Funció trigonomètrica - Viquipèdia, l'enciclopèdia lliure

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-sticky-header-enabled vector-toc-available" lang="ca" dir="ltr"> <head> <meta charset="UTF-8"> <title>Funció trigonomètrica - Viquipèdia, l'enciclopèdia lliure</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-sticky-header-enabled vector-toc-available";var cookie=document.cookie.match(/(?:^|; )cawikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":[",\t.",".\t,"],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"dmy","wgMonthNames":["","gener","febrer","març","abril","maig","juny","juliol","agost","setembre","octubre","novembre","desembre"],"wgRequestId":"eca4b834-736f-468f-902e-70b942b2d9bd","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Funció_trigonomètrica","wgTitle":"Funció trigonomètrica","wgCurRevisionId":34633281,"wgRevisionId":34633281,"wgArticleId":160783,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Articles amb enllaços externs no actius","Articles amb la plantilla Webarchive amb enllaç wayback","Articles amb referències sense format","Pàgines amb enllaç commonscat des de Wikidata","Llista d'articles de qualitat","Control d'autoritats","Articles de qualitat de matemàtiques","Trigonometria","Funcions"],"wgPageViewLanguage":"ca","wgPageContentLanguage":"ca","wgPageContentModel":"wikitext","wgRelevantPageName":"Funció_trigonomètrica","wgRelevantArticleId":160783,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgRedirectedFrom":"Tangent_(trigonometria)","wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":true,"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"ca","pageLanguageDir":"ltr","pageVariantFallbacks":"ca"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":true,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":60000,"wgInternalRedirectTargetUrl":"/wiki/Funci%C3%B3_trigonom%C3%A8trica#Tangent","wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q93344","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGELevelingUpEnabledForUser":false}; RLSTATE={"ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.math.styles":"ready","ext.cite.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","wikibase.client.data-bridge.externalModifiers":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["mediawiki.action.view.redirect","ext.cite.ux-enhancements","mediawiki.page.media","site","mediawiki.page.ready","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.UkensKonkurranse","ext.gadget.refToolbar","ext.gadget.charinsert","ext.gadget.AltresViccionari","ext.gadget.purgetab","ext.gadget.DocTabs","ext.gadget.switcher","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","wikibase.client.data-bridge.init","ext.checkUser.clientHints","ext.quicksurveys.init","ext.growthExperiments.SuggestedEditSession","oojs-ui.styles.icons-media","oojs-ui-core.icons"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=ca&modules=ext.cite.styles%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.data-bridge.externalModifiers%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=ca&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=ca&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.22"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Circle-trig6.svg/1200px-Circle-trig6.svg.png"> <meta property="og:image:width" content="1200"> <meta property="og:image:height" content="768"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Circle-trig6.svg/800px-Circle-trig6.svg.png"> <meta property="og:image:width" content="800"> <meta property="og:image:height" content="512"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Circle-trig6.svg/640px-Circle-trig6.svg.png"> <meta property="og:image:width" content="640"> <meta property="og:image:height" content="410"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Funció trigonomètrica - Viquipèdia, l'enciclopèdia lliure"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//ca.m.wikipedia.org/wiki/Funci%C3%B3_trigonom%C3%A8trica#Tangent"> <link rel="alternate" type="application/x-wiki" title="Modifica" href="/w/index.php?title=Funci%C3%B3_trigonom%C3%A8trica&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Viquipèdia (ca)"> <link rel="EditURI" type="application/rsd+xml" href="//ca.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://ca.wikipedia.org/wiki/Funci%C3%B3_trigonom%C3%A8trica#Tangent"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.ca"> <link rel="alternate" type="application/atom+xml" title="Canal de sindicació Atom Viquipèdia" href="/w/index.php?title=Especial:Canvis_recents&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="auth.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Funció_trigonomètrica rootpage-Funció_trigonomètrica skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Vés al contingut</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Lloc"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" title="Menú principal" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Menú principal" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">Menú principal</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Menú principal</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">mou a la barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">amaga</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navegació </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Portada" title="Visiteu la pàgina principal [z]" accesskey="z"><span>Portada</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Especial:Article_aleatori" title="Carrega una pàgina a l’atzar [x]" accesskey="x"><span>Article a l'atzar</span></a></li><li id="n-Articles-de-qualitat" class="mw-list-item"><a href="/wiki/Viquip%C3%A8dia:Articles_de_qualitat"><span>Articles de qualitat</span></a></li><li id="n-specialpages" class="mw-list-item"><a href="/wiki/Especial:P%C3%A0gines_especials"><span>Pàgines especials</span></a></li> </ul> </div> </div> <div id="p-Comunitat" class="vector-menu mw-portlet mw-portlet-Comunitat" > <div class="vector-menu-heading"> Comunitat </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-portal" class="mw-list-item"><a href="/wiki/Viquip%C3%A8dia:Portal" title="Sobre el projecte, què podeu fer, on trobareu les coses"><span>Portal viquipedista</span></a></li><li id="n-Agenda-d'actes" class="mw-list-item"><a href="/wiki/Viquip%C3%A8dia:Trobades"><span>Agenda d'actes</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Especial:Canvis_recents" title="Una llista dels canvis recents al wiki [r]" accesskey="r"><span>Canvis recents</span></a></li><li id="n-La-taverna" class="mw-list-item"><a href="/wiki/Viquip%C3%A8dia:La_taverna"><span>La taverna</span></a></li><li id="n-contactpage" class="mw-list-item"><a href="/wiki/Viquip%C3%A8dia:Contacte"><span>Contacte</span></a></li><li id="n-Xat" class="mw-list-item"><a href="/wiki/Viquip%C3%A8dia:Canals_IRC"><span>Xat</span></a></li><li id="n-help" class="mw-list-item"><a href="/wiki/Viquip%C3%A8dia:Ajuda" title="El lloc per a saber més coses"><span>Ajuda</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Portada" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Viquipèdia" src="/static/images/mobile/copyright/wikipedia-wordmark-ca.svg" style="width: 7.5em; height: 1.4375em;"> <img class="mw-logo-tagline" alt="l'Enciclopèdia Lliure" src="/static/images/mobile/copyright/wikipedia-tagline-ca.svg" width="120" height="14" style="width: 7.5em; height: 0.875em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Especial:Cerca" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Cerca a la Viquipèdia [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Cerca</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Cerca a Viquipèdia" aria-label="Cerca a Viquipèdia" autocapitalize="sentences" title="Cerca a la Viquipèdia [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Especial:Cerca"> </div> <button class="cdx-button cdx-search-input__end-button">Cerca</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Eines personals"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Aparença"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Modificar la dimension, la largor e la color del tèxte." > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Aparença" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Aparença</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=ca.wikipedia.org&uselang=ca" class=""><span>Donatius</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Especial:Crea_compte&returnto=Funci%C3%B3+trigonom%C3%A8trica" title="Us animem a crear un compte i iniciar una sessió, encara que no és obligatori" class=""><span>Crea un compte</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Especial:Registre_i_entrada&returnto=Funci%C3%B3+trigonom%C3%A8trica" title="Us animem a registrar-vos, però no és obligatori [o]" accesskey="o" class=""><span>Inicia la sessió</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Més opcions" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Eines personals" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Eines personals</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="Menú d'usuari" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=ca.wikipedia.org&uselang=ca"><span>Donatius</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Especial:Crea_compte&returnto=Funci%C3%B3+trigonom%C3%A8trica" title="Us animem a crear un compte i iniciar una sessió, encara que no és obligatori"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Crea un compte</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Especial:Registre_i_entrada&returnto=Funci%C3%B3+trigonom%C3%A8trica" title="Us animem a registrar-vos, però no és obligatori [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Inicia la sessió</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Pàgines per a editors no registrats <a href="/wiki/Ajuda:Introducci%C3%B3" aria-label="Vegeu més informació sobre l'edició"><span>més informació</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Especial:Contribucions_pr%C3%B2pies" title="Una llista de les modificacions fetes des d'aquesta adreça IP [y]" accesskey="y"><span>Contribucions</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Especial:Discussi%C3%B3_personal" title="Discussió sobre les edicions per aquesta adreça ip. [n]" accesskey="n"><span>Discussió per aquest IP</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Lloc"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Contingut" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Contingut</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">mou a la barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">amaga</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Inici</div> </a> </li> <li id="toc-Història" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Història"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Història</span> </div> </a> <ul id="toc-Història-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Etimologia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Etimologia"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Etimologia</span> </div> </a> <ul id="toc-Etimologia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Definició_de_les_funcions_trigonomètriques" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definició_de_les_funcions_trigonomètriques"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Definició de les funcions trigonomètriques</span> </div> </a> <button aria-controls="toc-Definició_de_les_funcions_trigonomètriques-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Commuta la subsecció Definició de les funcions trigonomètriques</span> </button> <ul id="toc-Definició_de_les_funcions_trigonomètriques-sublist" class="vector-toc-list"> <li id="toc-Basant-se_en_el_triangle_rectangle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Basant-se_en_el_triangle_rectangle"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Basant-se en el triangle rectangle</span> </div> </a> <ul id="toc-Basant-se_en_el_triangle_rectangle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Basant-se_en_la_circumferència_goniomètrica" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Basant-se_en_la_circumferència_goniomètrica"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Basant-se en la circumferència goniomètrica</span> </div> </a> <ul id="toc-Basant-se_en_la_circumferència_goniomètrica-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Basant-se_en_sèries" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Basant-se_en_sèries"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Basant-se en sèries</span> </div> </a> <ul id="toc-Basant-se_en_sèries-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Emprant_equacions_diferencials" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Emprant_equacions_diferencials"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Emprant equacions diferencials</span> </div> </a> <ul id="toc-Emprant_equacions_diferencials-sublist" class="vector-toc-list"> <li id="toc-Conseqüències_de_fer_servir_radiants" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Conseqüències_de_fer_servir_radiants"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4.1</span> <span>Conseqüències de fer servir radiants</span> </div> </a> <ul id="toc-Conseqüències_de_fer_servir_radiants-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Emprant_equacions_funcionals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Emprant_equacions_funcionals"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Emprant equacions funcionals</span> </div> </a> <ul id="toc-Emprant_equacions_funcionals-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Relació_de_les_funcions_trigonomètriques_amb_la_funció_exponencial_i_els_nombres_complexos" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Relació_de_les_funcions_trigonomètriques_amb_la_funció_exponencial_i_els_nombres_complexos"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Relació de les funcions trigonomètriques amb la funció exponencial i els nombres complexos</span> </div> </a> <ul id="toc-Relació_de_les_funcions_trigonomètriques_amb_la_funció_exponencial_i_els_nombres_complexos-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Identitats_trigonomètriques" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Identitats_trigonomètriques"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Identitats trigonomètriques</span> </div> </a> <button aria-controls="toc-Identitats_trigonomètriques-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Commuta la subsecció Identitats trigonomètriques</span> </button> <ul id="toc-Identitats_trigonomètriques-sublist" class="vector-toc-list"> <li id="toc-Identitats_trigonomètriques_en_càlcul_infinitesimal" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Identitats_trigonomètriques_en_càlcul_infinitesimal"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Identitats trigonomètriques en càlcul infinitesimal</span> </div> </a> <ul id="toc-Identitats_trigonomètriques_en_càlcul_infinitesimal-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Càlcul_de_les_funcions_trigonomètriques" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Càlcul_de_les_funcions_trigonomètriques"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Càlcul de les funcions trigonomètriques</span> </div> </a> <ul id="toc-Càlcul_de_les_funcions_trigonomètriques-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Funcions_inverses" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Funcions_inverses"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Funcions inverses</span> </div> </a> <ul id="toc-Funcions_inverses-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Propietats_i_aplicacions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Propietats_i_aplicacions"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Propietats i aplicacions</span> </div> </a> <button aria-controls="toc-Propietats_i_aplicacions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Commuta la subsecció Propietats i aplicacions</span> </button> <ul id="toc-Propietats_i_aplicacions-sublist" class="vector-toc-list"> <li id="toc-Teorema_del_sinus" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Teorema_del_sinus"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>Teorema del sinus</span> </div> </a> <ul id="toc-Teorema_del_sinus-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Teorema_del_cosinus" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Teorema_del_cosinus"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.2</span> <span>Teorema del cosinus</span> </div> </a> <ul id="toc-Teorema_del_cosinus-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Teorema_de_la_tangent" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Teorema_de_la_tangent"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.3</span> <span>Teorema de la tangent</span> </div> </a> <ul id="toc-Teorema_de_la_tangent-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Funcions_periòdiques" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Funcions_periòdiques"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.4</span> <span>Funcions periòdiques</span> </div> </a> <ul id="toc-Funcions_periòdiques-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Vegeu_també" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Vegeu_també"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Vegeu també</span> </div> </a> <ul id="toc-Vegeu_també-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Referències" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Referències"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Referències</span> </div> </a> <ul id="toc-Referències-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Enllaços_externs" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Enllaços_externs"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>Enllaços externs</span> </div> </a> <ul id="toc-Enllaços_externs-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="Contingut" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Taula de continguts" > <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="Commuta la taula de continguts." > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Commuta la taula de continguts.</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Funció trigonomètrica</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Vés a un article en una altra llengua. Disponible en 74 llengües" > <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-74" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">74 llengües</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar badge-Q17437796 badge-featuredarticle mw-list-item" title="article de qualitat"><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 - àrab" lang="ar" hreflang="ar" data-title="دوال مثلثية" data-language-autonym="العربية" data-language-local-name="àrab" class="interlanguage-link-target"><span>العربية</span></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 - asturià" lang="ast" hreflang="ast" data-title="Función trigonométrica" data-language-autonym="Asturianu" data-language-local-name="asturià" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Triqonometrik_funksiyalar" title="Triqonometrik funksiyalar - azerbaidjanès" lang="az" hreflang="az" data-title="Triqonometrik funksiyalar" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaidjanès" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%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 - baixkir" lang="ba" hreflang="ba" data-title="Тригонометрик функциялар" data-language-autonym="Башҡортса" data-language-local-name="baixkir" class="interlanguage-link-target"><span>Башҡортса</span></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 - belarús" lang="be" hreflang="be" data-title="Трыганаметрычныя функцыі" data-language-autonym="Беларуская" data-language-local-name="belarús" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%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 - búlgar" lang="bg" hreflang="bg" data-title="Тригонометрична функция" data-language-autonym="Български" data-language-local-name="búlgar" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%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 - bengalí" lang="bn" hreflang="bn" data-title="ত্রিকোণমিতিক অপেক্ষক" data-language-autonym="বাংলা" data-language-local-name="bengalí" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Trigonometrijska_funkcija" title="Trigonometrijska funkcija - bosnià" lang="bs" hreflang="bs" data-title="Trigonometrijska funkcija" data-language-autonym="Bosanski" data-language-local-name="bosnià" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%81%D8%A7%D9%86%DA%A9%D8%B4%D9%86%DB%95_%D8%B3%DB%8E%DA%AF%DB%86%D8%B4%DB%95%DB%8C%DB%8C%DB%8C%DB%95%DA%A9%D8%A7%D9%86" title="فانکشنە سێگۆشەیییەکان - kurd central" lang="ckb" hreflang="ckb" data-title="فانکشنە سێگۆشەیییەکان" data-language-autonym="کوردی" data-language-local-name="kurd central" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Goniometrick%C3%A1_funkce" title="Goniometrická funkce - txec" lang="cs" hreflang="cs" data-title="Goniometrická funkce" data-language-autonym="Čeština" data-language-local-name="txec" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A2%D1%80%D0%B8%D0%B3%D0%BE%D0%BD%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D0%BB%D0%BB%D0%B5_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%81%D0%B5%D0%BC" title="Тригонометрилле функцисем - txuvaix" lang="cv" hreflang="cv" data-title="Тригонометрилле функцисем" data-language-autonym="Чӑвашла" data-language-local-name="txuvaix" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Ffwythiannau_trigonometrig" title="Ffwythiannau trigonometrig - gal·lès" lang="cy" hreflang="cy" data-title="Ffwythiannau trigonometrig" data-language-autonym="Cymraeg" data-language-local-name="gal·lès" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Trigonometrisk_funktion" title="Trigonometrisk funktion - danès" lang="da" hreflang="da" data-title="Trigonometrisk funktion" data-language-autonym="Dansk" data-language-local-name="danès" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Trigonometrische_Funktion" title="Trigonometrische Funktion - alemany" lang="de" hreflang="de" data-title="Trigonometrische Funktion" data-language-autonym="Deutsch" data-language-local-name="alemany" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%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 - grec" lang="el" hreflang="el" data-title="Τριγωνομετρική συνάρτηση" data-language-autonym="Ελληνικά" data-language-local-name="grec" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Trigonometric_functions" title="Trigonometric functions - anglès" lang="en" hreflang="en" data-title="Trigonometric functions" data-language-autonym="English" data-language-local-name="anglès" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/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"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Funci%C3%B3n_trigonom%C3%A9trica" title="Función trigonométrica - espanyol" lang="es" hreflang="es" data-title="Función trigonométrica" data-language-autonym="Español" data-language-local-name="espanyol" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Trigonomeetrilised_funktsioonid" title="Trigonomeetrilised funktsioonid - estonià" lang="et" hreflang="et" data-title="Trigonomeetrilised funktsioonid" data-language-autonym="Eesti" data-language-local-name="estonià" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Funtzio_trigonometriko" title="Funtzio trigonometriko - basc" lang="eu" hreflang="eu" data-title="Funtzio trigonometriko" data-language-autonym="Euskara" data-language-local-name="basc" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa badge-Q17437796 badge-featuredarticle mw-list-item" title="article de qualitat"><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 - persa" lang="fa" hreflang="fa" data-title="توابع مثلثاتی" data-language-autonym="فارسی" data-language-local-name="persa" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Trigonometrinen_funktio" title="Trigonometrinen funktio - finès" lang="fi" hreflang="fi" data-title="Trigonometrinen funktio" data-language-autonym="Suomi" data-language-local-name="finès" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Fonction_trigonom%C3%A9trique" title="Fonction trigonométrique - francès" lang="fr" hreflang="fr" data-title="Fonction trigonométrique" data-language-autonym="Français" data-language-local-name="francès" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Funci%C3%B3n_trigonom%C3%A9trica" title="Función trigonométrica - gallec" lang="gl" hreflang="gl" data-title="Función trigonométrica" data-language-autonym="Galego" data-language-local-name="gallec" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%95%D7%AA_%D7%98%D7%A8%D7%99%D7%92%D7%95%D7%A0%D7%95%D7%9E%D7%98%D7%A8%D7%99%D7%95%D7%AA" title="פונקציות טריגונומטריות - hebreu" lang="he" hreflang="he" data-title="פונקציות טריגונומטריות" data-language-autonym="עברית" data-language-local-name="hebreu" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%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"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Trigonometrijske_funkcije" title="Trigonometrijske funkcije - croat" lang="hr" hreflang="hr" data-title="Trigonometrijske