=V.I.&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%2FTrigonometric_functions&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrigonometric+functions" class="Z3988"> </li> <li id="cite_note-Larson_2013-10"><a href="#cite_ref-Larson_2013_10-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLarson2013" class="citation book cs1">Larson, Ron (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=zbgWAAAAQBAJ">Trigonometry</a> (9th ed.). Cengage Learning. p. 153. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-285-60718-4" title="Special:BookSources/978-1-285-60718-4"><bdi>978-1-285-60718-4</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20180215144848/https://books.google.com/books?id=zbgWAAAAQBAJ">Archived</a> from the original on 2018-02-15.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Trigonometry&rft.pages=153&rft.edition=9th&rft.pub=Cengage+Learning&rft.date=2013&rft.isbn=978-1-285-60718-4&rft.aulast=Larson&rft.aufirst=Ron&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DzbgWAAAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrigonometric+functions" class="Z3988"> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=zbgWAAAAQBAJ&pg=PA153">Extract of page 153</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20180215144848/https://books.google.com/books?id=zbgWAAAAQBAJ&pg=PA153">Archived</a> 15 February 2018 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> </li> <li id="cite_note-Hardy-12">^ <a href="#cite_ref-Hardy_12-0">a</a> <a href="#cite_ref-Hardy_12-1">b</a> <a href="#cite_ref-Hardy_12-2">c</a> <a href="#cite_ref-Hardy_12-3">d</a> <a href="#cite_ref-Hardy_12-4">e</a> <a href="#cite_ref-Hardy_12-5">f</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHardy1950" class="citation cs2">Hardy, G.H. (1950), A course of pure mathematics (8th ed.), pp. 432–438</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+course+of+pure+mathematics&rft.pages=432-438&rft.edition=8th&rft.date=1950&rft.aulast=Hardy&rft.aufirst=G.H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrigonometric+functions" class="Z3988"> </li> <li id="cite_note-WW-13"><a href="#cite_ref-WW_13-0">^</a> Whittaker, E. T., & Watson, G. N. (1920). A course of modern analysis: an introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions. University press. </li> <li id="cite_note-BS-14"><a href="#cite_ref-BS_14-0">^</a> Bartle, R. G., & Sherbert, D. R. (2000). Introduction to real analysis (3rd ed). Wiley. </li> <li id="cite_note-FOOTNOTEBartleSherbert1999247-15"><a href="#cite_ref-FOOTNOTEBartleSherbert1999247_15-0">^</a> <a href="#CITEREFBartleSherbert1999">Bartle & Sherbert 1999</a>, p. 247. </li> <li id="cite_note-16"><a href="#cite_ref-16">^</a> Whitaker and Watson, p 584 </li> <li id="cite_note-17"><a href="#cite_ref-17">^</a> Stanley, Enumerative Combinatorics, Vol I., p. 149 </li> <li id="cite_note-18"><a href="#cite_ref-18">^</a> Abramowitz; Weisstein. </li> <li id="cite_note-19"><a href="#cite_ref-19">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLambert2004" class="citation cs2">Lambert, Johann Heinrich (2004) [1768], "Mémoire sur quelques propriétés remarquables des quantités transcendantes circulaires et logarithmiques", in Berggren, Lennart; <a href="/wiki/Jonathan_M._Borwein" class="mw-redirect" title="Jonathan M. Borwein">Borwein, Jonathan M.</a>; <a href="/wiki/Peter_B._Borwein" class="mw-redirect" title="Peter B. Borwein">Borwein, Peter B.</a> (eds.), Pi, a source book (3rd ed.), New York: <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer-Verlag</a>, pp. 129–140, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-20571-3" title="Special:BookSources/0-387-20571-3"><bdi>0-387-20571-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=M%C3%A9moire+sur+quelques+propri%C3%A9t%C3%A9s+remarquables+des+quantit%C3%A9s+transcendantes+circulaires+et+logarithmiques&rft.btitle=Pi%2C+a+source+book&rft.place=New+York&rft.pages=129-140&rft.edition=3rd&rft.pub=Springer-Verlag&rft.date=2004&rft.isbn=0-387-20571-3&rft.aulast=Lambert&rft.aufirst=Johann+Heinrich&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrigonometric+functions" class="Z3988"> </li> <li id="cite_note-Aigner_2000-20"><a href="#cite_ref-Aigner_2000_20-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAignerZiegler2000" class="citation book cs1"><a href="/wiki/Martin_Aigner" title="Martin Aigner">Aigner, Martin</a>; <a href="/wiki/G%C3%BCnter_Ziegler" class="mw-redirect" title="Günter Ziegler">Ziegler, Günter M.</a> (2000). <a rel="nofollow" class="external text" href="https://www.springer.com/mathematics/book/978-3-642-00855-9">Proofs from THE BOOK</a> (Second ed.). <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. p. 149. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-00855-9" title="Special:BookSources/978-3-642-00855-9"><bdi>978-3-642-00855-9</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20140308034453/http://www.springer.com/mathematics/book/978-3-642-00855-9">Archived</a> from the original on 2014-03-08.