funkcije" data-language-autonym="Hrvatski" data-language-local-name="croat" class="interlanguage-link-target"><span>Hrvatski</span></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 - hongarès" lang="hu" hreflang="hu" data-title="Szögfüggvények" data-language-autonym="Magyar" data-language-local-name="hongarès" class="interlanguage-link-target"><span>Magyar</span></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 - armeni" lang="hy" hreflang="hy" data-title="Եռանկյունաչափական ֆունկցիաներ" data-language-autonym="Հայերեն" data-language-local-name="armeni" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Fungsi_trigonometri" title="Fungsi trigonometri - indonesi" lang="id" hreflang="id" data-title="Fungsi trigonometri" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesi" class="interlanguage-link-target"><span>Bahasa Indonesia</span></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"><span>Ido</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Hornafall" title="Hornafall - islandès" lang="is" hreflang="is" data-title="Hornafall" data-language-autonym="Íslenska" data-language-local-name="islandès" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Funzione_trigonometrica" title="Funzione trigonometrica - italià" lang="it" hreflang="it" data-title="Funzione trigonometrica" data-language-autonym="Italiano" data-language-local-name="italià" class="interlanguage-link-target"><span>Italiano</span></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 - japonès" lang="ja" hreflang="ja" data-title="三角関数" data-language-autonym="日本語" data-language-local-name="japonès" class="interlanguage-link-target"><span>日本語</span></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 - georgià" lang="ka" hreflang="ka" data-title="ტრიგონომეტრიული ფუნქციები" data-language-autonym="ქართული" data-language-local-name="georgià" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-km badge-Q17437796 badge-featuredarticle mw-list-item" title="article de qualitat"><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"><span>ភាសាខ្មែរ</span></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 - coreà" lang="ko" hreflang="ko" data-title="삼각 함수" data-language-autonym="한국어" data-language-local-name="coreà" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Functiones_trigonometricae" title="Functiones trigonometricae - llatí" lang="la" hreflang="la" data-title="Functiones trigonometricae" data-language-autonym="Latina" data-language-local-name="llatí" class="interlanguage-link-target"><span>Latina</span></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"><span>Lingua Franca Nova</span></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 - lituà" lang="lt" hreflang="lt" data-title="Trigonometrinės funkcijos" data-language-autonym="Lietuvių" data-language-local-name="lituà" class="interlanguage-link-target"><span>Lietuvių</span></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 - letó" lang="lv" hreflang="lv" data-title="Trigonometriskās funkcijas" data-language-autonym="Latviešu" data-language-local-name="letó" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%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 - macedoni" lang="mk" hreflang="mk" data-title="Тригонометриски функции" data-language-autonym="Македонски" data-language-local-name="macedoni" class="interlanguage-link-target"><span>Македонски</span></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"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Fungsi_trigonometri" title="Fungsi trigonometri - malai" lang="ms" hreflang="ms" data-title="Fungsi trigonometri" data-language-autonym="Bahasa Melayu" data-language-local-name="malai" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Goniometrische_functie" title="Goniometrische functie - neerlandès" lang="nl" hreflang="nl" data-title="Goniometrische functie" data-language-autonym="Nederlands" data-language-local-name="neerlandès" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Trigonometrisk_funksjon" title="Trigonometrisk funksjon - noruec nynorsk" lang="nn" hreflang="nn" data-title="Trigonometrisk funksjon" data-language-autonym="Norsk nynorsk" data-language-local-name="noruec nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Trigonometrisk_funksjon" title="Trigonometrisk funksjon - noruec bokmål" lang="nb" hreflang="nb" data-title="Trigonometrisk funksjon" data-language-autonym="Norsk bokmål" data-language-local-name="noruec bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Foncions_trigonometricas" title="Foncions trigonometricas - occità" lang="oc" hreflang="oc" data-title="Foncions trigonometricas" data-language-autonym="Occitan" data-language-local-name="occità" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-pl badge-Q17437798 badge-goodarticle mw-list-item" title="article bo"><a href="https://pl.wikipedia.org/wiki/Funkcje_trygonometryczne" title="Funkcje trygonometryczne - polonès" lang="pl" hreflang="pl" data-title="Funkcje trygonometryczne" data-language-autonym="Polski" data-language-local-name="polonès" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Fun%C3%A7%C3%A3o_trigonom%C3%A9trica" title="Função trigonométrica - portuguès" lang="pt" hreflang="pt" data-title="Função trigonométrica" data-language-autonym="Português" data-language-local-name="portuguès" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Func%C8%9Bie_trigonometric%C4%83" title="Funcție trigonometrică - romanès" lang="ro" hreflang="ro" data-title="Funcție trigonometrică" data-language-autonym="Română" data-language-local-name="romanès" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%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 - rus" lang="ru" hreflang="ru" data-title="Тригонометрические функции" data-language-autonym="Русский" data-language-local-name="rus" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Trigonometrijske_funkcije" title="Trigonometrijske funkcije - serbocroat" lang="sh" hreflang="sh" data-title="Trigonometrijske funkcije" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbocroat" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%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 - singalès" lang="si" hreflang="si" data-title="ත්‍රිකෝණමිතික ශ්‍රිත" data-language-autonym="සිංහල" data-language-local-name="singalès" class="interlanguage-link-target"><span>සිංහල</span></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"><span>Simple English</span></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 - eslovac" lang="sk" hreflang="sk" data-title="Goniometrická funkcia" data-language-autonym="Slovenčina" data-language-local-name="eslovac" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Trigonometri%C4%8Dna_funkcija" title="Trigonometrična funkcija - eslovè" lang="sl" hreflang="sl" data-title="Trigonometrična funkcija" data-language-autonym="Slovenščina" data-language-local-name="eslovè" class="interlanguage-link-target"><span>Slovenščina</span></a></li><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 - sami d’Inari" lang="smn" hreflang="smn" data-title="Trigonometrisâš funktio" data-language-autonym="Anarâškielâ" data-language-local-name="sami d’Inari" class="interlanguage-link-target"><span>Anarâškielâ</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Funksionet_trigonometrike" title="Funksionet trigonometrike - albanès" lang="sq" hreflang="sq" data-title="Funksionet trigonometrike" data-language-autonym="Shqip" data-language-local-name="albanès" class="interlanguage-link-target"><span>Shqip</span></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 - serbi" lang="sr" hreflang="sr" data-title="Тригонометријске функције" data-language-autonym="Српски / srpski" data-language-local-name="serbi" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Trigonometrisk_funktion" title="Trigonometrisk funktion - suec" lang="sv" hreflang="sv" data-title="Trigonometrisk funktion" data-language-autonym="Svenska" data-language-local-name="suec" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AE%E0%AF%81%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AF%8B%E0%AE%A3%E0%AE%B5%E0%AE%BF%E0%AE%AF%E0%AE%B2%E0%AF%8D_%E0%AE%9A%E0%AE%BE%E0%AE%B0%E0%AF%8D%E0%AE%AA%E0%AF%81%E0%AE%95%E0%AE%B3%E0%AF%8D" title="முக்கோணவியல் சார்புகள் - tàmil" lang="ta" hreflang="ta" data-title="முக்கோணவியல் சார்புகள்" data-language-autonym="தமிழ்" data-language-local-name="tàmil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%81%D0%B8%D1%8F%D2%B3%D0%BE%D0%B8_%D1%82%D1%80%D0%B8%D0%B3%D0%BE%D0%BD%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D3%A3" title="Функсияҳои тригонометрӣ - tadjik" lang="tg" hreflang="tg" data-title="Функсияҳои тригонометрӣ" data-language-autonym="Тоҷикӣ" data-language-local-name="tadjik" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%9F%E0%B8%B1%E0%B8%87%E0%B8%81%E0%B9%8C%E0%B8%8A%E0%B8%B1%E0%B8%99%E0%B8%95%E0%B8%A3%E0%B8%B5%E0%B9%82%E0%B8%81%E0%B8%93%E0%B8%A1%E0%B8%B4%E0%B8%95%E0%B8%B4" title="ฟังก์ชันตรีโกณมิติ - tai" lang="th" hreflang="th" data-title="ฟังก์ชันตรีโกณมิติ" data-language-autonym="ไทย" data-language-local-name="tai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Trigonometrik_fonksiyonlar" title="Trigonometrik fonksiyonlar - turc" lang="tr" hreflang="tr" data-title="Trigonometrik fonksiyonlar" data-language-autonym="Türkçe" data-language-local-name="turc" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.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%D1%96_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%97" title="Тригонометричні функції - ucraïnès" lang="uk" hreflang="uk" data-title="Тригонометричні функції" data-language-autonym="Українська" data-language-local-name="ucraïnès" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%85%D8%AB%D9%84%D8%AB%DB%8C%D8%A7%D8%AA%DB%8C_%D8%AF%D8%A7%D9%84%DB%81" title="مثلثیاتی دالہ - urdú" lang="ur" hreflang="ur" data-title="مثلثیاتی دالہ" data-language-autonym="اردو" data-language-local-name="urdú" class="interlanguage-link-target"><span>اردو</span></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"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/H%C3%A0m_l%C6%B0%E1%BB%A3ng_gi%C3%A1c" title="Hàm lượng giác - vietnamita" lang="vi" hreflang="vi" data-title="Hàm lượng giác" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamita" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E4%B8%89%E8%A7%92%E5%87%BD%E6%95%B0" title="三角函数 - xinès wu" lang="wuu" hreflang="wuu" data-title="三角函数" data-language-autonym="吴语" data-language-local-name="xinès wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh badge-Q17437798 badge-goodarticle mw-list-item" title="article bo"><a href="https://zh.wikipedia.org/wiki/%E4%B8%89%E8%A7%92%E5%87%BD%E6%95%B0" title="三角函数 - xinès" lang="zh" hreflang="zh" data-title="三角函数" data-language-autonym="中文" data-language-local-name="xinès" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E4%B8%89%E8%A7%92%E5%87%BD%E6%95%B8" title="三角函數 - xinès clàssic" lang="lzh" hreflang="lzh" data-title="三角函數" data-language-autonym="文言" data-language-local-name="xinès clàssic" class="interlanguage-link-target"><span>文言</span></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 - xinès min del sud" lang="nan" hreflang="nan" data-title="Saⁿ-kak hâm-sò͘" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="xinès min del sud" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E4%B8%89%E8%A7%92%E5%87%BD%E6%95%B8" title="三角函數 - cantonès" lang="yue" hreflang="yue" data-title="三角函數" data-language-autonym="粵語" data-language-local-name="cantonès" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q93344#sitelinks-wikipedia" title="Modificau enllaços interlingües" class="wbc-editpage">Modifica els enllaços</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Espais de noms"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Funci%C3%B3_trigonom%C3%A8trica" title="Vegeu el contingut de la pàgina [c]" accesskey="c"><span>Pàgina</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Discussi%C3%B3:Funci%C3%B3_trigonom%C3%A8trica" rel="discussion" title="Discussió sobre el contingut d'aquesta pàgina [t]" accesskey="t"><span>Discussió</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Canvia la variant de llengua" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">català</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Vistes"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Funci%C3%B3_trigonom%C3%A8trica"><span>Mostra</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Funci%C3%B3_trigonom%C3%A8trica&action=edit" title="Modifica el codi font d'aquesta pàgina [e]" accesskey="e"><span>Modifica</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Funci%C3%B3_trigonom%C3%A8trica&action=history" title="Versions antigues d'aquesta pàgina [h]" accesskey="h"><span>Mostra l'historial</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Eines de la pàgina"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Eines" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Eines</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Eines</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">mou a la barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">amaga</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Més opcions" > <div class="vector-menu-heading"> Accions </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Funci%C3%B3_trigonom%C3%A8trica"><span>Mostra</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Funci%C3%B3_trigonom%C3%A8trica&action=edit" title="Modifica el codi font d'aquesta pàgina [e]" accesskey="e"><span>Modifica</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Funci%C3%B3_trigonom%C3%A8trica&action=history"><span>Mostra l'historial</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Especial:Enlla%C3%A7os/Funci%C3%B3_trigonom%C3%A8trica" title="Una llista de totes les pàgines wiki que enllacen amb aquesta [j]" accesskey="j"><span>Què hi enllaça</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Especial:Seguiment/Funci%C3%B3_trigonom%C3%A8trica" rel="nofollow" title="Canvis recents a pàgines enllaçades des d'aquesta pàgina [k]" accesskey="k"><span>Canvis relacionats</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Funci%C3%B3_trigonom%C3%A8trica&oldid=34633281" title="Enllaç permanent a aquesta revisió de la pàgina"><span>Enllaç permanent</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Funci%C3%B3_trigonom%C3%A8trica&action=info" title="Més informació sobre aquesta pàgina"><span>Informació de la pàgina</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Especial:Citau&page=Funci%C3%B3_trigonom%C3%A8trica&id=34633281&wpFormIdentifier=titleform" title="Informació sobre com citar aquesta pàgina"><span>Citau aquest article</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Especial:UrlShortener&url=https%3A%2F%2Fca.wikipedia.org%2Fwiki%2FFunci%25C3%25B3_trigonom%25C3%25A8trica%23Tangent"><span>Obtén una URL abreujada</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Especial:QrCode&url=https%3A%2F%2Fca.wikipedia.org%2Fwiki%2FFunci%25C3%25B3_trigonom%25C3%25A8trica%23Tangent"><span>Descarrega el codi QR</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Imprimeix/exporta </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=Especial:Llibre&bookcmd=book_creator&referer=Funci%C3%B3+trigonom%C3%A8trica"><span>Crea un llibre</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Especial:DownloadAsPdf&page=Funci%C3%B3_trigonom%C3%A8trica&action=show-download-screen"><span>Baixa com a PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Funci%C3%B3_trigonom%C3%A8trica&printable=yes" title="Versió per a impressió d'aquesta pàgina [p]" accesskey="p"><span>Versió per a impressora</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> En altres projectes </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Trigonometric_functions" hreflang="en"><span>Commons</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q93344" title="Enllaç a l'element del repositori de dades connectat [g]" accesskey="g"><span>Element a Wikidata</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Eines de la pàgina"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Aparença"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Aparença</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">mou a la barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">amaga</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> <div id="mw-indicator-ind-qualitat" class="mw-indicator"><div class="mw-parser-output"><span typeof="mw:File"><a href="/wiki/Viquip%C3%A8dia:Articles_de_qualitat" title="Article de qualitat"><img alt="Article de qualitat" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Segell_AdQ_unificat_Viquip%C3%A8dia.svg/40px-Segell_AdQ_unificat_Viquip%C3%A8dia.svg.png" decoding="async" width="30" height="36" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Segell_AdQ_unificat_Viquip%C3%A8dia.svg/60px-Segell_AdQ_unificat_Viquip%C3%A8dia.svg.png 1.5x" data-file-width="2408" data-file-height="2896" /></a></span></div></div> </div> <div id="siteSub" class="noprint">De la Viquipèdia, l'enciclopèdia lliure</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">(S'ha redirigit des de: <a href="/w/index.php?title=Tangent_(trigonometria)&redirect=no" class="mw-redirect" title="Tangent (trigonometria)">Tangent (trigonometria)</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="ca" dir="ltr"><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fitxer:Circle-trig6.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Circle-trig6.svg/400px-Circle-trig6.svg.png" decoding="async" width="400" height="256" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Circle-trig6.svg/600px-Circle-trig6.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Circle-trig6.svg/800px-Circle-trig6.svg.png 2x" data-file-width="1250" data-file-height="800" /></a><figcaption>Totes les funcions trigonomètriques d'un angle θ es poden construir geomètricament en termes de la circumferència goniomètrica.</figcaption></figure> <p>En <a href="/wiki/Matem%C3%A0tiques" title="Matemàtiques">matemàtiques</a>, les <b>funcions trigonomètriques</b> són <a href="/wiki/Funci%C3%B3_(matem%C3%A0tiques)" class="mw-redirect" title="Funció (matemàtiques)">funcions</a> d'un <a href="/wiki/Angle" title="Angle">angle</a>. Són la base per l'estudi de la <a href="/wiki/Trigonometria" title="Trigonometria">trigonometria</a>, els <a href="/wiki/Triangle" title="Triangle">triangles</a> i per la modelització dels <a href="/wiki/Funci%C3%B3_peri%C3%B2dica" title="Funció periòdica">fenòmens periòdics</a>, entre moltes altres aplicacions. Les funcions trigonomètriques es defineixen habitualment com a <a href="/wiki/Quocient" class="mw-redirect" title="Quocient">quocients</a> entre les longituds de dos costats d'un triangle rectangle que contingui l'angle, i de forma equivalent es poden definir a partir de les longituds de diversos segments a partir de la <a href="/wiki/Circumfer%C3%A8ncia_goniom%C3%A8trica" title="Circumferència goniomètrica">circumferència goniomètrica</a> (circumferència de radi unitat, el centre de la qual és l'origen d'un sistema de <a href="/wiki/Coordenades_cartesianes" class="mw-redirect" title="Coordenades cartesianes">coordenades cartesianes</a>). Hi ha definicions més modernes que les expressen com a <a href="/wiki/S%C3%A8rie_infinita" class="mw-redirect" title="Sèrie infinita">sèries infinites</a> o com a solucions d'<a href="/wiki/Equaci%C3%B3_diferencial" title="Equació diferencial">equacions diferencials</a>; l'avantatge d'aquestes definicions és que permeten estendre les funcions trigonomètriques a cossos arbitraris com per exemple els <a href="/wiki/Nombres_complexos" class="mw-redirect" title="Nombres complexos">nombres complexos</a>. </p><p>Actualment es fan servir les sis funcions trigonomètriques que es presenten a la taula de la dreta, juntament amb algunes de les identitats que permeten calcular-ne unes a partir de les altres. En el cas de les últimes quatre funcions trigonomètriques, sovint es prenen aquestes identitats com a "definicions" de les mateixes funcions, però es poden definir perfectament de manera geomètrica, o per altres mitjans, i llavors demostrar aquestes identitats. De fet, tal com s'aprecia a les identitats de la taula, només cal definir-ne una qualsevol i després es poden emprar unes o altres identitats per definir i calcular tota la resta. <br style="clear:both;" /> </p> <table class="wikitable" align="right" style="margin-left:1em"> <tbody><tr> <th style="text-align:left">Funció </th> <th style="text-align:left">Abreviació </th> <th style="text-align:left"><a href="/wiki/Llista_d%27identitats_trigonom%C3%A8triques" title="Llista d'identitats trigonomètriques">Identitats</a> (emprant <a href="/wiki/Radiant_(angle)" class="mw-redirect" title="Radiant (angle)">radiants</a>) </th></tr> <tr style="background-color:#FFFFFF"> <td><b>Sinus</b> </td> <td>sin </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \theta =\cos \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\csc \theta }}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea0da91c69a769be76f11b49e698feeb46e7f2f0" 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 \sin \theta =\cos \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\csc \theta }}\,}" /></span> </td></tr> <tr style="background-color:#FFFFFF"> <td><b>Cosinus</b> </td> <td>cos </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \theta =\sin \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\sec \theta }}\,}</annotation> </semantics> </math></span><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 }}\,}" /></span> </td></tr> <tr style="background-color:#FFFFFF"> <td><b>Tangent</b> </td> <td>tan<br />(o tg) </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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> <mspace width="thinmathspace"></mspace> </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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cf01ed4d3956c6652a77289c89ef12d9b13cbe0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:37.332ex; height:5.509ex;" alt="{\displaystyle \tan \theta ={\frac {\sin \theta }{\cos \theta }}=\cot \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\cot \theta }}\,}" /></span> </td></tr> <tr style="background-color:#FFFFFF"> <td><b>Cosecant</b> </td> <td>csc<br />(o cosec) </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \csc \theta =\sec \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\sin \theta }}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3d8cf121e63145b656c464034eb9b8859bb054a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:28.067ex; height:5.343ex;" alt="{\displaystyle \csc \theta =\sec \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\sin \theta }}\,}" /></span> </td></tr> <tr style="background-color:#FFFFFF"> <td><b>Secant</b> </td> <td>sec </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sec \theta =\csc \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\cos \theta }}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7f1d3507eb0fc844a2b285766127c7994b0304b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:28.323ex; height:5.343ex;" alt="{\displaystyle \sec \theta =\csc \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\cos \theta }}\,}" /></span> </td></tr> <tr style="background-color:#FFFFFF"> <td><b>Cotangent</b> </td> <td>cot<br />(o ctg o ctn) </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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> <mspace width="thinmathspace"></mspace> </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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f42ad4dea9c31b770f1173ea761a743d6b4ffe61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:37.592ex; height:5.509ex;" alt="{\displaystyle \cot \theta ={\frac {\cos \theta }{\sin \theta }}=\tan \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\tan \theta }}\,}" /></span> </td></tr></tbody></table> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Història"><span id="Hist.C3.B2ria"></span>Història</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funci%C3%B3_trigonom%C3%A8trica&action=edit&section=1" title="Modifica la secció: Història"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r30997230">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}</style><div role="note" class="hatnote navigation-not-searchable">Article principal: <a href="/wiki/Hist%C3%B2ria_de_les_funcions_trigonom%C3%A8triques" title="Història de les funcions trigonomètriques">Història de les funcions trigonomètriques</a></div> <p>La noció que hi ha d'haver alguna correspondència bàsica entre la longitud de les cares d'un triangle i els angles apareix tan aviat com en reconeix que els <a href="/wiki/Triangles_semblants" class="mw-redirect" title="Triangles semblants">triangles semblants</a> mantenen constants les relacions entre les seves cares. És a dir, per a qualsevol triangle rectangle semblant a un de donat, la relació (el quocient entre les longituds) entre la hipotenusa (per exemple) i una altra de les seves cares es manté constant. Si la hipotenusa és el doble de llarga, també ho són les altres dues cares. Són precisament aquestes relacions les que es fan servir per a expressar les funcions trigonomètriques. </p><p>Les funcions trigonomètriques varen ser estudiades per: </p> <ul><li>El matemàtic <a href="/wiki/Gr%C3%A8cia" title="Grècia">grec</a> <a href="/wiki/Hiparc_de_Nicea" title="Hiparc de Nicea">Hiparc de Nicea</a> (180-125 AC) va crear taules on relacionava la longitud de l'arc amb la longitud de la <a href="/wiki/Corda_(geometria)" title="Corda (geometria)">corda</a> corresponent,<sup id="cite_ref-oconnor1996_1-0" class="reference"><a href="#cite_note-oconnor1996-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></li> <li>A <a href="/wiki/Egipte" title="Egipte">Egipte</a>, <a href="/wiki/Claudi_Ptolemeu" title="Claudi Ptolemeu">Claudi Ptolemeu</a> (90–180 AD) va escriure l'<a href="/wiki/Almagest" title="Almagest">almagest</a> on desenvolupa les fórmules equivalents a les actuals pel sinus de la suma de dos angles però per la funció corda, i una fórmula pel càlcul de la corda de l'angle meitat, a partir d'aquí va crear una taula trigonomètrica<sup id="cite_ref-boyer1991_2-0" class="reference"><a href="#cite_note-boyer1991-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></li> <li>El matemàtic <a href="/wiki/%C3%8Dndia" title="Índia">Indi</a> <a href="/wiki/Aryabhata" title="Aryabhata">Aryabhata</a> (476–550) definí per primera vegada les funcions sinus (la meitat de la corda) i cosinus. Els seus treballs contenen les taules més antigues que existeixin actualment dels valors del sinus de tots els angles compresos entre 0° i 90° a intervals de 3,75°, amb una precisió de 4 decimals. Aquesta taula fou reproduïda per <a href="/wiki/Brahmagupta" title="Brahmagupta">Brahmagupta</a> (628)<sup id="cite_ref-oconnor1996_1-1" class="reference"><a href="#cite_note-oconnor1996-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></li> <li>Segons l'obra d'<a href="/wiki/Abu_l-Wafa" class="mw-redirect" title="Abu l-Wafa">Abu l-Wafa</a>, sembla que els matemàtics musulmans utilitzaven cadascuna de les sis funcions trigonomètriques, i disposaven de taules amb intervals de 0,25°, amb 8 decimals exactes.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></li> <li>El matemàtic <a href="/wiki/Principat_de_Catalunya" title="Principat de Catalunya">català</a> <a href="/wiki/Savasorda" class="mw-redirect" title="Savasorda">Savasorda</a> (1070-1136) escriu la primera taula trigonomètrica en llatí. Aquesta taula fa servir la corda seguint la tradició clàssica en comptes del sinus que empraven els àrabs, divideix la <a href="/wiki/Circumfer%C3%A8ncia_goniom%C3%A8trica" title="Circumferència goniomètrica">circumferència goniomètrica</a> en 88 parts (en comptes de 360 graus) i al diàmetre li assigna 28 unitats. Això fa que la unitat d'angle sigui molt aproximadament el <a href="/wiki/Radian" title="Radian">radian</a>. La taula dona l'angle en graus, minuts i segons i és exacta fins a l'últim decimal.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></li> <li>El matemàtic indi <a href="/wiki/Bhaskara_II" title="Bhaskara II">Bhaskara II</a>, l'any 1150 explica un mètode detallat per construir les taules de sinus per a qualsevol angle.<sup id="cite_ref-oconnor1996_1-2" class="reference"><a href="#cite_note-oconnor1996-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Madhava_de_Sangamagrama" title="Madhava de Sangamagrama">Madhava de Sangamagrama</a> (1400) Va fer uns primers passos en l'<a href="/wiki/An%C3%A0lisi_matem%C3%A0tica" title="Anàlisi matemàtica">anàlisi matemàtica</a> de funcions trigonomètriques basada en <a href="/wiki/S%C3%A8rie_matem%C3%A0tica" class="mw-redirect" title="Sèrie matemàtica">sèries</a>.</li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> amb l'obra <i>Introductio in analysin infinitorum</i> (1748) va ser el principal responsable d'establir el tractament de les funcions trigonomètriques a Europa, també les va definir com a sèries infinites i va presentar la <a href="/wiki/F%C3%B3rmula_d%27Euler" title="Fórmula d'Euler">fórmula d'Euler</a>", així com les abreviatures gairebé igual que les modernes <i>sin., cos., tang., cot., sec.</i>, i <i>cosec.</i></li></ul> <p>Històricament era comú de fer servir unes quantes funcions més que les d'avui en dia (i aquestes funcions apareixen en les taules més antigues de funcions trigonomètriques), però en l'actualitat s'han deixat de fer servir. Són funcions com ara la <a href="/wiki/Corda_(geometria)" title="Corda (geometria)">corda</a> (crd(θ) = 2 sin(θ/2)), el <a href="/wiki/Versinus" title="Versinus">versinus</a> (versin(θ) = 1 − cos(θ) = 2 sin²(θ/2)), el <a href="/wiki/Semiversinus" class="mw-redirect" title="Semiversinus">semiversinus</a> (semiversin(θ) = versin(θ) / 2 = sin²(θ/2)), l'<a href="/wiki/Exsecant" title="Exsecant">exsecant</a> (exsec(θ) = sec(θ) − 1) i l'<a href="/wiki/Excosecant" class="mw-redirect" title="Excosecant">excosecant</a> (excsc(θ) = exsec(π/2 − θ) = csc(θ) − 1). Moltes més relacions, entre les quals es troben aquestes, es troben a l'article sobre la <a href="/wiki/Llista_d%27identitats_trigonom%C3%A8triques" title="Llista d'identitats trigonomètriques">llista d'identitats trigonomètriques</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Etimologia">Etimologia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funci%C3%B3_trigonom%C3%A8trica&action=edit&section=2" title="Modifica la secció: Etimologia"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>L'<a href="/wiki/Etimologia" title="Etimologia">etimologia</a> de la paraula <i>sinus</i> prové de la paraula en <a href="/wiki/S%C3%A0nscrit" title="Sànscrit">sànscrit</a> <i>jya-ardha</i> que vol dir "mitja corda", abreujat en <i>jiva</i>. Això es <a href="/wiki/Transliteraci%C3%B3" title="Transliteració">transliterava</a> en àrab com <i>jiba</i>, i s'escrivia <i>jb</i>, en no escriure les vocals en àrab. Després, aquesta transliteració va ser traduïda per error al <a href="/wiki/Segle_XII" title="Segle XII">segle <span title="Nombre escrit en xifres romanes" style="font-variant:small-caps;">xii</span></a> al <a href="/wiki/Llat%C3%AD" title="Llatí">llatí</a> com a <i>sinus</i>, sota la impressió equivocada que <i>jb</i> corresponia a la paraula <i>jaib</i>, que significa "pit" o "badia" o "plec" en àrab, tal com <i>sinus</i> en <a href="/wiki/Llat%C3%AD" title="Llatí">llatí</a>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> La paraula llatina és la que s'ha conservat en català. </p><p>La paraula <i>tangent</i> ve del llatí <i>tangens</i>: que "toca", ja que la recta emprada per definir-la <i>toca</i> la circumferència goniomètrica, mentre que <i>secant</i> prové de <i>secans</i> en llatí - "tallant" - ja que la recta emprada per definir-la <i>talla</i> la circumferència goniomètrica: ambdues van ser introduïdes pel matemàtic danès <a href="/wiki/Thomas_Fincke" title="Thomas Fincke">Thomas Fincke</a> el 1583.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Definició_de_les_funcions_trigonomètriques"><span id="Definici.C3.B3_de_les_funcions_trigonom.C3.A8triques"></span>Definició de les funcions trigonomètriques</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funci%C3%B3_trigonom%C3%A8trica&action=edit&section=3" title="Modifica la secció: Definició de les funcions trigonomètriques"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Les diferents definicions de les funcions trigonomètriques que es presenten a continuació són equivalents en el sentit que, per als angles en què es poden aplicar, el resultat coincideix. Però es diferencien en el sentit que es poden aplicar a angles en un sentit cada cop més ampli de la paraula. La definició basant-se en el <a href="/wiki/Triangle_rectangle" title="Triangle rectangle">triangle rectangle</a>, estrictament parlant, només es pot aplicar a <a href="/wiki/Angle_agut" class="mw-redirect" title="Angle agut">angles aguts</a>. La definició basant-se en la circumferència goniomètrica permet atribuir valor a les funcions trigonomètriques d'angles que tinguin qualsevol valor dins del conjunt dels <a href="/wiki/Nombres_reals" class="mw-redirect" title="Nombres reals">nombres reals</a>. La definició basada en <a href="/wiki/S%C3%A8rie_(matem%C3%A0tiques)" title="Sèrie (matemàtiques)">sèries</a> es pot fer servir per calcular les funcions trigonomètriques d'arguments <a href="/wiki/Nombre_complex" title="Nombre complex">complexos</a>. </p><p>A més, les definicions basades en sèries, en <a href="/wiki/Equaci%C3%B3_diferencial" title="Equació diferencial">equacions diferencials</a> o en <a href="/wiki/Equaci%C3%B3_funcional" title="Equació funcional">equacions funcionals</a>, permeten abordar l'estudi de les funcions trigonomètriques sense fer referència a consideracions geomètriques. Simplement acceptant la definició basada purament en conceptes propis de l'<a href="/wiki/An%C3%A0lisi_matem%C3%A0tica" title="Anàlisi matemàtica">anàlisi matemàtica</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Basant-se_en_el_triangle_rectangle">Basant-se en el triangle rectangle</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funci%C3%B3_trigonom%C3%A8trica&action=edit&section=4" title="Modifica la secció: Basant-se en el triangle rectangle"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fitxer:Trigonometry_triangle.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Trigonometry_triangle.svg/220px-Trigonometry_triangle.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Trigonometry_triangle.svg/330px-Trigonometry_triangle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Trigonometry_triangle.svg/440px-Trigonometry_triangle.svg.png 2x" data-file-width="1000" data-file-height="1000" /></a><figcaption>Un <a href="/wiki/Triangle_rectangle" title="Triangle rectangle">triangle rectangle</a> té sempre un angle de 90° (π/2 radiants), en aquest cas el <i>C</i>. Els angles <i>A</i> i <i>B</i> poden variar. Les funcions trigonomètriques especifiquen les relacions entre les longituds dels costats dels angles interiors d'un triangle rectangle.</figcaption></figure> <p>Per tal de definir les funcions trigonomètriques de l'angle <i>A</i>, es comença amb un <a href="/wiki/Triangle_rectangle" title="Triangle rectangle">triangle rectangle</a> arbitrari que contingui l'angle <i>A</i>. Es fan servir els següents noms pels costats del triangle: </p> <ul><li>La <a href="/wiki/Hipotenusa" title="Hipotenusa">hipotenusa</a> és el costat oposat a l'angle recte, o també es pot definir com el costat més llarg del triangle rectangle, en aquest cas <i>h</i>.</li> <li>El catet oposat és el costat oposat a l'angle que es pretén estudiar, en aquest cas <i>a</i>.</li> <li>El catet adjacent és el costat que està en contacte amb l'angle que s'està estudiant i l'angle recte. En aquest cas el catet adjacent és <i>b</i>.</li></ul> <p>Tots els triangles es prenen en el <a href="/wiki/Geometria_euclidiana" title="Geometria euclidiana">pla euclidià</a>; d'aquesta forma els angles interns de qualsevol triangle sumen π <a href="/wiki/Radiant_(angle)" class="mw-redirect" title="Radiant (angle)">radiants</a> (o 180<a href="/wiki/Grau_sexagesimal" title="Grau sexagesimal">°</a>). Per tant, per a qualsevol triangle rectangle els angles no rectes són entre zero i π/2 radiants (o 90°). Cal observar que les següents definicions, parlant estrictament, només defineixen les funcions trigonomètriques per angles dins d'aquest interval. S'estenen al conjunt sencer dels arguments reals a base de fer servir la <a href="/wiki/Circumfer%C3%A8ncia_goniom%C3%A8trica" title="Circumferència goniomètrica">circumferència goniomètrica</a>, o a base d'imposar certes simetries i que siguin <a href="/wiki/Funci%C3%B3_peri%C3%B2dica" title="Funció periòdica">funcions periòdiques</a>. </p><p>En la següent taula es recullen les definicions de les sis funcions trigonomètriques sobre la base del triangle rectangle:<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <table class="wikitable"> <tbody><tr> <th>Nom </th> <th>Definició </th> <th>Fórmula </th> <th>Triangle de <br />la figura </th></tr> <tr> <td><b>Sinus</b> </td> <td>El sinus d'un angle és el <b>quocient</b> entre la longitud del <b>catet oposat</b> i la longitud de la <b>hipotenusa</b>. </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin A={\frac {\textrm {oposat}}{\textrm {hipotenusa}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mtext>oposat</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>hipotenusa</mtext> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin A={\frac {\textrm {oposat}}{\textrm {hipotenusa}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61e6ac7aec1085f15733ba0db07d5a3da7a9f5b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.916ex; height:5.676ex;" alt="{\displaystyle \sin A={\frac {\textrm {oposat}}{\textrm {hipotenusa}}}}" /></span> </td> <td><center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {a}{h}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>h</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a}{h}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf1773f8807a5e5d1025fbadb5285c4bc64dc688" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:2.175ex; height:4.843ex;" alt="{\displaystyle {\frac {a}{h}}}" /></span> </center></td></tr> <tr bgcolor="#EFEFEF"> <td><b>Cosinus</b> </td> <td>El cosinus d'un angle és el <b>quocient</b> entre la longitud del <b>catet adjacent</b> i la <b>hipotenusa</b>. </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos A={\frac {\textrm {adjacent}}{\textrm {hipotenusa}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mtext>adjacent</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>hipotenusa</mtext> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos A={\frac {\textrm {adjacent}}{\textrm {hipotenusa}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1370e8ad143974d5ceda249ac3a464ebc5660375" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.171ex; height:5.843ex;" alt="{\displaystyle \cos A={\frac {\textrm {adjacent}}{\textrm {hipotenusa}}}}" /></span> </td> <td><center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {b}{h}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mi>h</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {b}{h}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c19ccac9ef7f254056a264dcb23fa8abe8e04eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:2.175ex; height:5.509ex;" alt="{\displaystyle {\frac {b}{h}}}" /></span> </center></td></tr> <tr> <td><b>Tangent</b> </td> <td>La tangent d'un angle és el <b>quocient</b> entre la longitud del <b>catet oposat</b> i la longitud del <b>catet adjacent</b>. </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan A={\frac {\textrm {oposat}}{\textrm {adjacent}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mtext>oposat</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>adjacent</mtext> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan A={\frac {\textrm {oposat}}{\textrm {adjacent}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f15e4352076de4b56d50ac05b043c67a2f6fd3d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.016ex; height:5.676ex;" alt="{\displaystyle \tan A={\frac {\textrm {oposat}}{\textrm {adjacent}}}}" /></span> </td> <td><center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {a}{b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a}{b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fbb66e57f89debc3cde3213de12228971148a93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:2.066ex; height:4.843ex;" alt="{\displaystyle {\frac {a}{b}}}" /></span> </center></td></tr> <tr bgcolor="#EFEFEF"> <td><b>Cosecant</b> </td> <td>La cosecant és la <b><a href="/wiki/Invers" class="mw-redirect" title="Invers">inversa</a> del sinus</b>, és a dir, el quocient entre la longitud de la hipotenusa i la longitud del catet oposat. </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \csc A={\frac {\textrm {hipotenusa}}{\textrm {oposat}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>csc</mi> <mo>⁡</mo> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mtext>hipotenusa</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>oposat</mtext> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \csc A={\frac {\textrm {hipotenusa}}{\textrm {oposat}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2171b551147a9a6368ee5281eec2d0063a119352" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.041ex; height:5.843ex;" alt="{\displaystyle \csc A={\frac {\textrm {hipotenusa}}{\textrm {oposat}}}}" /></span> </td> <td><center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {h}{a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>h</mi> <mi>a</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {h}{a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29ce625a015ca05914570ab27366bf3bd8506c39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.175ex; height:5.343ex;" alt="{\displaystyle {\frac {h}{a}}}" /></span> </center></td></tr> <tr> <td><b>Secant</b> </td> <td>La secant és la <b>inversa del cosinus</b>, és a dir, el quocient entre la longitud de la hipotenusa i la longitud del catet adjacent. </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sec A={\frac {\textrm {hipotenusa}}{\textrm {adjacent}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sec</mi> <mo>⁡</mo> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mtext>hipotenusa</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>adjacent</mtext> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sec A={\frac {\textrm {hipotenusa}}{\textrm {adjacent}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19ba3aa862cf4dc817f0da868e466015f5136f77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.041ex; height:5.843ex;" alt="{\displaystyle \sec A={\frac {\textrm {hipotenusa}}{\textrm {adjacent}}}}" /></span> </td> <td><center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {h}{b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>h</mi> <mi>b</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {h}{b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6f2cfafbd9d24eda06ce97dcf83ba95a80147a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:2.175ex; height:5.509ex;" alt="{\displaystyle {\frac {h}{b}}}" /></span> </center></td></tr> <tr bgcolor="#EFEFEF"> <td><b>Cotangent</b> </td> <td>La cotangent és la <b>inversa de la tangent</b>, és a dir, el quocient entre la longitud del catet adjacent i la longitud del catet oposat. </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cot A={\frac {\textrm {adjacent}}{\textrm {oposat}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cot</mi> <mo>⁡</mo> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mtext>adjacent</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>oposat</mtext> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cot A={\frac {\textrm {adjacent}}{\textrm {oposat}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d4e75f83b44c5b7f8b88aa884c51788ea761d55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.756ex; height:5.843ex;" alt="{\displaystyle \cot A={\frac {\textrm {adjacent}}{\textrm {oposat}}}}" /></span> </td> <td><center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {b}{a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mi>a</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {b}{a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30304b82a801ef33eaf4c0c0306aa6966e83d2f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.066ex; height:5.343ex;" alt="{\displaystyle {\frac {b}{a}}}" /></span> </center></td></tr></tbody></table> <p>Es pot veure que aquests quocients no depenen del triangle rectangle concret que s'hagi escollit, atès que tots els triangles rectangles que tinguin l'angle <i>A</i> són <a href="/wiki/Semblan%C3%A7a" title="Semblança">semblants</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Basant-se_en_la_circumferència_goniomètrica"><span id="Basant-se_en_la_circumfer.C3.A8ncia_goniom.C3.A8trica"></span>Basant-se en la circumferència goniomètrica</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funci%C3%B3_trigonom%C3%A8trica&action=edit&section=5" title="Modifica la secció: Basant-se en la circumferència goniomètrica"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fitxer:Unit_circle.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Unit_circle.svg/220px-Unit_circle.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Unit_circle.svg/330px-Unit_circle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Unit_circle.svg/440px-Unit_circle.svg.png 2x" data-file-width="352" data-file-height="352" /></a><figcaption>El sinus i el cosinus d'un angle <i>t</i> es defineixen respectivament com el valor de la coordenada <i>y</i> i la coordenada <i>x</i> del punt on la circumferència de radi unitat interseca el radi girat un angle <i>t</i> respecte de l'eix <i>x</i> positiu.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fitxer:Unit_circle_angles.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Unit_circle_angles.svg/220px-Unit_circle_angles.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Unit_circle_angles.svg/330px-Unit_circle_angles.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Unit_circle_angles.svg/440px-Unit_circle_angles.svg.png 2x" data-file-width="720" data-file-height="720" /></a><figcaption>La circumferència goniomètrica.</figcaption></figure> <p>Les sis funcions trigonomètriques també es poden definir en base la <a href="/wiki/Circumfer%C3%A8ncia_goniom%C3%A8trica" title="Circumferència goniomètrica">circumferència goniomètrica</a>, la <a href="/wiki/Circumfer%C3%A8ncia" title="Circumferència">circumferència</a> de radi unitat el centre de la qual és l'origen d'un sistema de <a href="/wiki/Coordenades_cartesianes" class="mw-redirect" title="Coordenades cartesianes">coordenades cartesianes</a>. La circumferència goniomètrica aporta poc en el camí cap als càlculs pràctics, si no que es recolza en els triangles rectangles per a la majoria d'angles. En canvi, la definició basada en la circumferència goniomètrica, permet la definició de les funcions trigonomètriques per a tots els arguments reals, tant positius com negatius, no només per a angles entre 0 i π/2 radiants. També dona una imatge visual única que conté de cop tots els angles rellevants. A partir del <a href="/wiki/Teorema_de_Pit%C3%A0gores" title="Teorema de Pitàgores">Teorema de Pitàgores</a> l'equació de la circumferència de radi unitat centrada a l'origen és: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+y^{2}=1\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b18c5b9d2c895a55c7449c48e832b893c3c21df4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.087ex; height:3.009ex;" alt="{\displaystyle x^{2}+y^{2}=1\,}" /></span></dd></dl> <p>Observant la figura de la dreta. Sia una segment de línia recta que va de l'origen fins a la circumferència goniomètrica i forma un angle positiu <i>t</i> amb la meitat positiva de l'eix <i>x</i>. Les coordenades <i>x</i> i <i>y</i> de l'extrem d'aquest segment que toca la circumferència goniomètrica són, respectivament, el cos <i>t</i> i el sin <i>t</i>.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Traçant una perpendicular a l'eix <i>x</i> que passi per l'extrem del segment s'obté un triangle rectangle format pel segment, aquesta perpendicular i l'eix x. Aquest triangle rectangle permet comprovar que pels angles del primer quadrant la definició coincideix amb la definició basada en el triangle rectangle. Com que el radi del cercle és igual a la hipotenusa i té longitud 1, resulta que sin θ = <i>y</i>/1 i cos θ = <i>x</i>/1. La circumferència goniomètrica, es pot entendre com una forma de representar un nombre infinit de triangles rectangles, en els que varien les longituds dels catets, però que la longitud de la hipotenusa es conserva constant igual a 1. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fitxer:Trigonometric_functions.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Trigonometric_functions.svg/220px-Trigonometric_functions.svg.png" decoding="async" width="220" height="143" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Trigonometric_functions.svg/330px-Trigonometric_functions.