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Proofs+from+THE+BOOK&rft.pages=149&rft.edition=Second&rft.pub=Springer-Verlag&rft.date=2000&rft.isbn=978-3-642-00855-9&rft.aulast=Aigner&rft.aufirst=Martin&rft.au=Ziegler%2C+G%C3%BCnter+M.&rft_id=https%3A%2F%2Fwww.springer.com%2Fmathematics%2Fbook%2F978-3-642-00855-9&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrigonometric+functions" class="Z3988"> </li> <li id="cite_note-Remmert_1991-21"><a href="#cite_ref-Remmert_1991_21-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRemmert1991" class="citation book cs1">Remmert, Reinhold (1991). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=CC0dQxtYb6kC">Theory of complex functions</a>. Springer. p. 327. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-97195-7" title="Special:BookSources/978-0-387-97195-7"><bdi>978-0-387-97195-7</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150320010718/http://books.google.com/books?id=CC0dQxtYb6kC">Archived</a> from the original on 2015-03-20.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theory+of+complex+functions&rft.pages=327&rft.pub=Springer&rft.date=1991&rft.isbn=978-0-387-97195-7&rft.aulast=Remmert&rft.aufirst=Reinhold&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DCC0dQxtYb6kC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrigonometric+functions" class="Z3988"> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=CC0dQxtYb6kC&pg=PA327">Extract of page 327</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150320010448/http://books.google.com/books?id=CC0dQxtYb6kC&pg=PA327">Archived</a> 20 March 2015 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> </li> <li id="cite_note-22"><a href="#cite_ref-22">^</a> Whittaker and Watson, p 137 </li> <li id="cite_note-23"><a href="#cite_ref-23">^</a> Ahlfors, p 197 </li> <li id="cite_note-24"><a href="#cite_ref-24">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBourbaki1981" class="citation book cs1"><a href="/wiki/Nicolas_Bourbaki" title="Nicolas Bourbaki">Bourbaki, Nicolas</a> (1981). Topologie generale. Springer. §VIII.2.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topologie+generale&rft.pages=%C2%A7VIII.2&rft.pub=Springer&rft.date=1981&rft.aulast=Bourbaki&rft.aufirst=Nicolas&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrigonometric+functions" class="Z3988"> </li> <li id="cite_note-25"><a href="#cite_ref-25">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBartle1964" class="citation cs2">Bartle (1964), Elements of real analysis, pp. 315–316</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elements+of+real+analysis&rft.pages=315-316&rft.date=1964&rft.au=Bartle&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrigonometric+functions" class="Z3988"> </li> <li id="cite_note-26"><a href="#cite_ref-26">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeierstrass1841" class="citation book cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Karl_Weierstrass" title="Karl Weierstrass">Weierstrass, Karl</a> (1841). <a rel="nofollow" class="external text" href="https://archive.org/details/mathematischewer01weieuoft/page/51/">"Darstellung einer analytischen Function einer complexen Veränderlichen, deren absoluter Betrag zwischen zwei gegebenen Grenzen liegt"</a> [Representation of an analytical function of a complex variable, whose absolute value lies between two given limits]. Mathematische Werke (in German). Vol. 1. Berlin: Mayer & Müller (published 1894). pp. 51–66.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Darstellung+einer+analytischen+Function+einer+complexen+Ver%C3%A4nderlichen%2C+deren+absoluter+Betrag+zwischen+zwei+gegebenen+Grenzen+liegt&rft.btitle=Mathematische+Werke&rft.place=Berlin&rft.pages=51-66&rft.pub=Mayer+%26+M%C3%BCller&rft.date=1841&rft.aulast=Weierstrass&rft.aufirst=Karl&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmathematischewer01weieuoft%2Fpage%2F51%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrigonometric+functions" class="Z3988"> </li> <li id="cite_note-Kannappan_2009-27"><a href="#cite_ref-Kannappan_2009_27-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKannappan2009" class="citation book cs1">Kannappan, Palaniappan (2009). Functional Equations and Inequalities with Applications. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0387894911" title="Special:BookSources/978-0387894911"><bdi>978-0387894911</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Functional+Equations+and+Inequalities+with+Applications&rft.pub=Springer&rft.date=2009&rft.isbn=978-0387894911&rft.aulast=Kannappan&rft.aufirst=Palaniappan&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrigonometric+functions" class="Z3988"> </li> <li id="cite_note-Allen_1976-28"><a href="#cite_ref-Allen_1976_28-0">^</a> The Universal Encyclopaedia of Mathematics, Pan Reference Books, 1976, pp. 529–530. English version George Allen and Unwin, 1964. Translated from the German version Meyers Rechenduden, 1960. </li> <li id="cite_note-Farlow_1993-29"><a href="#cite_ref-Farlow_1993_29-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFarlow1993" class="citation book cs1"><a href="/wiki/Stanley_Farlow" title="Stanley Farlow">Farlow, Stanley J.