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/19/Trigonometric_functions.svg/440px-Trigonometric_functions.svg.png 2x" data-file-width="744" data-file-height="485" /></a><figcaption>Funcions trigonomètriques:<span style="color:#00A">Sinus</span>, <span style="color:#0A0">Cosinus</span>, <span style="color:#A00">Tangent</span>, <span style="color:#AA0">Cosecant</span>, <span style="color:#A0A">Secant</span>, <span style="color:#0AA">Cotangent</span></figcaption></figure> <p>Per angles més grans que π/2 i més petits que π la coordenada <i>x</i> del punt passa a ser negativa i la coordenada <i>y</i> és la mateixa que la del triangle obtingut per simetria especular respecte de l'eix vertical, per tant per a aquests angles la definició basant-se en la circumferència goniomètrica és el mateix que estendre la definició a partir del triangle rectangle imposant les següents identitats: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&\sin \left(\pi -\theta \right)=\sin \left(\theta \right)\\&\cos \left(\pi -\theta \right)=-\cos \left(\theta \right)\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd></mtd> <mtd> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>π</mi> <mo>−</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>θ</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>π</mi> <mo>−</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>θ</mi> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&\sin \left(\pi -\theta \right)=\sin \left(\theta \right)\\&\cos \left(\pi -\theta \right)=-\cos \left(\theta \right)\\\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/660ce6a5ad3ec6f00764d3cc8a3ef2e7390f5951" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.239ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}&\sin \left(\pi -\theta \right)=\sin \left(\theta \right)\\&\cos \left(\pi -\theta \right)=-\cos \left(\theta \right)\\\end{aligned}}}" /></span></dd></dl> <p>Seguint el mateix raonament amb simetries respecte de l'eix horitzontal resulta que entre π i 3π/2 el sinus també esdevé negatiu i la definició basant-se en la circumferència goniomètrica és equivalent a estendre la definició a partir del triangle rectangle amb les següents imposicions: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&\sin \left(\pi +\theta \right)=-\sin \left(\theta \right)\\&\cos \left(\pi +\theta \right)=-\cos \left(\theta \right)\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd></mtd> <mtd> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>π</mi> <mo>+</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>θ</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>π</mi> <mo>+</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>θ</mi> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&\sin \left(\pi +\theta \right)=-\sin \left(\theta \right)\\&\cos \left(\pi +\theta \right)=-\cos \left(\theta \right)\\\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f01eba0bb1975f780d77ab4734195dbe226ed015" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.239ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}&\sin \left(\pi +\theta \right)=-\sin \left(\theta \right)\\&\cos \left(\pi +\theta \right)=-\cos \left(\theta \right)\\\end{aligned}}}" /></span></dd></dl> <p>I al tercer quadrant (és a dir entre 3π/2 i 2π): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&\sin \left(2\pi -\theta \right)=\sin \left(\theta \right)\\&\cos \left(2\pi -\theta \right)=-\cos \left(\theta \right)\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd></mtd> <mtd> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>π</mi> <mo>−</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>θ</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>π</mi> <mo>−</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>θ</mi> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&\sin \left(2\pi -\theta \right)=\sin \left(\theta \right)\\&\cos \left(2\pi -\theta \right)=-\cos \left(\theta \right)\\\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3491d16be39400d8f37991dc4aa2a12f2b8040d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.402ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}&\sin \left(2\pi -\theta \right)=\sin \left(\theta \right)\\&\cos \left(2\pi -\theta \right)=-\cos \left(\theta \right)\\\end{aligned}}}" /></span></dd></dl> <p>A la figura de la dreta es presenten els valors del sinus i el cosinus dels angles més habituals en els quatre quadrants. </p><p>Per angles més grans que 2π o més petits que −2π, senzillament es continua girant al voltant del cercle. D'aquesta forma, el sinus i el cosinus esdevenen <a href="/wiki/Funci%C3%B3_peri%C3%B2dica" title="Funció periòdica">funcions periòdiques</a> amb període 2π: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \theta =\sin \left(\theta +2\pi k\right)}"> <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>π</mi> <mi>k</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \theta =\sin \left(\theta +2\pi k\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d675b0b43b975ba3c36b76443339e11af63ff085" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.733ex; height:2.843ex;" alt="{\displaystyle \sin \theta =\sin \left(\theta +2\pi k\right)}" /></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \theta =\cos \left(\theta +2\pi k\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>θ</mi> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mi>k</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \theta =\cos \left(\theta +2\pi k\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc08773259eea763d7fde9af7ec759e14c17d091" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.244ex; height:2.843ex;" alt="{\displaystyle \cos \theta =\cos \left(\theta +2\pi k\right)}" /></span></dd></dl> <p>Per a qualsevol angle θ i qualsevol <a href="/wiki/Nombre_enter" title="Nombre enter">enter</a> <i>k</i>. </p><p>Del període positiu <i>més petit</i> d'una funció periòdica se'n diu el <i>període fonamental</i> de la funció.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> El període fonamental del sinus, el cosinus, la secant, i la cosecant, és el cercle complet, és a dir, 2π radiants o 360 graus; el període fonamental de la tangent i la cotangent és només la meitat del cercle, és a dir π radiants o 180 graus. Més amunt, només s'han definit, a través de la circumferència goniomètrica, el sinus i el cosinus, però les altres quatre funcions trigonomètriques es poden definir per: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan \theta ={\frac {\sin \theta }{\cos \theta }}\quad \sec \theta ={\frac {1}{\cos \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> <mspace width="1em"></mspace> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan \theta ={\frac {\sin \theta }{\cos \theta }}\quad \sec \theta ={\frac {1}{\cos \theta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7df1788a001c1835b5bed2305e74809828df51f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:29.052ex; height:5.509ex;" alt="{\displaystyle \tan \theta ={\frac {\sin \theta }{\cos \theta }}\quad \sec \theta ={\frac {1}{\cos \theta }}}" /></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \csc \theta ={\frac {1}{\sin \theta }}\quad \cot \theta ={\frac {\cos \theta }{\sin \theta }}}"> <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> <mn>1</mn> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> <mspace width="1em"></mspace> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \csc \theta ={\frac {1}{\sin \theta }}\quad \cot \theta ={\frac {\cos \theta }{\sin \theta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74428cc62818d6fa27b68ef9a64aa52035475f6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:28.536ex; height:5.509ex;" alt="{\displaystyle \csc \theta ={\frac {1}{\sin \theta }}\quad \cot \theta ={\frac {\cos \theta }{\sin \theta }}}" /></span></dd></dl> <p>A la figura de la dreta es presenten les gràfiques de les funcions trigonomètriques esteses al llarg dels nombres reals. </p><p>De forma alternativa, <i>totes</i> les funcions trigonomètriques bàsiques es poden definir en termes de la circumferència goniomètrica (que es presenta a la dreta al començament de la pàgina), i definicions geomètriques d'aquest estil són les que es varen fer servir històricament i són les que justifiquen el nom de les funcions. En particular, la corda <i>AB</i> del cercle, on θ és la meitat de l'angle, sin(θ) és <i>AC</i> (la meitat de la corda), una definició que va ser introduïda a l'<a href="/wiki/%C3%8Dndia" title="Índia">Índia</a> (vegeu més amunt la història de les funcions trigonomètriques). cos(θ) és la distància horitzontal <i>OC</i>, i el <a href="/wiki/Versinus" title="Versinus">versin</a> (θ) = 1 − cos(θ) és <i>CD</i>. tan(θ) és la longitud del segment <i>AE</i> de la línia tangent a la circumferència goniomètrica al punt <i>A</i>, d'aquí la paraula <i><a href="/wiki/Tangent" title="Tangent">tangent</a></i> per aquesta funció. cot(θ) és un altre segment tangent, <i>AF</i>. sec(θ) = <i>OE</i> i csc(θ) = <i>OF</i> són segments de <a href="/w/index.php?title=L%C3%ADnia_secant&action=edit&redlink=1" class="new" title="Línia secant (encara no existeix)">línies secants</a> (que intersequen el cercle a dos punts), i també es poden veure com projeccions de <i>OA</i> al llarg de la tangent a <i>A</i> sobre els eixos horitzontal i vertical respectivament. <i>DE</i> és la <a href="/wiki/Exsecant" title="Exsecant">exsec</a> (θ) = sec(θ) − 1 (la porció de la secant fora de, o <i>ex</i>, el cercle). A partir d'aquestes construccions, és fàcil de veure que les funcions secant i tangent divergeixen a mesura que θ tendeix a π/2 (90 graus) i que la cosecant i la cotangent divergeixen a mesura que θ tendeix a zero. (Hi ha moltes construccions semblants possibles, i les identitats trigonomètriques bàsiques, també es poden demostrar gràficament.) </p> <div class="mw-heading mw-heading3"><h3 id="Basant-se_en_sèries"><span id="Basant-se_en_s.C3.A8ries"></span>Basant-se en sèries</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funci%C3%B3_trigonom%C3%A8trica&action=edit&section=6" title="Modifica la secció: Basant-se en sèries"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fitxer:Taylorsine.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Taylorsine.svg/250px-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/500px-Taylorsine.svg.png 2x" data-file-width="1600" data-file-height="1200" /></a><figcaption>La funció sinus (blau) és aproximada amb molta exactitud pel seu <a href="/wiki/Polinomi_de_Taylor" class="mw-redirect" title="Polinomi de Taylor">polinomi de Taylor</a> de grau 5 (rosa) per un cercle complet centrat a l'origen.</figcaption></figure> <p>Fent servir només geometria i les propietats dels <a href="/wiki/L%C3%ADmit_d%27una_funci%C3%B3" class="mw-redirect" title="Límit d'una funció">límits</a>, es pot demostrar que, la <a href="/wiki/Derivada" title="Derivada">derivada</a> del sinus és el cosinus i la derivada del cosinus és menys el sinus (sempre que els angles es mesurin en <a href="/wiki/Radiant_(angle)" class="mw-redirect" title="Radiant (angle)">radiants</a>, vegeu <a href="/wiki/Derivaci%C3%B3_de_les_funcions_trigonom%C3%A8triques" title="Derivació de les funcions trigonomètriques">derivació de les funcions trigonomètriques</a>). Llavors es pot fer servir la teoria de les <a href="/wiki/S%C3%A8rie_de_Taylor" title="Sèrie de Taylor">sèries de Taylor</a> per demostrar que les següents identitats es donen per a tots els <a href="/wiki/Nombres_reals" class="mw-redirect" title="Nombres reals">nombres reals</a> <i>x</i>:<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin x=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots =\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n+1}}{(2n+1)!}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>x</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> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin x=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots =\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n+1}}{(2n+1)!}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20a1d4be32dec2312bf54e5f02a8a5f7c39ffcca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:51.891ex; height:7.009ex;" alt="{\displaystyle \sin x=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots =\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n+1}}{(2n+1)!}}}" /></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos x=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots =\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n}}{(2n)!}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>x</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> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos x=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots =\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n}}{(2n)!}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93f5a69670cc506456056f1853d2863ab2ca5fa0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:49.879ex; height:7.009ex;" alt="{\displaystyle \cos x=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots =\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n}}{(2n)!}}}" /></span></dd></dl> <p>Aquestes identitats sovint s'agafen com a <i>definicions</i> de les funcions sinus i cosinus.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> Sovint es fan servir com un punt de partida per a un tractament rigorós de les funcions trigonomètriques i les seves aplicacions (per exemple, en les <a href="/wiki/S%C3%A8rie_de_Fourier" title="Sèrie de Fourier">sèries de Fourier</a>), atès que la teoria de les <a href="/wiki/S%C3%A8rie_(matem%C3%A0tiques)" title="Sèrie (matemàtiques)">sèries</a> es pot desenvolupar a partir dels fonaments dels <a href="/wiki/Nombres_reals" class="mw-redirect" title="Nombres reals">nombres reals</a>, de forma independent de qualsevol mena de consideracions geomètriques. Llavors, la <a href="/wiki/Derivabilitat" title="Derivabilitat">derivabilitat</a> i la <a href="/wiki/Funci%C3%B3_cont%C3%ADnua" title="Funció contínua">continuïtat</a> d'aquestes funcions s'estableixen només a partir de les definicions de les sèries. </p><p>A partir d'un teorema d'<a href="/wiki/An%C3%A0lisi_complexa" title="Anàlisi complexa">anàlisi complexa</a>, hi ha una única <a href="/wiki/Continuaci%C3%B3_anal%C3%ADtica" title="Continuació analítica">extensió analítica</a> d'aquestes funcions reals al conjunt dels nombres complexos. Aquestes extensions tenen les mateixes sèries de Taylor, d'aquesta forma, les funcions trigonomètriques es defineixen en el conjunt dels nombres complexos emprant les sèries de Taylor de més amunt. </p><p>En les següents gràfiques, el domini és el <a href="/wiki/Pla_complex" title="Pla complex">pla complex</a>. Per representar el <a href="/wiki/Nombre_complex" title="Nombre complex">nombre complex</a> imatge de cada punt es fa servir el color per indicar l'argument, i la lluentor per indicar el mòdul. El color negre correspon al zero. Vegeu (<a href="/wiki/Fitxer:Complex_coloring.jpg" title="Fitxer:Complex coloring.jpg">conveni de colors</a>) </p> <table style="text-align:center"> <caption><b>Funcions trigonomètriques al pla complex</b> </caption> <tbody><tr> <td><figure class="mw-halign-none" typeof="mw:File"><a href="/wiki/Fitxer:Complex_sin.jpg" class="mw-file-description" title="sin z"><img alt="sin z" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/45/Complex_sin.jpg/122px-Complex_sin.jpg" decoding="async" width="122" height="140" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/45/Complex_sin.jpg/184px-Complex_sin.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/45/Complex_sin.jpg/245px-Complex_sin.jpg 2x" data-file-width="944" data-file-height="1079" /></a><figcaption>sin z</figcaption></figure> </td> <td><figure class="mw-halign-none" typeof="mw:File"><a href="/wiki/Fitxer:Complex_cos.jpg" class="mw-file-description" title="cos z"><img alt="cos z" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Complex_cos.jpg/250px-Complex_cos.jpg" decoding="async" width="129" height="140" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Complex_cos.jpg/330px-Complex_cos.jpg 2x" data-file-width="945" data-file-height="1026" /></a><figcaption>cos z</figcaption></figure> </td> <td><figure class="mw-halign-none" typeof="mw:File"><a href="/wiki/Fitxer:Complex_tan.jpg" class="mw-file-description" title="tan z"><img alt="tan z" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Complex_tan.jpg/250px-Complex_tan.jpg" decoding="async" width="127" height="140" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Complex_tan.jpg/330px-Complex_tan.jpg 2x" data-file-width="944" data-file-height="1041" /></a><figcaption>tan z</figcaption></figure> </td> <td><figure class="mw-halign-none" typeof="mw:File"><a href="/wiki/Fitxer:Complex_Cot.jpg" class="mw-file-description" title="cot z"><img alt="cot z" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Complex_Cot.jpg/250px-Complex_Cot.jpg" decoding="async" width="140" height="140" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Complex_Cot.jpg/330px-Complex_Cot.jpg 2x" data-file-width="1059" data-file-height="1059" /></a><figcaption>cot z</figcaption></figure> </td> <td><figure class="mw-halign-none" typeof="mw:File"><a href="/wiki/Fitxer:Complex_Sec.jpg" class="mw-file-description" title="sec z"><img alt="sec z" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Complex_Sec.jpg/140px-Complex_Sec.jpg" decoding="async" width="140" height="140" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Complex_Sec.jpg/210px-Complex_Sec.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Complex_Sec.jpg/280px-Complex_Sec.jpg 2x" data-file-width="951" data-file-height="951" /></a><figcaption>sec z</figcaption></figure> </td> <td><figure class="mw-halign-none" typeof="mw:File"><a href="/wiki/Fitxer:Complex_Csc.jpg" class="mw-file-description" title="csc z"><img alt="csc z" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/90/Complex_Csc.jpg/140px-Complex_Csc.jpg" decoding="async" width="140" height="140" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/90/Complex_Csc.jpg/210px-Complex_Csc.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/90/Complex_Csc.jpg/280px-Complex_Csc.jpg 2x" data-file-width="951" data-file-height="951" /></a><figcaption>csc z</figcaption></figure> </td></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin z\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>z</mi> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin z\,}</annotation> </semantics> </math></span><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\,}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos z\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>z</mi> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos z\,}</annotation> </semantics> </math></span><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\,}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan z\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mi>z</mi> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan z\,}</annotation> </semantics> </math></span><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\,}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cot z\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cot</mi> <mo>⁡</mo> <mi>z</mi> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cot z\,}</annotation> </semantics> </math></span><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\,}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sec z\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sec</mi> <mo>⁡</mo> <mi>z</mi> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sec z\,}</annotation> </semantics> </math></span><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\,}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \csc z\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>csc</mi> <mo>⁡</mo> <mi>z</mi> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \csc z\,}</annotation> </semantics> </math></span><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\,}" /></span> </td></tr></tbody></table> <p>També es poden trobar altres sèries:<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><table> <tbody><tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan x\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mi>x</mi> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan x\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ba406adf958555b3bdf27e00bfabe7f12c00473" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.464ex; height:2.009ex;" alt="{\displaystyle \tan x\,}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}=\sum _{n=0}^{\infty }{\frac {U_{2n+1}x^{2n+1}}{(2n+1)!}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> <mrow> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}=\sum _{n=0}^{\infty }{\frac {U_{2n+1}x^{2n+1}}{(2n+1)!}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7619bde32263db5dc8cba87d2e3d08615650b77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.876ex; height:6.843ex;" alt="{\displaystyle {}=\sum _{n=0}^{\infty }{\frac {U_{2n+1}x^{2n+1}}{(2n+1)!}}}" /></span> </td></tr> <tr> <td> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}=\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}2^{2n}(2^{2n}-1)B_{2n}x^{2n-1}}{(2n)!}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}=\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}2^{2n}(2^{2n}-1)B_{2n}x^{2n-1}}{(2n)!}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d76c9989adaddd6ff9b7e76592369434763cfa6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:37.269ex; height:7.009ex;" alt="{\displaystyle {}=\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}2^{2n}(2^{2n}-1)B_{2n}x^{2n-1}}{(2n)!}}}" /></span> </td></tr> <tr> <td> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}=x+{\frac {x^{3}}{3}}+{\frac {2x^{5}}{15}}+{\frac {17x^{7}}{315}}+\cdots ,\qquad {\mbox{per }}|x|<{\frac {\pi }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <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> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> <mn>15</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>17</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> </mrow> <mn>315</mn> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>,</mo> <mspace width="2em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>per </mtext> </mstyle> </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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}=x+{\frac {x^{3}}{3}}+{\frac {2x^{5}}{15}}+{\frac {17x^{7}}{315}}+\cdots ,\qquad {\mbox{per }}|x|<{\frac {\pi }{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/607b66622045bc8737be58f86e109a6526663e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:49.434ex; height:5.676ex;" alt="{\displaystyle {}=x+{\frac {x^{3}}{3}}+{\frac {2x^{5}}{15}}+{\frac {17x^{7}}{315}}+\cdots ,\qquad {\mbox{per }}|x|<{\frac {\pi }{2}}}" /></span> </td></tr></tbody></table></dd></dl> <p>on </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{n}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{n}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82bdcb51b914aec152a6daec97144a9536a90f58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.193ex; height:2.509ex;" alt="{\displaystyle U_{n}\,}" /></span> és el <i>n</i>-èsim nombre de la <a href="/w/index.php?title=Transformada_de_Boustrophedon&action=edit&redlink=1" class="new" title="Transformada de Boustrophedon (encara no existeix)">transformada de Boustrophedon</a>,</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{n}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{n}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d352186a495a156ca173e351226973b84706a165" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.37ex; height:2.509ex;" alt="{\displaystyle B_{n}\,}" /></span> és el <i>n</i>-èsim <a href="/wiki/Nombres_de_Bernoulli" title="Nombres de Bernoulli">nombre de Bernoulli</a>, i</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{n}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{n}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c621cabe4418802f7f26e069a046cd2270bff41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.321ex; height:2.509ex;" alt="{\displaystyle E_{n}\,}" /></span> (davall) és l'<i>n</i>-èsim <a href="/wiki/Nombre_d%27Euler" title="Nombre d'Euler">nombre d'Euler</a>.</dd></dl> <p>Quan això s'expressa de forma que els denominadors són els corresponents factorials i el numeradors, anomenats els "nombres tangents" tenen una interpretació <a href="/wiki/Combinat%C3%B2ria" title="Combinatòria">combinatòria</a>: enumeren <a href="/w/index.php?title=Permutaci%C3%B3_alternada&action=edit&redlink=1" class="new" title="Permutació alternada (encara no existeix)">permutacions alternades</a> de conjunts finits de cardinalitat senar. </p> <dl><dd><table> <tbody><tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \csc x\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>csc</mi> <mo>⁡</mo> <mi>x</mi> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \csc x\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b122f73f49996dd6408924fafb617eb8923c4b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.085ex; height:1.676ex;" alt="{\displaystyle \csc x\,}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}=\sum _{n=0}^{\infty }{\frac {(-1)^{n+1}2(2^{2n-1}-1)B_{2n}x^{2n-1}}{(2n)!}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> <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> <mo stretchy="false">(</mo> <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> <mo stretchy="false">)</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}=\sum _{n=0}^{\infty }{\frac {(-1)^{n+1}2(2^{2n-1}-1)B_{2n}x^{2n-1}}{(2n)!}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/618a924198ead07e7e67d7d78f54bc1991ff9b9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:37.329ex; height:7.009ex;" alt="{\displaystyle {}=\sum _{n=0}^{\infty }{\frac {(-1)^{n+1}2(2^{2n-1}-1)B_{2n}x^{2n-1}}{(2n)!}}}" /></span> </td></tr> <tr> <td> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}={\frac {1}{x}}+{\frac {x}{6}}+{\frac {7x^{3}}{360}}+{\frac {31x^{5}}{15120}}+\cdots ,\qquad {\mbox{per }}0<|x|<\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mn>6</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>7</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mn>360</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>31</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> <mn>15120</mn> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>,</mo> <mspace width="2em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>per </mtext> </mstyle> </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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}={\frac {1}{x}}+{\frac {x}{6}}+{\frac {7x^{3}}{360}}+{\frac {31x^{5}}{15120}}+\cdots ,\qquad {\mbox{per }}0<|x|<\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32c1562857ff9e0af7defb5c699a10344231e7d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:53.744ex; height:5.676ex;" alt="{\displaystyle {}={\frac {1}{x}}+{\frac {x}{6}}+{\frac {7x^{3}}{360}}+{\frac {31x^{5}}{15120}}+\cdots ,\qquad {\mbox{per }}0<|x|<\pi }" /></span> </td></tr></tbody></table></dd></dl> <dl><dd><table> <tbody><tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sec x\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sec</mi> <mo>⁡</mo> <mi>x</mi> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sec x\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/927fd84c0cb9aeb05db60c82d83bd735d9c6564e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.085ex; height:1.676ex;" alt="{\displaystyle \sec x\,}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}=\sum _{n=0}^{\infty }{\frac {U_{2n}x^{2n}}{(2n)!