</a> (1993). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=DLUYeSb49eAC&pg=PA82">Partial differential equations for scientists and engineers</a> (Reprint of Wiley 1982 ed.). Courier Dover Publications. p. 82. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-67620-3" title="Special:BookSources/978-0-486-67620-3"><bdi>978-0-486-67620-3</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150320011420/http://books.google.com/books?id=DLUYeSb49eAC&pg=PA82">Archived</a> from the original on 2015-03-20.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Partial+differential+equations+for+scientists+and+engineers&rft.pages=82&rft.edition=Reprint+of+Wiley+1982&rft.pub=Courier+Dover+Publications&rft.date=1993&rft.isbn=978-0-486-67620-3&rft.aulast=Farlow&rft.aufirst=Stanley+J.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DDLUYeSb49eAC%26pg%3DPA82&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrigonometric+functions" class="Z3988"> </li> <li id="cite_note-Folland_1992-30"><a href="#cite_ref-Folland_1992_30-0">^</a> See for example, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFolland2009" class="citation book cs1">Folland, Gerald B. (2009). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=idAomhpwI8MC&pg=PA77">"Convergence and completeness"</a>. Fourier Analysis and its Applications (Reprint of Wadsworth & Brooks/Cole 1992 ed.). American Mathematical Society. pp. 77ff. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-4790-9" title="Special:BookSources/978-0-8218-4790-9"><bdi>978-0-8218-4790-9</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150319230954/http://books.google.com/books?id=idAomhpwI8MC&pg=PA77">Archived</a> from the original on 2015-03-19.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Convergence+and+completeness&rft.btitle=Fourier+Analysis+and+its+Applications&rft.pages=77ff&rft.edition=Reprint+of+Wadsworth+%26+Brooks%2FCole+1992&rft.pub=American+Mathematical+Society&rft.date=2009&rft.isbn=978-0-8218-4790-9&rft.aulast=Folland&rft.aufirst=Gerald+B.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DidAomhpwI8MC%26pg%3DPA77&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrigonometric+functions" class="Z3988"> </li> <li id="cite_note-Boyer_1991-31">^ <a href="#cite_ref-Boyer_1991_31-0">a</a> <a href="#cite_ref-Boyer_1991_31-1">b</a> <a href="#cite_ref-Boyer_1991_31-2">c</a> Boyer, Carl B. (1991). A History of Mathematics (Second ed.). John Wiley & Sons, Inc. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-54397-7" title="Special:BookSources/0-471-54397-7">0-471-54397-7</a>, p. 210. </li> <li id="cite_note-Gingerich_1986-32">^ <a href="#cite_ref-Gingerich_1986_32-0">a</a> <a href="#cite_ref-Gingerich_1986_32-1">b</a> <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>. <a href="/wiki/Scientific_American" title="Scientific American">Scientific American</a>. 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. Retrieved 2010-07-13.</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%3ATrigonometric+functions" class="Z3988"> </li> <li id="cite_note-Sesiano-33"><a href="#cite_ref-Sesiano_33-0">^</a> 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). Mathematics Across Cultures: The History of Non-western Mathematics. <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%3ATrigonometric+functions" class="Z3988"> </li> <li id="cite_note-Britannica-34">^ <a href="#cite_ref-Britannica_34-0">a</a> <a href="#cite_ref-Britannica_34-1">b</a> <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. 2023-11-17.</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=2023-11-17&rft_id=http%3A%2F%2Fwww.britannica.com%2FEBchecked%2Ftopic%2F605281%2Ftrigonometry&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrigonometric+functions" class="Z3988"> </li> <li id="cite_note-mact-biog-35"><a href="#cite_ref-mact-biog_35-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFO'ConnorRobertson" class="citation web cs1">O'Connor, J. J.; Robertson, E. F. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20060514012903/http://www-gap.dcs.st-and.ac.uk/~history/Biographies/Madhava.html">"Madhava of Sangamagrama"</a>. School of Mathematics and Statistics University of St Andrews, Scotland. Archived from <a rel="nofollow" class="external text" href="http://www-gap.dcs.st-and.ac.uk/~history/Biographies/Madhava.html">the original</a> on 2006-05-14. Retrieved 2007-09-08.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Madhava+of+Sangamagrama&rft.pub=School+of+Mathematics+and+Statistics+University+of+St+Andrews%2C+Scotland&rft.aulast=O%27Connor&rft.aufirst=J.