}}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}E_{2n}x^{2n}}{(2n)!}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> <mrow> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <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> <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> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}=\sum _{n=0}^{\infty }{\frac {U_{2n}x^{2n}}{(2n)!}}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}E_{2n}x^{2n}}{(2n)!}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f109b288fe48eacfc147e7d1460809ff376e0ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:35.476ex; height:7.009ex;" alt="{\displaystyle {}=\sum _{n=0}^{\infty }{\frac {U_{2n}x^{2n}}{(2n)!}}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}E_{2n}x^{2n}}{(2n)!}}}" /></span> </td></tr> <tr> <td> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}=1+{\frac {x^{2}}{2}}+{\frac {5x^{4}}{24}}+{\frac {61x^{6}}{720}}+\cdots ,\qquad {\mbox{per }}|x|<{\frac {\pi }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <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> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>5</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <mn>24</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>61</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> </mrow> <mn>720</mn> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>,</mo> <mspace width="2em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>per </mtext> </mstyle> </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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}=1+{\frac {x^{2}}{2}}+{\frac {5x^{4}}{24}}+{\frac {61x^{6}}{720}}+\cdots ,\qquad {\mbox{per }}|x|<{\frac {\pi }{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf0dfa8348e190d2762d802365815b3485ae8fd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:49.267ex; height:5.843ex;" alt="{\displaystyle {}=1+{\frac {x^{2}}{2}}+{\frac {5x^{4}}{24}}+{\frac {61x^{6}}{720}}+\cdots ,\qquad {\mbox{per }}|x|<{\frac {\pi }{2}}}" /></span> </td></tr></tbody></table></dd></dl> <p>Quant això s'expressa en una forma en què els denominadors són els corresponents factorials, els numeradors, anomenats els "nombres secants", tenen una interpretació combinatòria: enumeren les permutacions alternades de conjunts de cardinalitat parell. </p> <dl><dd><table> <tbody><tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cot x\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cot</mi> <mo>⁡</mo> <mi>x</mi> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cot x\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a30d94c8dbc87f9487485b0915c571088c8ffbba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.203ex; height:2.009ex;" alt="{\displaystyle \cot x\,}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}2^{2n}B_{2n}x^{2n-1}}{(2n)!}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> <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> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}2^{2n}B_{2n}x^{2n-1}}{(2n)!}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7299016e42e48c19f66d7a43052e5fef5703d812" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.153ex; height:7.009ex;" alt="{\displaystyle {}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}2^{2n}B_{2n}x^{2n-1}}{(2n)!}}}" /></span> </td></tr> <tr> <td> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}={\frac {1}{x}}-{\frac {x}{3}}-{\frac {x^{3}}{45}}-{\frac {2x^{5}}{945}}-\cdots ,\qquad {\mbox{per }}0<|x|<\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mn>3</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mn>45</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> <mn>945</mn> </mfrac> </mrow> <mo>−</mo> <mo>⋯</mo> <mo>,</mo> <mspace width="2em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>per </mtext> </mstyle> </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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}={\frac {1}{x}}-{\frac {x}{3}}-{\frac {x^{3}}{45}}-{\frac {2x^{5}}{945}}-\cdots ,\qquad {\mbox{per }}0<|x|<\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62a5924c3f85b82af66e6c8e752314fcaefa6966" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:50.316ex; height:5.843ex;" alt="{\displaystyle {}={\frac {1}{x}}-{\frac {x}{3}}-{\frac {x^{3}}{45}}-{\frac {2x^{5}}{945}}-\cdots ,\qquad {\mbox{per }}0<|x|<\pi }" /></span> </td></tr></tbody></table></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Emprant_equacions_diferencials">Emprant equacions diferencials</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funci%C3%B3_trigonom%C3%A8trica&action=edit&section=7" title="Modifica la secció: Emprant equacions diferencials"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Tant la funció sinus com la funció cosinus satisfan l'<a href="/wiki/Equaci%C3%B3_diferencial" title="Equació diferencial">equació diferencial</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y''=-y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mo>″</mo> </msup> <mo>=</mo> <mo>−</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y''=-y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc7293b16d52f2a42cf102e32153c63e341ef7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.36ex; height:2.843ex;" alt="{\displaystyle y''=-y}" /></span>.</dd></dl> <p>És a dir, cada una és el negatiu de la seva pròpia <a href="/wiki/Derivada_segona" title="Derivada segona">derivada segona</a>. En l'<a href="/wiki/Espai_funcional" title="Espai funcional">espai funcional</a> de dues dimensions <i>V</i> consistent en totes les solucions d'aquesta equació, la funció sinus és l'única solució que satisfà les condicions inicials <i>y</i>(0) = 0 i <i>y</i>′(0) = 1, i la funció cosinus és l'única solució que satisfà les condicions inicials <i>y</i>(0) = 1 i <i>y</i>′(0) = 0. Atès que les funcions sinus i cosinus són linealment independents, juntes formen una <a href="/wiki/Base_(%C3%A0lgebra_lineal)" class="mw-redirect" title="Base (àlgebra lineal)">base</a> de <i>V</i>. Aquest mètode de definir les funcions sinus i cosinus, és essencialment equivalent a fer servir la fórmula d'Euler. (<i>Vegeu: <a href="/wiki/Equaci%C3%B3_diferencial_lineal" title="Equació diferencial lineal">Equació diferencial lineal</a></i>.) Resulta que aquesta equació diferencial no només es pot fer servir per a definir les funcions sinus i cosinus, sinó que també es pot fer servir per a demostrar les <a href="/wiki/Identitat_trigonom%C3%A8trica" class="mw-redirect" title="Identitat trigonomètrica">identitats trigonomètriques</a> de les funcions sinus i cosinus. A més, l'observació que les funcions sinus i cosinus satisfan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y''=-y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mo>″</mo> </msup> <mo>=</mo> <mo>−</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y''=-y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc7293b16d52f2a42cf102e32153c63e341ef7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.36ex; height:2.843ex;" alt="{\displaystyle y''=-y}" /></span> significa que són les <a href="/wiki/Funci%C3%B3_pr%C3%B2pia" title="Funció pròpia">funcions pròpies</a> de l'operador derivada segona. </p><p>La funció tangent és l'única solució de l'equació diferencial no lineal </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y'=1+y^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f451ecc1e08531466d4663a99ddc326fc663eb4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.161ex; height:3.009ex;" alt="{\displaystyle y'=1+y^{2}}" /></span></dd></dl> <p>que satisfà la condició inicial <i>y</i>(0) = 0. Hi ha una demostració visual molt interessant que la funció tangent satisfà aquesta equació diferencial; vegeu Needham's <i>Visual Complex Analysis.</i><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Conseqüències_de_fer_servir_radiants"><span id="Conseq.C3.BC.C3.A8ncies_de_fer_servir_radiants"></span>Conseqüències de fer servir radiants</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funci%C3%B3_trigonom%C3%A8trica&action=edit&section=8" title="Modifica la secció: Conseqüències de fer servir radiants"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Els <a href="/wiki/Radiant_(angle)" class="mw-redirect" title="Radiant (angle)">radiants</a> especifiquen un angle a base de mesurar la longitud de l'arc sobre el cercle de radi unitat i constitueixen un argument especial per a les funcions sinus i cosinus. En particular, només aquestes sinus i cosinus que apliquen radiants a quocients satisfan les equacions diferencials que els descriuen de forma clàssica. Si a l'argument del sinus o del cosinus en radiants se li aplica un factor d'escala (es multiplica per una freqüència), </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\sin(kx);k\neq 0,k\neq 1\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>;</mo> <mi>k</mi> <mo>≠</mo> <mn>0</mn> <mo>,</mo> <mi>k</mi> <mo>≠</mo> <mn>1</mn> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=\sin(kx);k\neq 0,k\neq 1\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b53a0e19e74a3787de96fa383bd1ee96dbd8b10f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.121ex; height:2.843ex;" alt="{\displaystyle f(x)=\sin(kx);k\neq 0,k\neq 1\,}" /></span></dd></dl> <p>Llavors les <a href="/wiki/Derivada" title="Derivada">derivades</a> resulten multiplicades per una factor d'escala en l'<i>amplitud</i>. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f'(x)=k\cos(kx)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>k</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f'(x)=k\cos(kx)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d1b4bfa5c887eeb6a42fefbf9ff08739b351b86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.689ex; height:3.009ex;" alt="{\displaystyle f'(x)=k\cos(kx)\,}" /></span>.</dd></dl> <p>Aquí, <i>k</i> és una constant que representa un factor de canvi d'unitats. Si <i>x</i> està en graus, llavors </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k={\frac {\pi }{180^{\circ }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k={\frac {\pi }{180^{\circ }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c658129e3d324b3e887e2aea7f944d3a1222ddf7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.687ex; height:4.843ex;" alt="{\displaystyle k={\frac {\pi }{180^{\circ }}}}" /></span>.</dd></dl> <p>Això significa que la <a href="/wiki/Derivada_segona" title="Derivada segona">derivada segona</a> del sinus en graus no satisfà l'equació diferencial </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y''=-y\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mo>″</mo> </msup> <mo>=</mo> <mo>−</mo> <mi>y</mi> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y''=-y\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ec435b8e99a4fa765cf0fbfc7834f6ceffa4a48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.747ex; height:2.843ex;" alt="{\displaystyle y''=-y\,}" /></span>,</dd></dl> <p>sinó </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y''=-k^{2}y\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mo>″</mo> </msup> <mo>=</mo> <mo>−</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>y</mi> <mspace width="thickmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y''=-k^{2}y\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/598f73647db251c8fe25c84f19b16abff17d4621" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.27ex; height:3.009ex;" alt="{\displaystyle y''=-k^{2}y\;}" /></span>;</dd></dl> <p>la <a href="/wiki/Derivada_segona" title="Derivada segona">derivada segona</a> del cosinus es comporta de forma similar. </p><p>Això significa que aquests sinus i cosinus (mesurant els angles en una unitat diferent del radiant) són funcions diferents, i que les derivades quartes del sinus, només tornaran a ser altre cop sinus si l'argument ve donat en radiants. </p> <div class="mw-heading mw-heading3"><h3 id="Emprant_equacions_funcionals">Emprant equacions funcionals</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funci%C3%B3_trigonom%C3%A8trica&action=edit&section=9" title="Modifica la secció: Emprant equacions funcionals"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>En <a href="/wiki/An%C3%A0lisi_matem%C3%A0tica" title="Anàlisi matemàtica">anàlisi matemàtica</a>, les funcions trigonomètriques es poden definir emprant <a href="/wiki/Equaci%C3%B3_funcional" title="Equació funcional">equacions funcionals</a> basades en propietats com les fórmules de la suma i la diferència. Prenent com a dades aquestes fórmules i la identitat de Pitàgores, per exemple, es pot demostrar que només hi ha dues funcions reals que satisfan aquestes condicions. Simbòlicament es pot dir que existeix exactament un parell de funcions reals <i>sin</i> i <i>cos</i> tals que per a qualsevulla nombres reals <i>x</i> i <i>y</i>, es compleixen les següents equacions: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin ^{2}(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> <mo stretchy="false">(</mo> <mi>x</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>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin ^{2}(x)+\cos ^{2}(x)=1,\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24762c3810ab59df58d4aa7ffebd7853b4e170fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.488ex; height:3.176ex;" alt="{\displaystyle \sin ^{2}(x)+\cos ^{2}(x)=1,\,}" /></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(x+y)=\sin(x)\cos(y)+\cos(x)\sin(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> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(x+y)=\sin(x)\cos(y)+\cos(x)\sin(y),\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e17974e07a93e4894cfd367011cd3fdf1205f6c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.878ex; height:2.843ex;" alt="{\displaystyle \sin(x+y)=\sin(x)\cos(y)+\cos(x)\sin(y),\,}" /></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(x+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> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(x+y)=\cos(x)\cos(y)-\sin(x)\sin(y),\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3dae1631f40e2acb9a60285925e8757d1a26c690" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.134ex; height:2.843ex;" alt="{\displaystyle \cos(x+y)=\cos(x)\cos(y)-\sin(x)\sin(y),\,}" /></span></dd></dl> <p>amb la condició afegida que </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<x\cos(x)<\sin(x)<x\ \mathrm {per} \ 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> <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> <mi>x</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">r</mi> </mrow> <mtext> </mtext> <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\ \mathrm {per} \ 0<x<1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d23564d877a10149fedf42b4ea60fd2042b264f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.998ex; height:2.843ex;" alt="{\displaystyle 0<x\cos(x)<\sin(x)<x\ \mathrm {per} \ 0<x<1}" /></span>.</dd></dl> <p>També són possibles altres deduccions començant a partir d'altres equacions funcionals i aquestes definicions es poden estendre als nombres complexos. Com a exemple, aquest camí es pot fer servir per definir la <a href="/w/index.php?title=Trigonometria_en_cossos_de_Galois&action=edit&redlink=1" class="new" title="Trigonometria en cossos de Galois (encara no existeix)">trigonometria en cossos de Galois</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Relació_de_les_funcions_trigonomètriques_amb_la_funció_exponencial_i_els_nombres_complexos"><span id="Relaci.C3.B3_de_les_funcions_trigonom.C3.A8triques_amb_la_funci.C3.B3_exponencial_i_els_nombres_complexos"></span>Relació de les funcions trigonomètriques amb la funció exponencial i els nombres complexos</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funci%C3%B3_trigonom%C3%A8trica&action=edit&section=10" title="Modifica la secció: Relació de les funcions trigonomètriques amb la funció exponencial i els nombres complexos"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fitxer:Sine_and_Cosine_fundamental_relationship_to_Circle_(and_Helix).gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Sine_and_Cosine_fundamental_relationship_to_Circle_%28and_Helix%29.gif/250px-Sine_and_Cosine_fundamental_relationship_to_Circle_%28and_Helix%29.gif" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Sine_and_Cosine_fundamental_relationship_to_Circle_%28and_Helix%29.gif/330px-Sine_and_Cosine_fundamental_relationship_to_Circle_%28and_Helix%29.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/8/8e/Sine_and_Cosine_fundamental_relationship_to_Circle_%28and_Helix%29.gif 2x" data-file-width="420" data-file-height="420" /></a><figcaption><a href="/wiki/F%C3%B3rmula_d%27Euler" title="Fórmula d'Euler">Fórmula d'Euler</a> il·lustrada per l'<a href="/wiki/H%C3%A8lix" class="mw-redirect mw-disambig" title="Hèlix">hèlix</a> en tres dimensions, començant amb els components resultat de la <a href="/wiki/Projecci%C3%B3_ortogonal" title="Projecció ortogonal">projecció ortogonal</a> 2-D de la <a href="/wiki/Circumfer%C3%A8ncia_goniom%C3%A8trica" title="Circumferència goniomètrica">circumferència goniomètrica</a>, el <a href="/wiki/Ona_sinusoidal" class="mw-redirect" title="Ona sinusoidal">sinus</a> el cosinus (fent <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35e05afcda9311a37ac0765af7f54977f600d634" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.029ex; height:2.176ex;" alt="{\displaystyle \theta =t}" /></span>).</figcaption></figure> <p>A partir del desenvolupament en sèrie de la <a href="/wiki/Funci%C3%B3_exponencial" title="Funció exponencial">funció exponencial</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{x}=\sum \limits _{n=0}^{\infty }{\frac {x^{n}}{n!}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>=</mo> <munderover> <mo movablelimits="false">∑</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> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{x}=\sum \limits _{n=0}^{\infty }{\frac {x^{n}}{n!}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51bcf82e698ba823d24e6139a338bac68a18d420" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.481ex; height:6.843ex;" alt="{\displaystyle e^{x}=\sum \limits _{n=0}^{\infty }{\frac {x^{n}}{n!}}}" /></span></dd></dl> <p>Es calcula l'exponencial d'iθ: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{i\theta }=\sum \limits _{n=0}^{\infty }{\frac {i^{n}\theta ^{n}}{n!}}=1+{\frac {i\theta }{1}}+{\frac {(-1)\theta ^{2}}{2!}}+{\frac {(-1)i\theta ^{3}}{3!}}+{\frac {(-1)^{2}\theta ^{4}}{4!}}+...}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> <mo>=</mo> <munderover> <mo movablelimits="false">∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <mi>θ</mi> </mrow> <mn>1</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>i</mi> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mrow> <mn>3</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <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"> <mn>2</mn> </mrow> </msup> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <mrow> <mn>4</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{i\theta }=\sum \limits _{n=0}^{\infty }{\frac {i^{n}\theta ^{n}}{n!}}=1+{\frac {i\theta }{1}}+{\frac {(-1)\theta ^{2}}{2!}}+{\frac {(-1)i\theta ^{3}}{3!}}+{\frac {(-1)^{2}\theta ^{4}}{4!}}+...}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd708670da4b6fe769b3f3923eb41e3fbf3bed26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:62.674ex; height:7.009ex;" alt="{\displaystyle e^{i\theta }=\sum \limits _{n=0}^{\infty }{\frac {i^{n}\theta ^{n}}{n!}}=1+{\frac {i\theta }{1}}+{\frac {(-1)\theta ^{2}}{2!}}+{\frac {(-1)i\theta ^{3}}{3!}}+{\frac {(-1)^{2}\theta ^{4}}{4!}}+...}" /></span></dd></dl> <p>Tot seguit s'agrupen els termes parells i els senars (la reordenació dels termes es pot fer a causa del fet que la sèrie és uniformement convergent) i es treu factor comú i: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{i\theta }=\left(1+{\frac {(-1)\theta ^{2}}{2!}}+{\frac {(-1)^{2}\theta ^{4}}{4!}}...\right)+i\left({\frac {\theta }{1}}+{\frac {(-1)\theta ^{3}}{3!}}...\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <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"> <mn>2</mn> </mrow> </msup> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <mrow> <mn>4</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>i</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>1</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mrow> <mn>3</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{i\theta }=\left(1+{\frac {(-1)\theta ^{2}}{2!}}+{\frac {(-1)^{2}\theta ^{4}}{4!}}...\right)+i\left({\frac {\theta }{1}}+{\frac {(-1)\theta ^{3}}{3!}}...\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4f4e57b63e206a713d9c49c84462516001bb6b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:59.368ex; height:7.509ex;" alt="{\displaystyle e^{i\theta }=\left(1+{\frac {(-1)\theta ^{2}}{2!}}+{\frac {(-1)^{2}\theta ^{4}}{4!}}...\right)+i\left({\frac {\theta }{1}}+{\frac {(-1)\theta ^{3}}{3!}}...\right)}" /></span></dd></dl> <p>Observant el desenvolupament en sèrie de les funcions sinus i cosinus queda clar que això és igual a: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{i\theta }=\cos \left(\theta \right)+i\sin \left(\theta \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>θ</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>θ</mi> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{i\theta }=\cos \left(\theta \right)+i\sin \left(\theta \right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/835bc4c92a901486e6c2b95e99833ea3e3db6c62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.549ex; height:3.176ex;" alt="{\displaystyle e^{i\theta }=\cos \left(\theta \right)+i\sin \left(\theta \right)}" /></span></dd></dl> <p>D'aquesta identitat se'n diu la <a href="/wiki/F%C3%B3rmula_d%27Euler" title="Fórmula d'Euler">fórmula d'Euler</a>. D'aquesta forma les funcions trigonomètriques esdevenen essencials en la interpretació geomètrica de l'anàlisi complexa. Per exemple, amb la identitat de dalt, si es considera el cercle unitat en el <a href="/wiki/Pla_complex" title="Pla complex">pla complex</a>, definit per <i>e</i><sup>i<i>x</i></sup>, i tal com s'ha fet abans, es pot parametritzar aquest cercle en termes de cosinus i sinus, la relació entre l'exponencial complexa i les funcions trigonomètriques esdevé més clara. </p><p>A més, això permet la definició de les funcions trigonomètriques per arguments complexos <i>z</i>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin z\,=\,\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}z^{2n+1}\,=\,{e^{iz}-e^{-iz} \over 2i}=-i\sinh \left(iz\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>z</mi> <mspace width="thinmathspace"></mspace> <mo>=</mo> <mspace width="thinmathspace"></mspace> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mo>=</mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>z</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>z</mi> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>i</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mi>i</mi> <mi>sinh</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mi>z</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin z\,=\,\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}z^{2n+1}\,=\,{e^{iz}-e^{-iz} \over 2i}=-i\sinh \left(iz\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e33baf066c240ab80ca92137e80090aaa879345" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:55.106ex; height:6.843ex;" alt="{\displaystyle \sin z\,=\,\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}z^{2n+1}\,=\,{e^{iz}-e^{-iz} \over 2i}=-i\sinh \left(iz\right)}" /></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos z\,=\,\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}z^{2n}\,=\,{e^{iz}+e^{-iz} \over 2}=\cosh \left(iz\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>z</mi> <mspace width="thinmathspace"></mspace> <mo>=</mo> <mspace width="thinmathspace"></mspace> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mo>=</mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>z</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>z</mi> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mi>cosh</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mi>z</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos z\,=\,\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}z^{2n}\,=\,{e^{iz}+e^{-iz} \over 2}=\cosh \left(iz\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c975cebf8ca0b4cb690f2fbc87e12a5debb65c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:47.501ex; height:6.843ex;" alt="{\displaystyle \cos z\,=\,\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}z^{2n}\,=\,{e^{iz}+e^{-iz} \over 2}=\cosh \left(iz\right)}" /></span></dd></dl> <p>on <i>i</i>² = −1. També, per <i>x</i> reals purs, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos x\,=\,{\mbox{Re }}(e^{ix})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mo>=</mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>Re </mtext> </mstyle> </mrow> <mo stretchy="false">(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>x</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos x\,=\,{\mbox{Re }}(e^{ix})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8911ca0a7a3053ced2eb3468659ef6ed45a2c2f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.657ex; height:3.176ex;" alt="{\displaystyle \cos x\,=\,{\mbox{Re }}(e^{ix})}" /></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin x\,=\,{\mbox{Im }}(e^{ix})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mo>=</mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>Im </mtext> </mstyle> </mrow> <mo stretchy="false">(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>x</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin x\,=\,{\mbox{Im }}(e^{ix})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c8cac517ff4685c9d499070913418e7017f9349" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.434ex; height:3.176ex;" alt="{\displaystyle \sin x\,=\,{\mbox{Im }}(e^{ix})}" /></span></dd></dl> <p>També és conegut que els processos exponencials estan íntimament lligats al comportament periòdic. </p> <div class="mw-heading mw-heading2"><h2 id="Identitats_trigonomètriques"><span id="Identitats_trigonom.C3.A8triques"></span>Identitats trigonomètriques</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funci%C3%B3_trigonom%C3%A8trica&action=edit&section=11" title="Modifica la secció: Identitats trigonomètriques"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r30997230" /><div role="note" class="hatnote navigation-not-searchable">Articles principals: <a href="/wiki/Llista_d%27identitats_trigonom%C3%A8triques" title="Llista d'identitats trigonomètriques">Llista d'identitats trigonomètriques</a> i <a href="/wiki/Demostraci%C3%B3_de_les_identitats_trigonom%C3%A8triques" title="Demostració de les identitats trigonomètriques">Demostració de les identitats trigonomètriques</a></div> <p>Hi ha moltes identitats matemàtiques relacionades amb les funcions trigonomètriques. Entre les més utilitzades hi ha la <a href="/w/index.php?title=Identitat_de_Pit%C3%A0gores&action=edit&redlink=1" class="new" title="Identitat de Pitàgores (encara no existeix)">identitat de Pitàgores</a>, que estableix que per qualsevol angle, el sinus al quadrat més el cosinus al quadrat dona 1. És fàcil de veure, considerant el triangle rectangle amb la hipotenusa de longitud 1 i aplicant el <a href="/wiki/Teorema_de_Pit%C3%A0gores" title="Teorema de Pitàgores">teorema de Pitàgores</a>. La identitat de Pitàgores s'escriu, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\sin x\right)^{2}+\left(\cos x\right)^{2}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\sin x\right)^{2}+\left(\cos x\right)^{2}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edeb267350aed2b6efd0d089f2878e71ae004c9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.229ex; height:3.343ex;" alt="{\displaystyle \left(\sin x\right)^{2}+\left(\cos x\right)^{2}=1}" /></span>,</dd></dl> <p>O més freqüentment escrivint l'exponent "dos" al costat dels símbols sinus i cosinus: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin ^{2}\left(x\right)+\cos ^{2}\left(x\right)=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> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>+</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin ^{2}\left(x\right)+\cos ^{2}\left(x\right)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cda39b3cd32cefa24b1ad135fd93fa64d849633" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.454ex; height:3.176ex;" alt="{\displaystyle \sin ^{2}\left(x\right)+\cos ^{2}\left(x\right)=1}" /></span>.