+J.&rft.au=Robertson%2C+E.+F.&rft_id=http%3A%2F%2Fwww-gap.dcs.st-and.ac.uk%2F~history%2FBiographies%2FMadhava.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrigonometric+functions" class="Z3988"> </li> <li id="cite_note-36"><a href="#cite_ref-36">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVan_Brummelen2018" class="citation journal cs1">Van Brummelen, Glen (2018). <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/45211959">"The end of an error: Bianchini, Regiomontanus, and the tabulation of stellar coordinates"</a>. Archive for History of Exact Sciences. 72 (5): 547–563. <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%2Fs00407-018-0214-2">10.1007/s00407-018-0214-2</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/45211959">45211959</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:240294796">240294796</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Archive+for+History+of+Exact+Sciences&rft.atitle=The+end+of+an+error%3A+Bianchini%2C+Regiomontanus%2C+and+the+tabulation+of+stellar+coordinates&rft.volume=72&rft.issue=5&rft.pages=547-563&rft.date=2018&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A240294796%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F45211959%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.1007%2Fs00407-018-0214-2&rft.aulast=Van+Brummelen&rft.aufirst=Glen&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F45211959&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrigonometric+functions" class="Z3988"> </li> <li id="cite_note-Fincke-37"><a href="#cite_ref-Fincke_37-0">^</a> <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-history.mcs.st-andrews.ac.uk/Biographies/Fincke.html">"Fincke biography"</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170107035144/http://www-history.mcs.st-andrews.ac.uk/Biographies/Fincke.html">Archived</a> from the original on 2017-01-07. Retrieved 2017-03-15.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Fincke+biography&rft_id=http%3A%2F%2Fwww-history.mcs.st-andrews.ac.uk%2FBiographies%2FFincke.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrigonometric+functions" class="Z3988"> </li> <li id="cite_note-MacTutor-38"><a href="#cite_ref-MacTutor_38-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFO'ConnorRobertson" class="citation cs2">O'Connor, John J.; <a href="/wiki/Edmund_F._Robertson" title="Edmund F. Robertson">Robertson, Edmund F.</a>, <a rel="nofollow" class="external text" href="https://mathshistory.st-andrews.ac.uk/Biographies/Girard_Albert.html">"Trigonometric functions"</a>, <a href="/wiki/MacTutor_History_of_Mathematics_Archive" title="MacTutor History of Mathematics Archive">MacTutor History of Mathematics Archive</a>, <a href="/wiki/University_of_St_Andrews" title="University of St Andrews">University of St Andrews</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Trigonometric+functions&rft.btitle=MacTutor+History+of+Mathematics+Archive&rft.pub=University+of+St+Andrews&rft.aulast=O%27Connor&rft.aufirst=John+J.&rft.au=Robertson%2C+Edmund+F.&rft_id=https%3A%2F%2Fmathshistory.st-andrews.ac.uk%2FBiographies%2FGirard_Albert.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrigonometric+functions" class="Z3988"> </li> <li id="cite_note-Bourbaki_1994-39"><a href="#cite_ref-Bourbaki_1994_39-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBourbaki1994" class="citation book cs1">Bourbaki, Nicolás (1994). <a rel="nofollow" class="external text" href="https://archive.org/details/elementsofhistor0000bour">Elements of the History of Mathematics</a>. 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.aulast=Bourbaki&rft.aufirst=Nicol%C3%A1s&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Felementsofhistor0000bour&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrigonometric+functions" class="Z3988"> </li> <li id="cite_note-40"><a href="#cite_ref-40">^</a> <a href="#CITEREFNielsen1966">Nielsen (1966</a>, pp. xxiii–xxiv) </li> <li id="cite_note-Hammer_1897-41"><a href="#cite_ref-Hammer_1897_41-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFvon_Hammer1897" class="citation book cs1 cs1-prop-interwiki-linked-name cs1-prop-foreign-lang-source"><a href="https://de.wikipedia.org/wiki/Ernst_von_Hammer" class="extiw" title="de:Ernst von Hammer">von Hammer, Ernst Hermann Heinrich</a> [in German], ed. (1897). <a rel="nofollow" class="external text" href="https://quod.lib.umich.edu/u/umhistmath/ABN6964.0001.001/?view=toc">Lehrbuch der ebenen und sphärischen Trigonometrie. Zum Gebrauch bei Selbstunterricht und in Schulen, besonders als Vorbereitung auf Geodäsie und sphärische Astronomie</a> (in German) (2 ed.). Stuttgart, Germany: <a href="/wiki/J._B._Metzlerscher_Verlag" class="mw-redirect" title="J. B. Metzlerscher Verlag">J. B. Metzlerscher Verlag</a>. Retrieved 2024-02-06.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lehrbuch+der+ebenen+und+sph%C3%A4rischen+Trigonometrie.+Zum+Gebrauch+bei+Selbstunterricht+und+in+Schulen%2C+besonders+als+Vorbereitung+auf+Geod%C3%A4sie+und+sph%C3%A4rische+Astronomie&rft.place=Stuttgart%2C+Germany&rft.edition=2&rft.pub=J.+B.+Metzlerscher+Verlag&rft.date=1897&rft_id=https%3A%2F%2Fquod.lib.umich.edu%2Fu%2Fumhistmath%2FABN6964.0001.001%2F%3Fview%3Dtoc&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrigonometric+functions" class="Z3988"> </li> <li id="cite_note-Heß_1916-42"><a href="#cite_ref-Heß_1916_42-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHeß1926" class="citation book cs1 cs1-prop-foreign-lang-source">Heß, Adolf (1926) [1916]. Trigonometrie für Maschinenbauer und Elektrotechniker - Ein Lehr- und Aufgabenbuch für den Unterricht und zum Selbststudium (in German) (6 ed.). Winterthur, Switzerland: 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%2F978-3-662-36585-4">10.1007/978-3-662-36585-4</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-662-35755-2" title="Special:BookSources/978-3-662-35755-2"><bdi>978-3-662-35755-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Trigonometrie+f%C3%BCr+Maschinenbauer+und+Elektrotechniker+-+Ein+Lehr-+und+Aufgabenbuch+f%C3%BCr+den+Unterricht+und+zum+Selbststudium&rft.place=Winterthur%2C+Switzerland&rft.edition=6&rft.pub=Springer&rft.date=1926&rft_id=info%3Adoi%2F10.1007%2F978-3-662-36585-4&rft.isbn=978-3-662-35755-2&rft.aulast=He%C3%9F&rft.aufirst=Adolf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrigonometric+functions" class="Z3988"> </li> <li id="cite_note-Lötzbeyer_1950-43"><a href="#cite_ref-Lötzbeyer_1950_43-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLötzbeyer1950" class="citation book cs1 cs1-prop-foreign-lang-source">Lötzbeyer, Philipp (1950). <a rel="nofollow" class="external text" href="https://www.degruyter.com/document/doi/10.1515/9783111507545-015/html">"§ 14. Erläuterungen u. Beispiele zu T. 13: lg sin X; lg cos X und T. 14: lg tg x; lg ctg X"</a>. <a rel="nofollow" class="external text" href="https://www.degruyter.com/document/doi/10.1515/9783111507545/html">Erläuterungen und Beispiele für den Gebrauch der vierstelligen Tafeln zum praktischen Rechnen</a> (in German) (1 ed.). Berlin, Germany: <a href="/wiki/Walter_de_Gruyter_%26_Co." class="mw-redirect" title="Walter de Gruyter & Co.">Walter de Gruyter & Co.</a> <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1515%2F9783111507545">10.1515/9783111507545</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-11114038-4" title="Special:BookSources/978-3-11114038-4"><bdi>978-3-11114038-4</bdi></a>. Archive ID 541650. Retrieved 2024-02-06.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=%C2%A7+14.+Erl%C3%A4uterungen+u.+Beispiele+zu+T.+13%3A+lg+sin+X%3B+lg+cos+X+und+T.+14%3A+lg+tg+x%3B+lg+ctg+X&rft.btitle=Erl%C3%A4uterungen+und+Beispiele+f%C3%BCr+den+Gebrauch+der+vierstelligen+Tafeln+zum+praktischen+Rechnen&rft.place=Berlin%2C+Germany&rft.edition=1&rft.pub=Walter+de+Gruyter+%26+Co.&rft.date=1950&rft_id=info%3Adoi%2F10.1515%2F9783111507545&rft.isbn=978-3-11114038-4&rft.aulast=L%C3%B6tzbeyer&rft.aufirst=Philipp&rft_id=https%3A%2F%2Fwww.degruyter.com%2Fdocument%2Fdoi%2F10.1515%2F9783111507545-015%2Fhtml&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrigonometric+functions" class="Z3988"> </li> <li id="cite_note-Roegel_2016-44"><a href="#cite_ref-Roegel_2016_44-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRoegel2016" class="citation book cs1">Roegel, Denis, ed. (2016-08-30). <a rel="nofollow" class="external text" href="https://inria.hal.science/hal-01357842/document">A reconstruction of Peters's table of 7-place logarithms (volume 2, 1940)</a>. Vandoeuvre-lès-Nancy, France: <a href="/wiki/Universit%C3%A9_de_Lorraine" class="mw-redirect" title="Université de Lorraine">Université de Lorraine</a>. hal-01357842. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20240206211422/https://inria.hal.science/hal-01357842/document">Archived</a> from the original on 2024-02-06. Retrieved 2024-02-06.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+reconstruction+of+Peters%27s+table+of+7-place+logarithms+%28volume+2%2C+1940%29&rft.place=Vandoeuvre-l%C3%A8s-Nancy%2C+France&rft.pub=Universit%C3%A9+de+Lorraine&rft.date=2016-08-30&rft_id=https%3A%2F%2Finria.hal.science%2Fhal-01357842%2Fdocument&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrigonometric+functions" class="Z3988"> </li> <li id="cite_note-45"><a href="#cite_ref-45">^</a> The anglicized form is first recorded in 1593 in <a href="/wiki/Thomas_Fale" title="Thomas Fale">Thomas Fale</a>'s Horologiographia, the Art of Dialling. </li> <li id="cite_note-46"><a href="#cite_ref-46">^</a> Various sources credit the first use of sinus to either <ul><li><a href="/wiki/Plato_Tiburtinus" title="Plato Tiburtinus">Plato Tiburtinus</a>'s 1116 translation of the Astronomy 