</dd></dl> <p>En alguns casos els parèntesis interiors es poden ometre. </p><p>Aquesta identitat permet expressar el sinus en funció del cosinus o viceversa, conjuntament amb les identitats que sorgeixen de les definicions de les funcions trigonomètriques permet expressar qualsevol de les sis funcions trigonomètriques en funció de qualsevol de les altres (pel cas d'angles aguts sense cap mena d'ambigüitats, pels altres cal informació addicional per determinar el signe de les funcions). A la següent taula es resumeixen aquestes identitats: </p> <table class="wikitable" style="background-color:#FFFFFF;text-align:center"> <caption>Cada una de les funcions trigonomètriques expressada en funció de cada una de les altres cinc. </caption> <tbody><tr> <th>Funció </th> <th>sin </th> <th>cos </th> <th>tan </th> <th>csc </th> <th>sec </th> <th>cot </th></tr> <tr> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \theta =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \theta =}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85a567c5248097d657724e34cb8666208c3085ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.786ex; height:2.176ex;" alt="{\displaystyle \sin \theta =}" /></span> </th> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \theta \ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \theta \ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24199033500a02f8fc7e410e959aa622f8fa3ed2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.914ex; height:2.176ex;" alt="{\displaystyle \sin \theta \ }" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {1-\cos ^{2}\theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {1-\cos ^{2}\theta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b825d54fdfbfc9eff6f525efabccace541f6955b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.97ex; height:3.509ex;" alt="{\displaystyle {\sqrt {1-\cos ^{2}\theta }}}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\tan \theta }{\sqrt {1+\tan ^{2}\theta }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\tan \theta }{\sqrt {1+\tan ^{2}\theta }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55eb6456849e116e71c76d832ae05b8051816923" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:13.054ex; height:6.676ex;" alt="{\displaystyle {\frac {\tan \theta }{\sqrt {1+\tan ^{2}\theta }}}}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\csc \theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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 {\frac {1}{\csc \theta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fc4edd8913183d4072ff4afc02cae272602d49d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:5.295ex; height:5.343ex;" alt="{\displaystyle {\frac {1}{\csc \theta }}}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\sqrt {\sec ^{2}\theta -1}}{\sec \theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <msup> <mi>sec</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <mn>1</mn> </msqrt> <mrow> <mi>sec</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\sqrt {\sec ^{2}\theta -1}}{\sec \theta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f493a83afba5b4d2ca73d9c80a1235d59967b32c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.676ex; height:6.343ex;" alt="{\displaystyle {\frac {\sqrt {\sec ^{2}\theta -1}}{\sec \theta }}}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\sqrt {1+\cot ^{2}\theta }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>cot</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sqrt {1+\cot ^{2}\theta }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82e7642ece6a5d8db07485d76b5951b4ee2b297f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:12.794ex; height:6.509ex;" alt="{\displaystyle {\frac {1}{\sqrt {1+\cot ^{2}\theta }}}}" /></span> </td></tr> <tr> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \theta =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \theta =}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6182f1f03d1c1c76dfb4c6943877c788511980d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.042ex; height:2.176ex;" alt="{\displaystyle \cos \theta =}" /></span> </th> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {1-\sin ^{2}\theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {1-\sin ^{2}\theta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/806247fb8b459627650887f5f49b66c3fe9492df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.714ex; height:3.509ex;" alt="{\displaystyle {\sqrt {1-\sin ^{2}\theta }}}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \theta \ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \theta \ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf55491c9359c3cd40ec4c3cefdd92dc8ca937d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.169ex; height:2.176ex;" alt="{\displaystyle \cos \theta \ }" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\sqrt {1+\tan ^{2}\theta }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sqrt {1+\tan ^{2}\theta }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd3b870c8f1e715ec839d08937be28572476fd53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:13.054ex; height:6.509ex;" alt="{\displaystyle {\frac {1}{\sqrt {1+\tan ^{2}\theta }}}}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\sqrt {\csc ^{2}\theta -1}}{\csc \theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <msup> <mi>csc</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <mn>1</mn> </msqrt> <mrow> <mi>csc</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\sqrt {\csc ^{2}\theta -1}}{\csc \theta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a619942db34dc33a02b4fa1aa5e739e7f2bb6f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.676ex; height:6.343ex;" alt="{\displaystyle {\frac {\sqrt {\csc ^{2}\theta -1}}{\csc \theta }}}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\sec \theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>sec</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sec \theta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68a698c7710dad53b8f5fb67d0ade8c702f675f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:5.295ex; height:5.343ex;" alt="{\displaystyle {\frac {1}{\sec \theta }}}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\cot \theta }{\sqrt {1+\cot ^{2}\theta }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>cot</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>cot</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\cot \theta }{\sqrt {1+\cot ^{2}\theta }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/797427000906f615677ad84267a3b23d46e491dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:12.794ex; height:6.676ex;" alt="{\displaystyle {\frac {\cot \theta }{\sqrt {1+\cot ^{2}\theta }}}}" /></span> </td></tr> <tr> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan \theta =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan \theta =}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/811edbd828a6027e90de36901ee7a868a3315b4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.29ex; height:2.176ex;" alt="{\displaystyle \tan \theta =}" /></span> </th> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\sin \theta }{\sqrt {1-\sin ^{2}\theta }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\sin \theta }{\sqrt {1-\sin ^{2}\theta }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6490b5091f71ff1eeb809b885142f1e78dd14328" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:12.55ex; height:6.676ex;" alt="{\displaystyle {\frac {\sin \theta }{\sqrt {1-\sin ^{2}\theta }}}}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\sqrt {1-\cos ^{2}\theta }}{\cos \theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </msqrt> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\sqrt {1-\cos ^{2}\theta }}{\cos \theta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f0137a206adc458cada957c9cb89783efa49050" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.806ex; height:6.343ex;" alt="{\displaystyle {\frac {\sqrt {1-\cos ^{2}\theta }}{\cos \theta }}}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan \theta \ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan \theta \ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/479f18c0bc7e820332a7d26a8a01e7172ed68535" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.418ex; height:2.176ex;" alt="{\displaystyle \tan \theta \ }" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\sqrt {\csc ^{2}\theta -1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <msup> <mi>csc</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <mn>1</mn> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sqrt {\csc ^{2}\theta -1}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c39f4d108731bfa78bc1bef5a17c77d6b930b10a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:12.676ex; height:6.509ex;" alt="{\displaystyle {\frac {1}{\sqrt {\csc ^{2}\theta -1}}}}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\sec ^{2}\theta -1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>sec</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\sec ^{2}\theta -1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84ca394fd1ea88c7f36f62e5025a29c08152fbba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.839ex; height:3.509ex;" alt="{\displaystyle {\sqrt {\sec ^{2}\theta -1}}}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\cot \theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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 {\frac {1}{\cot \theta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f572723cb9aeffbb394cbcd84e394344885a3ec1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:5.413ex; height:5.343ex;" alt="{\displaystyle {\frac {1}{\cot \theta }}}" /></span> </td></tr> <tr> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \csc \theta =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>csc</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \csc \theta =}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5563ddf95d132c38ef38814ab3da18f5f1031cea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.912ex; height:2.176ex;" alt="{\displaystyle \csc \theta =}" /></span> </th> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {1 \over \sin \theta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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 {1 \over \sin \theta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a0c761b25b3a7f894f5265017e83ec3ac819470" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:5.169ex; height:5.343ex;" alt="{\displaystyle {1 \over \sin \theta }}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {1 \over {\sqrt {1-\cos ^{2}\theta }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </msqrt> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {1 \over {\sqrt {1-\cos ^{2}\theta }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/805004892e45b8bf1fdc4c0705cf582266212c0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:12.806ex; height:6.509ex;" alt="{\displaystyle {1 \over {\sqrt {1-\cos ^{2}\theta }}}}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {{\sqrt {1+\tan ^{2}\theta }} \over \tan \theta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </msqrt> </mrow> <mrow> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {{\sqrt {1+\tan ^{2}\theta }} \over \tan \theta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00ed1322bd995f59aad87a40a44d432cf43539a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.054ex; height:6.343ex;" alt="{\displaystyle {{\sqrt {1+\tan ^{2}\theta }} \over \tan \theta }}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \csc \theta \ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>csc</mi> <mo>⁡</mo> <mi>θ</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \csc \theta \ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf5a47fa43893b79d8efd400d3889773379fcca7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.039ex; height:2.176ex;" alt="{\displaystyle \csc \theta \ }" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sec \theta \over {\sqrt {\sec ^{2}\theta -1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sec</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>sec</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sec \theta \over {\sqrt {\sec ^{2}\theta -1}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bac554b31ff7ac7b42e6e9fc8b5dbbeaf2df2454" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:12.676ex; height:6.676ex;" alt="{\displaystyle {\sec \theta \over {\sqrt {\sec ^{2}\theta -1}}}}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {1+\cot ^{2}\theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>cot</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {1+\cot ^{2}\theta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7ca498631c3481c481baff2d908e3947f5fb2a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.958ex; height:3.509ex;" alt="{\displaystyle {\sqrt {1+\cot ^{2}\theta }}}" /></span> </td></tr> <tr> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sec \theta =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sec</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sec \theta =}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/542b3aa408c7cd3d04ab46f327ad1920db68335d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.912ex; height:2.176ex;" alt="{\displaystyle \sec \theta =}" /></span> </th> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {1 \over {\sqrt {1-\sin ^{2}\theta }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </msqrt> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {1 \over {\sqrt {1-\sin ^{2}\theta }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c8059fd205236cca59123984cb8de2ea5473065" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:12.55ex; height:6.509ex;" alt="{\displaystyle {1 \over {\sqrt {1-\sin ^{2}\theta }}}}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {1 \over \cos \theta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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 {1 \over \cos \theta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92cd0dfef3ee95e3fe6640596ecf89584b91b206" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:5.425ex; height:5.343ex;" alt="{\displaystyle {1 \over \cos \theta }}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {1+\tan ^{2}\theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {1+\tan ^{2}\theta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e456d733550c2645a3bd3c76296f3796db4398d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.218ex; height:3.509ex;" alt="{\displaystyle {\sqrt {1+\tan ^{2}\theta }}}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\csc \theta \over {\sqrt {\csc ^{2}\theta -1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>csc</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>csc</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\csc \theta \over {\sqrt {\csc ^{2}\theta -1}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/faa1b0c0d05e81be990e9ccf191ab315500db567" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:12.676ex; height:6.676ex;" alt="{\displaystyle {\csc \theta \over {\sqrt {\csc ^{2}\theta -1}}}}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sec \theta \ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sec</mi> <mo>⁡</mo> <mi>θ</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sec \theta \ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fe6e4b4234b1ca72888508c649f1da0fe80e432" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.039ex; height:2.176ex;" alt="{\displaystyle \sec \theta \ }" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {{\sqrt {1+\cot ^{2}\theta }} \over \cot \theta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>cot</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </msqrt> </mrow> <mrow> <mi>cot</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {{\sqrt {1+\cot ^{2}\theta }} \over \cot \theta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb0032fa47cc2a8f75deb7c00048624172a141e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.794ex; height:6.343ex;" alt="{\displaystyle {{\sqrt {1+\cot ^{2}\theta }} \over \cot \theta }}" /></span> </td></tr> <tr> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cot \theta =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cot</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cot \theta =}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/791414887be8a6e9b575d5915b886fdb98ffe0e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.03ex; height:2.176ex;" alt="{\displaystyle \cot \theta =}" /></span> </th> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {{\sqrt {1-\sin ^{2}\theta }} \over \sin \theta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </msqrt> </mrow> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {{\sqrt {1-\sin ^{2}\theta }} \over \sin \theta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ac0b8abdc9ec4959a35b0d61dd0ec0d4ba07d8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.55ex; height:6.343ex;" alt="{\displaystyle {{\sqrt {1-\sin ^{2}\theta }} \over \sin \theta }}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cos \theta \over {\sqrt {1-\cos ^{2}\theta }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </msqrt> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\cos \theta \over {\sqrt {1-\cos ^{2}\theta }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6411ae64dd942dd183d1f189e641d66ca3fbdd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:12.806ex; height:6.676ex;" alt="{\displaystyle {\cos \theta \over {\sqrt {1-\cos ^{2}\theta }}}}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {1 \over \tan \theta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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 {1 \over \tan \theta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb6fb3676c8c10e5be920b2bd44f58aea597f844" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:5.673ex; height:5.343ex;" alt="{\displaystyle {1 \over \tan \theta }}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\csc ^{2}\theta -1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>csc</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\csc ^{2}\theta -1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfed51fec74de73c365561977b6982a80716ac88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.839ex; height:3.509ex;" alt="{\displaystyle {\sqrt {\csc ^{2}\theta -1}}}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {1 \over {\sqrt {\sec ^{2}\theta -1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>sec</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {1 \over {\sqrt {\sec ^{2}\theta -1}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4d6ae413cb767e1a47ddd4189abef3ad5a5e6c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:12.676ex; height:6.509ex;" alt="{\displaystyle {1 \over {\sqrt {\sec ^{2}\theta -1}}}}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cot \theta \ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cot</mi> <mo>⁡</mo> <mi>θ</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cot \theta \ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/027634a29d1416d9261cfad75bd1eac287c72244" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.158ex; height:2.176ex;" alt="{\displaystyle \cot \theta \ }" /></span> </td></tr></tbody></table> <p>Altres relacions importants són les <i>fórmules de les funcions trigonomètriques de la suma i la diferència d'angles</i>, que donen el sinus i el cosinus de la suma o la diferència de dos angles en funció del sinus i del cosinus dels propis angles. Es poden obtenir per raonaments geomètrics, emprant arguments que es retrotreuen a l'època de <a href="/wiki/Claudi_Ptolemeu" title="Claudi Ptolemeu">Claudi Ptolemeu</a>. Aquestes fórmules són especialment importants perquè a partir d'elles es troba la <a href="/wiki/Derivaci%C3%B3_de_les_funcions_trigonom%C3%A8triques" title="Derivació de les funcions trigonomètriques">derivada de les funcions trigonomètriques</a> que és necessària per demostrar la <a href="/wiki/F%C3%B3rmula_d%27Euler" title="Fórmula d'Euler">fórmula d'Euler</a>. Un cop coneguda la fórmula d'Euler es pot emprar per memoritzar aquestes identitats a base de deduir-les ràpidament: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&\cos \left(x+y\right)+i\sin \left(x+y\right)=e^{i\left(x+y\right)}=e^{ix}e^{iy}\\&e^{ix}e^{iy}=\left[\cos \left(x\right)+i\sin \left(x\right)\right]\cdot \left[\cos \left(y\right)+i\sin \left(y\right)\right]=\\&=\left[\cos \left(x\right)\cos \left(y\right)-\sin \left(x\right)\sin \left(y\right)\right]+i\left[\cos \left(x\right)\sin \left(y\right)+\sin \left(x\right)\cos \left(y\right)\right]\\&\cos \left(x+y\right)=\cos \left(x\right)\cos \left(y\right)-\sin \left(x\right)\sin \left(y\right)\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd></mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>+</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>+</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>+</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mrow> </msup> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>x</mi> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>y</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>x</mi> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>y</mi> </mrow> </msup> <mo>=</mo> <mrow> <mo>[</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mo>⋅</mo> <mrow> <mo>[</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mo>=</mo> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>[</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mo>+</mo> <mi>i</mi> <mrow> <mo>[</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>+</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&\cos \left(x+y\right)+i\sin \left(x+y\right)=e^{i\left(x+y\right)}=e^{ix}e^{iy}\\&e^{ix}e^{iy}=\left[\cos \left(x\right)+i\sin \left(x\right)\right]\cdot \left[\cos \left(y\right)+i\sin \left(y\right)\right]=\\&=\left[\cos \left(x\right)\cos \left(y\right)-\sin \left(x\right)\sin \left(y\right)\right]+i\left[\cos \left(x\right)\sin \left(y\right)+\sin \left(x\right)\cos \left(y\right)\right]\\&\cos \left(x+y\right)=\cos \left(x\right)\cos \left(y\right)-\sin \left(x\right)\sin \left(y\right)\\\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8f658c68ac51dbda835a579f87fec6564e1705c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:65.978ex; height:12.843ex;" alt="{\displaystyle {\begin{aligned}&\cos \left(x+y\right)+i\sin \left(x+y\right)=e^{i\left(x+y\right)}=e^{ix}e^{iy}\\&e^{ix}e^{iy}=\left[\cos \left(x\right)+i\sin \left(x\right)\right]\cdot \left[\cos \left(y\right)+i\sin \left(y\right)\right]=\\&=\left[\cos \left(x\right)\cos \left(y\right)-\sin \left(x\right)\sin \left(y\right)\right]+i\left[\cos \left(x\right)\sin \left(y\right)+\sin \left(x\right)\cos \left(y\right)\right]\\&\cos \left(x+y\right)=\cos \left(x\right)\cos \left(y\right)-\sin \left(x\right)\sin \left(y\right)\\\end{aligned}}}" /></span></dd></dl> <p>Com que, perquè dos nombres complexos siguin iguals, han de ser iguals les seves parts reals i les seves parts imaginàries, resulten directament les identitats que es buscaven. Pel cas de la diferència només cal aplicar les mateixes fórmules i tenir en compte les simetries de les funcions sinus i cosinus en aplicar-les a angles negatius, el resultat són les quatre identitats: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \left(x+y\right)=\sin x\cos y+\cos x\sin y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>+</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \left(x+y\right)=\sin x\cos y+\cos x\sin y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e863739245ff9022e0766723a53d52f05fc06017" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.156ex; height:2.843ex;" alt="{\displaystyle \sin \left(x+y\right)=\sin x\cos y+\cos x\sin y}" /></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \left(x+y\right)=\cos x\cos y-\sin x\sin y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>+</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \left(x+y\right)=\cos x\cos y-\sin x\sin y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d6c1f8820258d3db350d332f76ccd8b12a3900c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.411ex; height:2.843ex;" alt="{\displaystyle \cos \left(x+y\right)=\cos x\cos y-\sin x\sin y}" /></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \left(x-y\right)=\sin x\cos y-\cos x\sin y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \left(x-y\right)=\sin x\cos y-\cos x\sin y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e881a8119150497d49cff66cd28e3bf03aa592d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.156ex; height:2.843ex;" alt="{\displaystyle \sin \left(x-y\right)=\sin x\cos y-\cos x\sin y}" /></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \left(x-y\right)=\cos x\cos y+\sin x\sin y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \left(x-y\right)=\cos x\cos y+\sin x\sin y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ef8ff180ff764621c697ccbce00d1f9042abf8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.411ex; height:2.843ex;" alt="{\displaystyle \cos \left(x-y\right)=\cos x\cos y+\sin x\sin y}" /></span></dd></dl> <p>Per a una demostració basada en raonaments geomètrics vegeu: <a href="/wiki/Demostraci%C3%B3_de_les_identitats_trigonom%C3%A8triques#Identitats_de_la_suma_d'angles" title="Demostració de les identitats trigonomètriques">Demostració de les identitats de la suma d'angles</a>. </p><p>Quan els dos angles són iguals, les fórmules de la suma es redueixen a identitats més senzilles conegudes amb el nom de <i>fórmules de l'angle doble</i>, o aplicant-les repetidament les de l'angle triple. A partir de les fórmules de l'angle doble es troben les fórmules d'angle meitat. </p> <table class="wikitable" style="background-color:#FFFFFF;"> <tbody><tr> <th colspan="4">Fórmules de l'angle doble<sup id="cite_ref-mathworld_double_angle_14-0" class="reference"><a href="#cite_note-mathworld_double_angle-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </th></tr> <tr> <td style="vertical-align:top"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\sin 2\theta &=2\sin \theta \cos \theta \ \\&={\frac {2\tan \theta }{1+\tan ^{2}\theta }}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mn>2</mn> <mi>θ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mtext> </mtext> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>tan</mi> <mo>⁡</mo> <mi>θ</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>θ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sin 2\theta &=2\sin \theta \cos \theta \ \\&={\frac {2\tan \theta }{1+\tan ^{2}\theta }}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0099e858d435fc05995057ef5638b128844c3f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:20.785ex; height:8.843ex;" alt="{\displaystyle {\begin{aligned}\sin 2\theta &=2\sin \theta \cos \theta \ \\&={\frac {2\tan \theta }{1+\tan ^{2}\theta }}\end{aligned}}}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\cos 2\theta &=\cos ^{2}\theta -\sin ^{2}\theta \\&=2\cos ^{2}\theta -1\\&=1-2\sin ^{2}\theta \\&={\frac {1-\tan ^{2}\theta }{1+\tan ^{2}\theta }}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mn>2</mn> <mi>θ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <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> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></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>θ</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>θ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\cos 2\theta &=\cos ^{2}\theta -\sin ^{2}\theta \\&=2\cos ^{2}\theta -1\\&=1-2\sin ^{2}\theta \\&={\frac {1-\tan ^{2}\theta }{1+\tan ^{2}\theta }}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0fcb55c3bbc5babe06acef179a7749c625a2976" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.338ex; width:23.472ex; height:15.843ex;" alt="{\displaystyle {\begin{aligned}\cos 2\theta &=\cos ^{2}\theta -\sin ^{2}\theta \\&=2\cos ^{2}\theta -1\\&=1-2\sin ^{2}\theta \\&={\frac {1-\tan ^{2}\theta }{1+\tan ^{2}\theta }}\end{aligned}}}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan 2\theta ={\frac {2\tan \theta }{1-\tan ^{2}\theta }}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mn>2</mn> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>tan</mi> <mo>⁡</mo> <mi>θ</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>θ</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan 2\theta ={\frac {2\tan \theta }{1-\tan ^{2}\theta }}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e3966cd4971d9fd0e4879cc8aee149dbc90eacd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.216ex; height:5.843ex;" alt="{\displaystyle \tan 2\theta ={\frac {2\tan \theta }{1-\tan ^{2}\theta }}\,}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cot 2\theta ={\frac {\cot \theta -\tan \theta }{2}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cot</mi> <mo>⁡</mo> <mn>2</mn> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>cot</mi> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cot 2\theta ={\frac {\cot \theta -\tan \theta }{2}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d94f7155adba7b4e96de3f0d9e6517ac7f9a025e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.316ex; height:5.343ex;" alt="{\displaystyle \cot 2\theta ={\frac {\cot \theta -\tan \theta }{2}}\,}" /></span> </td></tr> <tr> <th colspan="4">Fórmules de l'angle triple<sup id="cite_ref-mathworld_multiple_angle_15-0" class="reference"><a href="#cite_note-mathworld_multiple_angle-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </th></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin 3\theta =3\sin \theta -4\sin ^{3}\theta \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mn>3</mn> <mi>θ</mi> <mo>=</mo> <mn>3</mn> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <mn>4</mn> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin 3\theta =3\sin \theta -4\sin ^{3}\theta \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe2b8785f6f7d9ad1c0c0d3efe15a26f1700ad80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:24.641ex; height:2.843ex;" alt="{\displaystyle \sin 3\theta =3\sin \theta -4\sin ^{3}\theta \,}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos 3\theta =4\cos ^{3}\theta -3\cos \theta \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mn>3</mn> <mi>θ</mi> <mo>=</mo> <mn>4</mn> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <mn>3</mn> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos 3\theta =4\cos ^{3}\theta -3\cos \theta \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e005910f2963beb3ad6ee3831c244398b0c7d24f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:25.408ex; height:2.843ex;" alt="{\displaystyle \cos 3\theta =4\cos ^{3}\theta -3\cos \theta \,}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan 3\theta ={\frac {3\tan \theta -\tan ^{3}\theta }{1-3\tan ^{2}\theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mn>3</mn> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mn>3</mn> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan 3\theta ={\frac {3\tan \theta -\tan ^{3}\theta }{1-3\tan ^{2}\theta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da5608e031f3cafc680c80f37dce9899f869de81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.053ex; height:6.176ex;" alt="{\displaystyle \tan 3\theta ={\frac {3\tan \theta -\tan ^{3}\theta }{1-3\tan ^{2}\theta }}}" /></span> </td> <td> </td></tr> <tr> <th colspan="4">Fórmules de l'angle meitat<sup id="cite_ref-mathworld_half_angle_16-0" class="reference"><a href="#cite_note-mathworld_half_angle-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </th></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin {\tfrac {\theta }{2}}=\pm \,{\sqrt {\frac {1-\cos \theta }{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mo>±</mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mn>2</mn> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin {\tfrac {\theta }{2}}=\pm \,{\sqrt {\frac {1-\cos \theta }{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0689c3d823dd315af410117dc4434ab0f84c009" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.946ex; height:6.343ex;" alt="{\displaystyle \sin {\tfrac {\theta }{2}}=\pm \,{\sqrt {\frac {1-\cos \theta }{2}}}}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos {\tfrac {\theta }{2}}=\pm \,{\sqrt {\frac {1+\cos \theta }{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mo>±</mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mn>2</mn> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos {\tfrac {\theta }{2}}=\pm \,{\sqrt {\frac {1+\cos \theta }{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc837e1a78dcc3af91d878d37f02bbb410660532" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.201ex; height:6.343ex;" alt="{\displaystyle \cos {\tfrac {\theta }{2}}=\pm \,{\sqrt {\frac {1+\cos \theta }{2}}}}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\tan {\tfrac {\theta }{2}}&=\csc \theta -\cot \theta \\&=\pm \,{\sqrt {1-\cos \theta \over 1+\cos \theta }}\\&={\frac {\sin \theta }{1+\cos \theta }}\\&={\frac {1-\cos \theta }{\sin \theta }}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>csc</mi> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <mi>cot</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mo>±</mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\tan {\tfrac {\theta }{2}}&=\csc \theta -\cot \theta \\&=\pm \,{\sqrt {1-\cos \theta \over 1+\cos \theta }}\\&={\frac {\sin \theta }{1+\cos \theta }}\\&={\frac {1-\cos \theta }{\sin \theta }}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e92f1cc337240182b15898d6c0b2be9475c86741" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.838ex; width:23.201ex; height:22.843ex;" alt="{\displaystyle {\begin{aligned}\tan {\tfrac {\theta }{2}}&=\csc \theta -\cot \theta \\&=\pm \,{\sqrt {1-\cos \theta \over 1+\cos \theta }}\\&={\frac {\sin \theta }{1+\cos \theta }}\\&={\frac {1-\cos \theta }{\sin \theta }}\end{aligned}}}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cot {\tfrac {\theta }{2}}=\csc \theta +\cot \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cot</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mi>csc</mi> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <mi>cot</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cot {\tfrac {\theta }{2}}=\csc \theta +\cot \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53df7db345db00d4f160d50283a532a6a23a955d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:20.119ex; height:3.676ex;" alt="{\displaystyle \cot {\tfrac {\theta }{2}}=\csc \theta +\cot \theta }" /></span> </td></tr></tbody></table> <p>Aquestes identitats es poden fer servir per demostrar <a href="/wiki/Llista_d%27identitats_trigonom%C3%A8triques#Fórmules_de_Simpson" title="Llista d'identitats trigonomètriques">fórmules de Simpson</a> que permeten passar de productes a sumes i de sumes a productes. Antigament es feien servir en comptes dels <a href="/wiki/Logaritme" title="Logaritme">logaritmes</a> per a transformar el producte de dos nombres en una suma i augmentar la velocitat dels càlculs. </p> <div class="mw-heading mw-heading3"><h3 id="Identitats_trigonomètriques_en_càlcul_infinitesimal"><span id="Identitats_trigonom.C3.A8triques_en_c.C3.A0lcul_infinitesimal"></span>Identitats trigonomètriques en càlcul infinitesimal</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funci%C3%B3_trigonom%C3%A8trica&action=edit&section=12" title="Modifica la secció: Identitats trigonomètriques en càlcul infinitesimal"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Per a les identitats que sorgeixen en derivar i integrar les funcions trigonomètriques, vegeu: <a href="/wiki/Derivaci%C3%B3_de_les_funcions_trigonom%C3%A8triques" title="Derivació de les funcions trigonomètriques">Derivació de les funcions trigonomètriques</a>, <a href="/wiki/Taula_d%27integrals#Funcions_trigonomètriques" title="Taula d'integrals">taula d'integrals de funcions trigonomètriques</a>, <a href="/wiki/Llista_d%27integrals_de_funcions_trigonom%C3%A8triques" class="mw-redirect" title="Llista d'integrals de funcions trigonomètriques">Llista d'integrals de funcions trigonomètriques</a> i <a href="/wiki/Llista_d%27integrals_d%27inverses_de_funcions_trigonom%C3%A8triques" title="Llista d'integrals d'inverses de funcions trigonomètriques">Llista d'integrals d'inverses de funcions trigonomètriques</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Càlcul_de_les_funcions_trigonomètriques"><span id="C.C3.A0lcul_de_les_funcions_trigonom.C3.A8triques"></span>Càlcul de les funcions trigonomètriques</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funci%C3%B3_trigonom%C3%A8trica&action=edit&section=13" title="Modifica la secció: Càlcul de les funcions trigonomètriques"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>El càlcul de les funcions trigonomètriques és una qüestió complicada que avui en dia pot ser evitada per la majoria de la gent a causa de l'ampla disponibilitat dels <a href="/wiki/Ordinador" title="Ordinador">ordinadors</a> i de les <a href="/w/index.php?title=Calculadora_cient%C3%ADfica&action=edit&redlink=1" class="new" title="Calculadora científica (encara no existeix)">calculadores científiques</a> que ofereixen càlcul preprogramat de funcions trigonomètriques per a qualsevol angle. Ara bé, en aquesta secció es donen més detalls sobre el seu càlcul en tres contexts importants: l'ús històric de les taules trigonomètriques, les tècniques modernes que es fan servir per programar els ordinadors per a calcular-les, i en quant angles "importants" per als quals els valors exactes simplement es poden trobar de forma senzilla. </p><p>En tot el que segueix, n'hi ha prou de considerar un recorregut petit d'angles, de 0 a π/2, atès que els altres angles es poden reduir a aquests gràcies a les simetries i a la periodicitat de les funcions trigonomètriques. </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r30997230" /><div role="note" class="hatnote navigation-not-searchable">Article principal: <a href="/wiki/Construcci%C3%B3_de_les_taules_trigonom%C3%A8triques" title="Construcció de les taules trigonomètriques">Construcció de les taules trigonomètriques</a></div> <p>Abans de l'aparició dels ordinadors, la gent avaluava les funcions trigonomètriques per <a href="/wiki/Interpolaci%C3%B3" title="Interpolació">interpolació</a> a partir de les taules amb els valors calculats amb molts <a href="/wiki/D%C3%ADgit_significatiu" title="Dígit significatiu">dígits significatius</a>. Aquestes taules han estat construïdes des de tant antic com la descripció de les mateixes funcions trigonomètriques (<i>vegeu <a href="/wiki/Hist%C3%B2ria_de_les_funcions_trigonom%C3%A8triques" title="Història de les funcions trigonomètriques">història de les funcions trigonomètriques</a></i>). Normalment es generaven a base d'aplicar repetidament les fórmules de l'angle meitat i de la suma d'angles començant a partir d'un valor conegut, tal com per exemple sin(π/6)=1/2. </p><p>Els ordinadors moderns fan servir una gran varietat de tècniques.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> Un mètode habitual, especialment en processadors amb unitats de <a href="/wiki/Coma_flotant" title="Coma flotant">coma flotant</a>, és combinar una <a href="/wiki/Teoria_de_l%27aproximaci%C3%B3" title="Teoria de l'aproximació">aproximació</a> per un <a href="/wiki/Polinomi" title="Polinomi">polinomi</a> o una <a href="/wiki/Funci%C3%B3_racional" title="Funció racional">funció racional</a> (com ara l'<a href="/w/index.php?title=Aproximaci%C3%B3_de_Chebyshev&action=edit&redlink=1" class="new" title="Aproximació de Chebyshev (encara no existeix)">aproximació de Chebyshev</a>, l'aproximació uniforme òptima, i l'<a href="/wiki/Aproximant_de_Pad%C3%A9" title="Aproximant de Padé">aproximació de Padé</a>, i per a precisions més altes o precisions variables, la <a href="/wiki/S%C3%A8rie_de_Taylor" title="Sèrie de Taylor">sèrie de Taylor</a> i la <a href="/wiki/S%C3%A8rie_de_Laurent" title="Sèrie de Laurent">sèrie de Laurent</a>) amb una reducció del recorregut i una cerca en taula — primer busquen en una petita taula l'angle més proper, i llavors utilitzen el polinomi per a calcular la correcció.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> En dispositius més senzills als que els manquen unitats de maquinari per a fer multiplicacions, hi ha un algorisme anomenat <a href="/wiki/CORDIC" title="CORDIC">CORDIC</a> (amb les seves tècniques relacionades) que és més eficient perquè només fa servir sumes i l'<a href="/w/index.php?title=Operaci%C3%B3_despla%C3%A7ament&action=edit&redlink=1" class="new" title="Operació desplaçament (encara no existeix)">operació desplaçament</a>. Tots aquest mètodes es poden implementar en circuits especialitzats per motius d'eficiència. </p><p>Per a càlculs de molt alta precisió, quan la convergència de les sèries esdevé massa lenta, les funcions trigonomètriques es poden aproximar amb la <a href="/wiki/Mitjana_aritm%C3%A8tico-geom%C3%A8trica" title="Mitjana aritmètico-geomètrica">mitjana aritmètico-geomètrica</a>, que ella mateixa aproxima la funció trigonomètrica per la <a href="/wiki/Integral_el%C2%B7l%C3%ADptica" title="Integral el·líptica">integral el·líptica</a> (complexa).<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r30997230" /><div role="note" class="hatnote navigation-not-searchable">Article principal: <a href="/wiki/Constants_trigonom%C3%A8triques_exactes" title="Constants trigonomètriques exactes">Constants trigonomètriques exactes</a></div> <p>Finalment, per alguns angles senzills, els valors es poden calcular fàcilment a mà emprant el <a href="/wiki/Teorema_de_Pit%C3%A0gores" title="Teorema de Pitàgores">teorema de Pitàgores</a>, com en els exemples que es presentaran a continuació. De fet, el sinus, el cosinus i la tangent de qualsevol múltiple enter de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi /60}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>60</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi /60}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1794f2729816ff56ca0a009382a4b191f93455c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.819ex; height:2.843ex;" alt="{\displaystyle \pi /60}" /></span> <a href="/wiki/Radiant_(angle)" class="mw-redirect" title="Radiant (angle)">radiants</a> (3°) es pot calcular <a href="/wiki/Constants_trigonom%C3%A8triques_exactes" title="Constants trigonomètriques exactes">exactament a mà</a>. </p><p>Es considera el triangle rectangle que té iguals els altres dos angles diferents del recte, per tant tots dos valen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi /4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi /4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1707aa0fec2c8ef008b9e30b6045fbf95dab9e20" 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 /4}" /></span> radiants (45°). Llavors la longitud dels costats <i>b</i> i <i>a</i> són iguals; es pot definir <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=b=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>b</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=b=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc0bea7277de6846c7b9f35ba678e63d4e8ce100" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.587ex; height:2.176ex;" alt="{\displaystyle a=b=1}" /></span>. Els valors del sinus, del cosinus i de la tangent de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi /4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi /4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1707aa0fec2c8ef008b9e30b6045fbf95dab9e20" 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 /4}" /></span> radiants (45°) llavors es poden trobar emprant el teorema de Pitàgores: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c={\sqrt {a^{2}+b^{2}}}={\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <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> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c={\sqrt {a^{2}+b^{2}}}={\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a718269f0421243ef9c15e73491e8186f53712dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.802ex; height:3.509ex;" alt="{\displaystyle c={\sqrt {a^{2}+b^{2}}}={\sqrt {2}}}" /></span>.</dd></dl> <p>Per tant: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \left(\pi /4\right)=\sin \left(45^{\circ }\right)=\cos \left(\pi /4\right)=\cos \left(45^{\circ }\right)={1 \over {\sqrt {2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msup> <mn>45</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msup> <mn>45</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \left(\pi /4\right)=\sin \left(45^{\circ }\right)=\cos \left(\pi /4\right)=\cos \left(45^{\circ }\right)={1 \over {\sqrt {2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c8ff841539215b461798375c6fcc2b272afa09e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:49.571ex; height:6.176ex;" alt="{\displaystyle \sin \left(\pi /4\right)=\sin \left(45^{\circ }\right)=\cos \left(\pi /4\right)=\cos \left(45^{\circ }\right)={1 \over {\sqrt {2}}}}" /></span>,</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan \left(\pi /4\right)=\tan \left(45^{\circ }\right)={{\sin \left(\pi /4\right)} \over {\cos \left(\pi /4\right)}}={1 \over {\sqrt {2}}}\cdot {{\sqrt {2}} \over 1}={{\sqrt {2}} \over {\sqrt {2}}}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>tan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msup> <mn>45</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mn>1</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan \left(\pi /4\right)=\tan \left(45^{\circ }\right)={{\sin \left(\pi /4\right)} \over {\cos \left(\pi /4\right)}}={1 \over {\sqrt {2}}}\cdot {{\sqrt {2}} \over 1}={{\sqrt {2}} \over {\sqrt {2}}}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/397a24348e111ec94cc73f13e865ad6b5be47b91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:56.925ex; height:6.843ex;" alt="{\displaystyle \tan \left(\pi /4\right)=\tan \left(45^{\circ }\right)={{\sin \left(\pi /4\right)} \over {\cos \left(\pi /4\right)}}={1 \over {\sqrt {2}}}\cdot {{\sqrt {2}} \over 1}={{\sqrt {2}} \over {\sqrt {2}}}=1}" /></span>.