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 Algebra 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 Merlet, <a rel="nofollow" class="external text" href="https://link.springer.com/chapter/10.1007/1-4020-2204-2_16#page-1">A Note on the History of the Trigonometric Functions</a> in Ceccarelli (ed.), International Symposium on History of Machines and Mechanisms, Springer, 2004 See Maor (1998), chapter 3, for an earlier etymology crediting Gerard. See <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKatx2008" class="citation book cs1">Katx, Victor (July 2008). A history of mathematics (3rd ed.). Boston: <a href="/wiki/Pearson_(publisher)" class="mw-redirect" title="Pearson (publisher)">Pearson</a>. p. 210 (sidebar). <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0321387004" title="Special:BookSources/978-0321387004"><bdi>978-0321387004</bdi></a>.</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.pages=210+%28sidebar%29&rft.edition=3rd&rft.pub=Pearson&rft.date=2008-07&rft.isbn=978-0321387004&rft.aulast=Katx&rft.aufirst=Victor&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrigonometric+functions" class="Z3988"> </li> <li id="cite_note-Plofker_2009-47"><a href="#cite_ref-Plofker_2009_47-0">^</a> See Plofker, <a href="/wiki/Mathematics_in_India_(book)" title="Mathematics in India (book)">Mathematics in India</a>, Princeton University Press, 2009, p. 257 See <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.clarku.edu/~djoyce/trig/">"Clark University"</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20080615133310/http://www.clarku.edu/~djoyce/trig/">Archived</a> from the original on 2008-06-15.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Clark+University&rft_id=http%3A%2F%2Fwww.clarku.edu%2F~djoyce%2Ftrig%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrigonometric+functions" class="Z3988"> See Maor (1998), chapter 3, regarding the etymology. </li> <li id="cite_note-48"><a href="#cite_ref-48">^</a> Oxford English Dictionary </li> <li id="cite_note-Gunter_1620-49"><a href="#cite_ref-Gunter_1620_49-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGunter1620" class="citation book cs1"><a href="/wiki/Edmund_Gunter" title="Edmund Gunter">Gunter, Edmund</a> (1620). Canon triangulorum.</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%3ATrigonometric+functions" class="Z3988"> </li> <li id="cite_note-Roegel_2010-50"><a href="#cite_ref-Roegel_2010_50-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRoegel2010" class="citation web cs1">Roegel, Denis, ed. (2010-12-06). <a rel="nofollow" class="external text" href="https://hal.inria.fr/inria-00543938/document">"A reconstruction of Gunter's Canon triangulorum (1620)"</a> (Research report). HAL. inria-00543938. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170728192238/https://hal.inria.fr/inria-00543938/document">Archived</a> from the original on 2017-07-28. Retrieved 2017-07-28.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=A+reconstruction+of+Gunter%27s+Canon+triangulorum+%281620%29&rft.pub=HAL&rft.date=2010-12-06&rft_id=https%3A%2F%2Fhal.inria.fr%2Finria-00543938%2Fdocument&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrigonometric+functions" class="Z3988"> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Trigonometric_functions&action=edit&section=36" title="Edit section: References">edit</a>]</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" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAbramowitzStegun1983" class="citation book cs1"><a href="/wiki/Milton_Abramowitz" title="Milton Abramowitz">Abramowitz, Milton</a>; <a href="/wiki/Irene_Stegun" title="Irene Stegun">Stegun, Irene Ann</a>, eds. (1983) [June 1964]. <a href="/wiki/Abramowitz_and_Stegun" title="Abramowitz and Stegun">Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables</a>. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-61272-0" title="Special:BookSources/978-0-486-61272-0"><bdi>978-0-486-61272-0</bdi></a>. <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/64-60036">64-60036</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=0167642">0167642</a>. <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://www.loc.gov/item/65012253">65-12253</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+Mathematical+Functions+with+Formulas%2C+Graphs%2C+and+Mathematical+Tables&rft.place=Washington+D.C.%3B+New+York&rft.series=Applied+Mathematics+Series&rft.edition=Ninth+reprint+with+additional+corrections+of+tenth+original+printing+with+corrections+%28December+1972%29%3B+first&rft.pub=United+States+Department+of+Commerce%2C+National+Bureau+of+Standards%3B+Dover+Publications&rft.date=1983&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0167642%23id-name%3DMR&rft_id=info%3Alccn%2F64-60036&rft.isbn=978-0-486-61272-0&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrigonometric+functions" class="Z3988"></li> <li><a href="/wiki/Lars_Ahlfors" title="Lars Ahlfors">Lars Ahlfors</a>, Complex Analysis: an introduction to the theory of analytic functions of one complex variable, second edition, <a href="/wiki/McGraw-Hill_Book_Company" class="mw-redirect" title="McGraw-Hill Book Company">McGraw-Hill Book Company</a>, New York, 1966.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBartleSherbert1999" class="citation book cs1"><a href="/wiki/Robert_G._Bartle" title="Robert G. Bartle">Bartle, Robert G.</a>; Sherbert, Donald R. (1999). Introduction to Real Analysis (3rd ed.). Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780471321484" title="Special:BookSources/9780471321484"><bdi>9780471321484</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Real+Analysis&rft.edition=3rd&rft.pub=Wiley&rft.date=1999&rft.isbn=9780471321484&rft.aulast=Bartle&rft.aufirst=Robert+G.&rft.au=Sherbert%2C+Donald+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrigonometric+functions" class="Z3988"></li> <li><a href="/wiki/Carl_Benjamin_Boyer" title="Carl Benjamin Boyer">Boyer, Carl B.</a>, A History of Mathematics, John Wiley & Sons, Inc., 2nd edition. (1991). <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-54397-7" title="Special:BookSources/0-471-54397-7">0-471-54397-7</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCajori1929" class="citation book cs1"><a href="/wiki/Florian_Cajori" title="Florian Cajori">Cajori, Florian</a> (1929). <a rel="nofollow" class="external text" href="https://archive.org/details/b29980343_0002/page/142/">"§2.2.1. Trigonometric Notations"</a>. A History of Mathematical Notations. Vol. 2. Open Court. pp. 142–179 (¶511–537).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=%C2%A72.2.1.+Trigonometric+Notations&rft.btitle=A+History+of+Mathematical+Notations&rft.pages=142-179+%28%C2%B6511-537%29&rft.pub=Open+Court&rft.date=1929&rft.aulast=Cajori&rft.aufirst=Florian&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fb29980343_0002%2Fpage%2F142%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrigonometric+functions" class="Z3988"></li> <li>Gal, Shmuel and Bachelis, Boris. An accurate elementary mathematical library for the IEEE floating point standard, ACM Transactions on Mathematical Software (1991).</li> <li>Joseph, George G., The Crest of the Peacock: Non-European Roots of Mathematics, 2nd ed. <a href="/wiki/Penguin_Books" title="Penguin Books">Penguin Books</a>, London. (2000). <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-691-00659-8" title="Special:BookSources/0-691-00659-8">0-691-00659-8</a>.</li> <li>Kantabutra, Vitit, "On hardware for computing exponential and trigonometric functions," IEEE Trans. Computers 45 (3), 328–339 (1996).</li> <li>Maor, Eli, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20040404234808/http://www.pupress.princeton.edu/books/maor/">Trigonometric Delights</a>, Princeton Univ. Press. (1998). Reprint edition (2002): <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-691-09541-8" title="Special:BookSources/0-691-09541-8">0-691-09541-8</a>.</li> <li>Needham, Tristan, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20040602145226/http://www.usfca.edu/vca/PDF/vca-preface.pdf">"Preface"</a>" to <a rel="nofollow" class="external text" href="http://www.usfca.edu/vca/">Visual Complex Analysis</a>. Oxford University Press, (1999). <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-853446-9" title="Special:BookSources/0-19-853446-9">0-19-853446-9</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNielsen1966" class="citation cs2">Nielsen, Kaj L. (1966), Logarithmic and Trigonometric Tables to Five Places (2nd ed.), New York: <a href="/wiki/Barnes_%26_Noble" title="Barnes & Noble">Barnes & Noble</a>, <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/61-9103">61-9103</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Logarithmic+and+Trigonometric+Tables+to+Five+Places&rft.place=New+York&rft.edition=2nd&rft.pub=Barnes+%26+Noble&rft.date=1966&rft_id=info%3Alccn%2F61-9103&rft.aulast=Nielsen&rft.aufirst=Kaj+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrigonometric+functions" class="Z3988"></li> <li>O'Connor, J. J., and E. F. Robertson, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130120084848/http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Trigonometric_functions.html">"Trigonometric functions"</a>, <a href="/wiki/MacTutor_History_of_Mathematics_archive" class="mw-redirect" title="MacTutor History of Mathematics archive">MacTutor History of Mathematics archive</a>. (1996).</li> <li>O'Connor, J. J., and E. F. Robertson, <a rel="nofollow" class="external text" href="http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Madhava.html">"Madhava of Sangamagramma"</a>, <a href="/wiki/MacTutor_History_of_Mathematics_archive" class="mw-redirect" title="MacTutor History of Mathematics archive">MacTutor History of Mathematics archive</a>. (2000).</li> <li>Pearce, Ian G., <a rel="nofollow" class="external text" href="http://www-history.mcs.st-andrews.ac.uk/history/Projects/Pearce/Chapters/Ch9_3.html">"Madhava of Sangamagramma"</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20060505201342/http://www-history.mcs.st-andrews.ac.uk/history/Projects/Pearce/Chapters/Ch9_3.html">Archived</a> 2006-05-05 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, <a href="/wiki/MacTutor_History_of_Mathematics_archive" class="mw-redirect" title="MacTutor History of Mathematics archive">MacTutor History of Mathematics archive</a>. (2002).</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFProtterMorrey1970" class="citation cs2">Protter, Murray H.; Morrey, Charles B. Jr. (1970), College Calculus with Analytic Geometry (2nd ed.), Reading: <a href="/wiki/Addison-Wesley" title="Addison-Wesley">Addison-Wesley</a>, <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/76087042">76087042</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=College+Calculus+with+Analytic+Geometry&rft.place=Reading&rft.edition=2nd&rft.pub=Addison-Wesley&rft.date=1970&rft_id=info%3Alccn%2F76087042&rft.aulast=Protter&rft.aufirst=Murray+H.&rft.au=Morrey%2C+Charles+B.+Jr.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrigonometric+functions" class="Z3988"></li> <li>Weisstein, Eric W., <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Tangent.html">"Tangent"</a> from <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>, accessed 21 January 2006.</li></ul> </div> <div class="mw-heading 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src="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/40px-Wikibooks-logo-en-noslogan.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/60px-Wikibooks-logo-en-noslogan.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/80px-Wikibooks-logo-en-noslogan.svg.png 2x" data-file-width="400" data-file-height="400" /></div> <div class="side-box-text plainlist">Wikibooks has a book on the topic of: <a href="https://en.wikibooks.org/wiki/Trigonometry" class="extiw" title="wikibooks:Trigonometry">Trigonometry</a></div></div> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Trigonometric_functions">"Trigonometric functions"</a>, <a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Trigonometric+functions&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DTrigonometric_functions&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrigonometric+functions" class="Z3988"></li> <li><a rel="nofollow" class="external text" href="https://www.visionlearning.com/library/module_viewer.php?mid=131&l=&c3=">Visionlearning Module on Wave Mathematics</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20071006172054/http://glab.trixon.se/">GonioLab</a> Visualization of the unit circle, trigonometric and hyperbolic functions</li> <li><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/q-Sine.html">q-Sine</a> Article about the <a href="/wiki/Q-analog" title="Q-analog">q-analog</a> of sin at <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></li> <li><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/q-Cosine.html">q-Cosine</a> Article about the <a href="/wiki/Q-analog" title="Q-analog">q-analog</a> of cos at <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output 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style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><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_functions" 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 class="mw-selflink selflink">Trigonometric</a> <ul><li><a href="/wiki/Sine_and_cosine" title="Sine and cosine">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 class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"><style data-mw-deduplicate="TemplateStyles:r1038841319">.mw-parser-output .tooltip-dotted{border-bottom:1px dotted;cursor:help}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1038841319"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1038841319"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1038841319"><link 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/></a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">International</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><a rel="nofollow" class="external text" href="http://id.worldcat.org/fast/1156712/">FAST</a></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">National</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4186137-1">Germany</a></li><li><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/sh85137518">United States</a></li><li><a rel="nofollow" class="external text" href="https://catalogue.bnf.fr/ark:/12148/cb12168469v">France</a></li><li><a rel="nofollow" class="external text" href="https://data.bnf.fr/ark:/12148/cb12168469v">BnF data</a></li><li><a rel="nofollow" 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Trigonometric functions - Wikipedia