</dd></dl> <p>Per a determinar les funcions trigonomètriques dels angles de π/3 radiants (60 graus) i π/6 radiants (30 graus), es comença amb una triangle equilàter de longitud 1 de costat. Tots els seus angles són de π/3 radiants (60 graus). Partint-lo per la meitat, s'obté un triangle rectangle amb angles de π/6 radiants (30 graus) i π/3 radiants (60 graus). Per aquest triangle, el costat més petit té una longitud de 1/2, el costat següent té una longitud de (√3)/2 i la hipotenusa 1. Això dona: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \left(\pi /6\right)=\sin \left(30^{\circ }\right)=\cos \left(\pi /3\right)=\cos \left(60^{\circ }\right)={1 \over 2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>6</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msup> <mn>30</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msup> <mn>60</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>)</mo> </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 \left(\pi /6\right)=\sin \left(30^{\circ }\right)=\cos \left(\pi /3\right)=\cos \left(60^{\circ }\right)={1 \over 2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9dc06e73c52a41de1157b0bc57bfc516e61d987c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:47.635ex; height:5.176ex;" alt="{\displaystyle \sin \left(\pi /6\right)=\sin \left(30^{\circ }\right)=\cos \left(\pi /3\right)=\cos \left(60^{\circ }\right)={1 \over 2}}" /></span>,</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \left(\pi /6\right)=\cos \left(30^{\circ }\right)=\sin \left(\pi /3\right)=\sin \left(60^{\circ }\right)={{\sqrt {3}} \over 2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>6</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msup> <mn>30</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msup> <mn>60</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \left(\pi /6\right)=\cos \left(30^{\circ }\right)=\sin \left(\pi /3\right)=\sin \left(60^{\circ }\right)={{\sqrt {3}} \over 2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4feff321b9340fbc4997404b48a0451c1af1251" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:49.571ex; height:5.843ex;" alt="{\displaystyle \cos \left(\pi /6\right)=\cos \left(30^{\circ }\right)=\sin \left(\pi /3\right)=\sin \left(60^{\circ }\right)={{\sqrt {3}} \over 2}}" /></span>,</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan \left(\pi /6\right)=\tan \left(30^{\circ }\right)=\cot \left(\pi /3\right)=\cot \left(60^{\circ }\right)={1 \over {\sqrt {3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>6</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>tan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msup> <mn>30</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mi>cot</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>cot</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msup> <mn>60</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan \left(\pi /6\right)=\tan \left(30^{\circ }\right)=\cot \left(\pi /3\right)=\cot \left(60^{\circ }\right)={1 \over {\sqrt {3}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f555d1c8de219e70891b9d02a7b6bb85dc1877d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:50.556ex; height:6.176ex;" alt="{\displaystyle \tan \left(\pi /6\right)=\tan \left(30^{\circ }\right)=\cot \left(\pi /3\right)=\cot \left(60^{\circ }\right)={1 \over {\sqrt {3}}}}" /></span>.</dd></dl> <p>A la següent taula es resumeixen aquests resultats: </p> <table border="1" cellpadding="4" cellspacing="3" style="page-break-before: always"> <tbody><tr> <th>Funció </th> <th>0° <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" /></span> </p> </th> <th>30° <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\pi }{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></span><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}}}" /></span> </p> </th> <th>45° <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\pi }{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></span><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}}}" /></span> </p> </th> <th>60° <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\pi }{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></span><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}}}" /></span> </p> </th> <th>90° <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\pi }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\pi }{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98f98bef5d4981ff6e2aa827d4699e347fb30db2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.168ex; height:4.676ex;" alt="{\displaystyle {\frac {\pi }{2}}}" /></span> </p> </th></tr> <tr> <td>Sinus </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\frac {0}{4}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn>0</mn> <mn>4</mn> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\frac {0}{4}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52e8b075d4fa6d21ec4cfc0fa67e3b67f6ea611c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:4.322ex; height:6.176ex;" alt="{\displaystyle {\sqrt {\frac {0}{4}}}}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\frac {1}{4}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\frac {1}{4}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/022e4f7fa4cb01562cde4f3f9a8a1d600b287192" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:4.322ex; height:6.176ex;" alt="{\displaystyle {\sqrt {\frac {1}{4}}}}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\frac {2}{4}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn>2</mn> <mn>4</mn> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\frac {2}{4}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4099e51ad9f17562acd076f1420ed644d907229" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:4.322ex; height:6.176ex;" alt="{\displaystyle {\sqrt {\frac {2}{4}}}}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\frac {3}{4}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\frac {3}{4}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6225f5ddb8d6b3e7efc19cc9247e901703d8b31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:4.322ex; height:6.176ex;" alt="{\displaystyle {\sqrt {\frac {3}{4}}}}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\frac {4}{4}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn>4</mn> <mn>4</mn> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\frac {4}{4}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f18e0e9a9ed41604ac49bb08357e5aa7830929e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:4.322ex; height:6.176ex;" alt="{\displaystyle {\sqrt {\frac {4}{4}}}}" /></span> </td></tr> <tr> <td>Cosinus </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\frac {4}{4}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn>4</mn> <mn>4</mn> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\frac {4}{4}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f18e0e9a9ed41604ac49bb08357e5aa7830929e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:4.322ex; height:6.176ex;" alt="{\displaystyle {\sqrt {\frac {4}{4}}}}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\frac {3}{4}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\frac {3}{4}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6225f5ddb8d6b3e7efc19cc9247e901703d8b31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:4.322ex; height:6.176ex;" alt="{\displaystyle {\sqrt {\frac {3}{4}}}}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\frac {2}{4}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn>2</mn> <mn>4</mn> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\frac {2}{4}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4099e51ad9f17562acd076f1420ed644d907229" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:4.322ex; height:6.176ex;" alt="{\displaystyle {\sqrt {\frac {2}{4}}}}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\frac {1}{4}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\frac {1}{4}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/022e4f7fa4cb01562cde4f3f9a8a1d600b287192" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:4.322ex; height:6.176ex;" alt="{\displaystyle {\sqrt {\frac {1}{4}}}}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\frac {0}{4}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn>0</mn> <mn>4</mn> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\frac {0}{4}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52e8b075d4fa6d21ec4cfc0fa67e3b67f6ea611c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:4.322ex; height:6.176ex;" alt="{\displaystyle {\sqrt {\frac {0}{4}}}}" /></span> </td></tr> <tr> <td>Tangent </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\frac {0}{4}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn>0</mn> <mn>4</mn> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\frac {0}{4}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52e8b075d4fa6d21ec4cfc0fa67e3b67f6ea611c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:4.322ex; height:6.176ex;" alt="{\displaystyle {\sqrt {\frac {0}{4}}}}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\frac {1}{3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\frac {1}{3}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd95e1e4e775af4a5989069c2aa8360bae6a150b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:4.322ex; height:6.176ex;" alt="{\displaystyle {\sqrt {\frac {1}{3}}}}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\frac {2}{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn>2</mn> <mn>2</mn> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\frac {2}{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6c80a4549766496a9da3b4e8720890f030e8bd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:4.322ex; height:6.176ex;" alt="{\displaystyle {\sqrt {\frac {2}{2}}}}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\frac {3}{1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn>3</mn> <mn>1</mn> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\frac {3}{1}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25ee9427a7496c827abdb99ae62f303743a9b06b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:4.322ex; height:6.176ex;" alt="{\displaystyle {\sqrt {\frac {3}{1}}}}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\frac {4}{0}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn>4</mn> <mn>0</mn> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\frac {4}{0}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c942398c8b8aea22c681177a76927b8af05f661c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:4.322ex; height:6.176ex;" alt="{\displaystyle {\sqrt {\frac {4}{0}}}}" /></span> </td></tr></tbody></table> <p>Les expressions s'han posat sense simplificar perquè així és més fàcil memoritzar-les. </p> <div class="mw-heading mw-heading2"><h2 id="Funcions_inverses">Funcions inverses</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funci%C3%B3_trigonom%C3%A8trica&action=edit&section=14" title="Modifica la secció: Funcions inverses"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r30997230" /><div role="note" class="hatnote navigation-not-searchable">Article principal: <a href="/wiki/Inverses_de_les_funcions_trigonom%C3%A8triques" title="Inverses de les funcions trigonomètriques">Inverses de les funcions trigonomètriques</a></div> <p>Les funcions trigonomètriques són periòdiques i, per tant, no <a href="/wiki/Funci%C3%B3_injectiva" title="Funció injectiva">injectives</a>, així, estrictament parlant, no tenen <a href="/wiki/Funci%C3%B3_inversa" title="Funció inversa">funció inversa</a>. Per a definir una funció inversa cal restringir el domini de forma que les funcions trigonomètriques siguin <a href="/wiki/Funci%C3%B3_bijectiva" title="Funció bijectiva">bijectives</a>. En el que segueix, les funcions de l'esquerra són <i>definides</i> per l'equació de la dreta; no són identitats demostrades. Les inverses principals, es defineixen normalment com: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}{\mbox{per}}&-{\frac {\pi }{2}}\leq y\leq {\frac {\pi }{2}},&y=\arcsin(x)&{\mbox{si}}&x=\sin(y)\\\\{\mbox{per}}&0\leq y\leq \pi ,&y=\arccos(x)&{\mbox{si}}&x=\cos(y)\\\\{\mbox{per}}&-{\frac {\pi }{2}}<y<{\frac {\pi }{2}},&y=\arctan(x)&{\mbox{si}}&x=\tan(y)\\\\{\mbox{per}}&-{\frac {\pi }{2}}\leq y\leq {\frac {\pi }{2}},y\neq 0,&y=\operatorname {arccsc}(x)&{\mbox{si}}&x=\csc(y)\\\\{\mbox{per}}&0\leq y\leq \pi ,y\neq {\frac {\pi }{2}},&y=\operatorname {arcsec}(x)&{\mbox{si}}&x=\sec(y)\\\\{\mbox{per}}&0<y<\pi ,&y=\operatorname {arccot}(x)&{\mbox{si}}&x=\cot(y)\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>per</mtext> </mstyle> </mrow> </mtd> <mtd> <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> </mtd> <mtd> <mi>y</mi> <mo>=</mo> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>si</mtext> </mstyle> </mrow> </mtd> <mtd> <mi>x</mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd></mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>per</mtext> </mstyle> </mrow> </mtd> <mtd> <mn>0</mn> <mo>≤</mo> <mi>y</mi> <mo>≤</mo> <mi>π</mi> <mo>,</mo> </mtd> <mtd> <mi>y</mi> <mo>=</mo> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>si</mtext> </mstyle> </mrow> </mtd> <mtd> <mi>x</mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd></mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>per</mtext> </mstyle> </mrow> </mtd> <mtd> <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> </mtd> <mtd> <mi>y</mi> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>si</mtext> </mstyle> </mrow> </mtd> <mtd> <mi>x</mi> <mo>=</mo> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd></mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>per</mtext> </mstyle> </mrow> </mtd> <mtd> <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> <mi>y</mi> <mo>≠</mo> <mn>0</mn> <mo>,</mo> </mtd> <mtd> <mi>y</mi> <mo>=</mo> <mi>arccsc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>si</mtext> </mstyle> </mrow> </mtd> <mtd> <mi>x</mi> <mo>=</mo> <mi>csc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd></mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>per</mtext> </mstyle> </mrow> </mtd> <mtd> <mn>0</mn> <mo>≤</mo> <mi>y</mi> <mo>≤</mo> <mi>π</mi> <mo>,</mo> <mi>y</mi> <mo>≠</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> </mtd> <mtd> <mi>y</mi> <mo>=</mo> <mi>arcsec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>si</mtext> </mstyle> </mrow> </mtd> <mtd> <mi>x</mi> <mo>=</mo> <mi>sec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd></mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>per</mtext> </mstyle> </mrow> </mtd> <mtd> <mn>0</mn> <mo><</mo> <mi>y</mi> <mo><</mo> <mi>π</mi> <mo>,</mo> </mtd> <mtd> <mi>y</mi> <mo>=</mo> <mi>arccot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>si</mtext> </mstyle> </mrow> </mtd> <mtd> <mi>x</mi> <mo>=</mo> <mi>cot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}{\mbox{per}}&-{\frac {\pi }{2}}\leq y\leq {\frac {\pi }{2}},&y=\arcsin(x)&{\mbox{si}}&x=\sin(y)\\\\{\mbox{per}}&0\leq y\leq \pi ,&y=\arccos(x)&{\mbox{si}}&x=\cos(y)\\\\{\mbox{per}}&-{\frac {\pi }{2}}<y<{\frac {\pi }{2}},&y=\arctan(x)&{\mbox{si}}&x=\tan(y)\\\\{\mbox{per}}&-{\frac {\pi }{2}}\leq y\leq {\frac {\pi }{2}},y\neq 0,&y=\operatorname {arccsc}(x)&{\mbox{si}}&x=\csc(y)\\\\{\mbox{per}}&0\leq y\leq \pi ,y\neq {\frac {\pi }{2}},&y=\operatorname {arcsec}(x)&{\mbox{si}}&x=\sec(y)\\\\{\mbox{per}}&0<y<\pi ,&y=\operatorname {arccot}(x)&{\mbox{si}}&x=\cot(y)\end{matrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a4b8dcbf9c5e21112a135564fbaa6d3b8144614" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -18.171ex; width:59.267ex; height:37.509ex;" alt="{\displaystyle {\begin{matrix}{\mbox{per}}&-{\frac {\pi }{2}}\leq y\leq {\frac {\pi }{2}},&y=\arcsin(x)&{\mbox{si}}&x=\sin(y)\\\\{\mbox{per}}&0\leq y\leq \pi ,&y=\arccos(x)&{\mbox{si}}&x=\cos(y)\\\\{\mbox{per}}&-{\frac {\pi }{2}}<y<{\frac {\pi }{2}},&y=\arctan(x)&{\mbox{si}}&x=\tan(y)\\\\{\mbox{per}}&-{\frac {\pi }{2}}\leq y\leq {\frac {\pi }{2}},y\neq 0,&y=\operatorname {arccsc}(x)&{\mbox{si}}&x=\csc(y)\\\\{\mbox{per}}&0\leq y\leq \pi ,y\neq {\frac {\pi }{2}},&y=\operatorname {arcsec}(x)&{\mbox{si}}&x=\sec(y)\\\\{\mbox{per}}&0<y<\pi ,&y=\operatorname {arccot}(x)&{\mbox{si}}&x=\cot(y)\end{matrix}}}" /></span></dd></dl> <p>Per a les inverses de les funcions trigonomètriques, sovint es fan servir les notacions sin<sup>−1</sup> i cos<sup>−1</sup>en comptes d'arcsin i arccos, etc. Quan es fa servir aquesta notació, les inverses de les funcions es podrien confondre amb les inverses del valor de les funcions respecte de la multiplicació. La notació emprant el prefix "arc-" evita aquesta confusió. </p><p>De la mateixa manera que el sinus i el cosinus, les inverses de les funcions trigonomètriques també es poden definir en termes de sèries infinites. Per exemple, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arcsin z=z+\left({\frac {1}{2}}\right){\frac {z^{3}}{3}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {z^{5}}{5}}+\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right){\frac {z^{7}}{7}}+\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arcsin</mi> <mo>⁡</mo> <mi>z</mi> <mo>=</mo> <mi>z</mi> <mo>+</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>⋅</mo> <mn>3</mn> </mrow> <mrow> <mn>2</mn> <mo>⋅</mo> <mn>4</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mn>5</mn> </mfrac> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>⋅</mo> <mn>3</mn> <mo>⋅</mo> <mn>5</mn> </mrow> <mrow> <mn>2</mn> <mo>⋅</mo> <mn>4</mn> <mo>⋅</mo> <mn>6</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mn>7</mn> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arcsin z=z+\left({\frac {1}{2}}\right){\frac {z^{3}}{3}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {z^{5}}{5}}+\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right){\frac {z^{7}}{7}}+\cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47a1f63bb3820277533af3eb69a5e7b399350e05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:60.596ex; height:6.343ex;" alt="{\displaystyle \arcsin z=z+\left({\frac {1}{2}}\right){\frac {z^{3}}{3}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {z^{5}}{5}}+\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right){\frac {z^{7}}{7}}+\cdots }" /></span></dd></dl> <p>Aquestes funcions també es poden definir com les primitives d'altres funcions. Per exemple, l'arcsinus, es pot escriure com la següent integral: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arcsin \left(x\right)=\int _{0}^{x}{\frac {1}{\sqrt {1-z^{2}}}}\,\mathrm {d} z,\quad |x|<1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arcsin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>z</mi> <mo>,</mo> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arcsin \left(x\right)=\int _{0}^{x}{\frac {1}{\sqrt {1-z^{2}}}}\,\mathrm {d} z,\quad |x|<1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961b7018b84fd4e4b4b22c46c10ab094b5974694" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:38.539ex; height:6.676ex;" alt="{\displaystyle \arcsin \left(x\right)=\int _{0}^{x}{\frac {1}{\sqrt {1-z^{2}}}}\,\mathrm {d} z,\quad |x|<1}" /></span></dd></dl> <p>Es poden trobar fórmules anàlogues a aquestes a l'article <i><a href="/wiki/Inverses_de_les_funcions_trigonom%C3%A8triques" title="Inverses de les funcions trigonomètriques">Inverses de les funcions trigonomètriques</a></i>. </p><p>Emprant el <a href="/wiki/Logaritme#Nombres_complexos" title="Logaritme">logaritme complex</a>, es poden generalitzar totes aquestes funcions a arguments complexos: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arcsin(z)=-i\log \left(iz+{\sqrt {1-z^{2}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>i</mi> <mi>log</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mi>z</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arcsin(z)=-i\log \left(iz+{\sqrt {1-z^{2}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eafa04feff4aca837520736109eb2556d8e2081" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:33.905ex; height:4.843ex;" alt="{\displaystyle \arcsin(z)=-i\log \left(iz+{\sqrt {1-z^{2}}}\right)}" /></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arccos(z)=-i\log \left(z+{\sqrt {z^{2}-1}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>i</mi> <mi>log</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>z</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arccos(z)=-i\log \left(z+{\sqrt {z^{2}-1}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/296ba5459c3ce73b1274265cd74965c10f307e2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:33.358ex; height:4.843ex;" alt="{\displaystyle \arccos(z)=-i\log \left(z+{\sqrt {z^{2}-1}}\right)}" /></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arctan(z)={\frac {i}{2}}\log \left({\frac {1-iz}{1+iz}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mn>2</mn> </mfrac> </mrow> <mi>log</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>i</mi> <mi>z</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>i</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arctan(z)={\frac {i}{2}}\log \left({\frac {1-iz}{1+iz}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3c8108bf087ab975301ed67e92e656fa98d786e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.97ex; height:6.176ex;" alt="{\displaystyle \arctan(z)={\frac {i}{2}}\log \left({\frac {1-iz}{1+iz}}\right)}" /></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Propietats_i_aplicacions">Propietats i aplicacions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funci%C3%B3_trigonom%C3%A8trica&action=edit&section=15" title="Modifica la secció: Propietats i aplicacions"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fitxer:Triangulaci%C3%B3_dels_Paisos_Catalans.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Triangulaci%C3%B3_dels_Paisos_Catalans.jpg/220px-Triangulaci%C3%B3_dels_Paisos_Catalans.jpg" decoding="async" width="220" height="250" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Triangulaci%C3%B3_dels_Paisos_Catalans.jpg/330px-Triangulaci%C3%B3_dels_Paisos_Catalans.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/63/Triangulaci%C3%B3_dels_Paisos_Catalans.jpg/440px-Triangulaci%C3%B3_dels_Paisos_Catalans.jpg 2x" data-file-width="442" data-file-height="502" /></a><figcaption>Triangulació dels <a href="/wiki/Pa%C3%AFsos_Catalans" title="Països Catalans">Països Catalans</a>, el triangle base està a <a href="/wiki/Salses" title="Salses">Salses</a> on es mesura una distància amb alta precisió, llavors pel mètode de la <a href="/wiki/Triangulaci%C3%B3" title="Triangulació">triangulació</a> es troben totes les altres distàncies mesurant únicament els angles. La línia que va de nord a sud correspon el <a href="/wiki/Meridi%C3%A0" title="Meridià">meridià</a> de <a href="/wiki/Par%C3%ADs" title="París">París</a> basant-se en la longitud del qual es va definir el <a href="/wiki/Metre" title="Metre">metre</a>.</figcaption></figure> <p>Les funcions trigonomètriques, tal com suggereix el seu nom, són d'importància crucial en <a href="/wiki/Trigonometria" title="Trigonometria">trigonometria</a> i en la <a href="/wiki/Resoluci%C3%B3_de_triangles" title="Resolució de triangles">resolució de triangles</a>, principalment a causa de l'aplicació del teorema del sinus, el del cosinus i el de la tangent. </p><p>La resolució de triangles és la tècnica en la que es basa la <a href="/wiki/Triangulaci%C3%B3" title="Triangulació">triangulació</a>. Aplicant sistemàticament aquesta tècnica es va fer la triangulació dels <a href="/wiki/Pa%C3%AFsos_Catalans" title="Països Catalans">Països Catalans</a> que es reprodueix a la figura de la dreta. La part que va des de <a href="/wiki/Salses" title="Salses">Salses</a> fins a <a href="/wiki/Barcelona" title="Barcelona">Barcelona</a> la va fer el científic <a href="/wiki/Fran%C3%A7a" title="França">francès</a> <a href="/wiki/Pierre_M%C3%A9chain" class="mw-redirect" title="Pierre Méchain">Pierre Méchain</a> i la part que va des de Barcelona fins a <a href="/wiki/Mallorca" title="Mallorca">Mallorca</a> la va començar Méchain juntament amb el científic <a href="/wiki/Catalunya_Nord" class="mw-redirect" title="Catalunya Nord">català</a> <a href="/wiki/Francesc_Arag%C3%B3" class="mw-redirect" title="Francesc Aragó">Francesc Aragó</a> i la va acabar aquest últim en morir Méchain a <a href="/wiki/Castell%C3%B3_de_la_Plana" title="Castelló de la Plana">Castelló de la Plana</a> sense haver pogut acabar els treballs. Aquesta part juntament amb la part francesa va servir per mesurar el meridià de <a href="/wiki/Par%C3%ADs" title="París">París</a> i aquesta mesura es la que es va fer servir per definir el <a href="/wiki/Metre" title="Metre">metre</a> patró.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Teorema_del_sinus">Teorema del sinus</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funci%C3%B3_trigonom%C3%A8trica&action=edit&section=16" title="Modifica la secció: Teorema del sinus"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r30997230" /><div role="note" class="hatnote navigation-not-searchable">Article principal: <a href="/wiki/Teorema_del_sinus" title="Teorema del sinus">Teorema del sinus</a></div> <p>El <a href="/wiki/Teorema_del_sinus" title="Teorema del sinus">teorema del sinus</a> estableix que per a qualsevol <a href="/wiki/Triangle" title="Triangle">triangle</a> de costats <i>a</i>, <i>b</i>, i <i>c</i> i angles oposats a aquests costats <i>A</i>, <i>B</i> i <i>C</i>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\sin A}{a}}={\frac {\sin B}{b}}={\frac {\sin C}{c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>A</mi> </mrow> <mi>a</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>B</mi> </mrow> <mi>b</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>C</mi> </mrow> <mi>c</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\sin A}{a}}={\frac {\sin B}{b}}={\frac {\sin C}{c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1e8929b4f629e8a791de1dedacddd8f7b6c9a7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:23.707ex; height:5.509ex;" alt="{\displaystyle {\frac {\sin A}{a}}={\frac {\sin B}{b}}={\frac {\sin C}{c}}}" /></span></dd></dl> <p>també conegut com: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {a}{\sin A}}={\frac {b}{\sin B}}={\frac {c}{\sin C}}=2R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>A</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>B</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>C</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a}{\sin A}}={\frac {b}{\sin B}}={\frac {c}{\sin C}}=2R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d80986c9d20c3eb3943d11776d454f6462b9b1ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:29.732ex; height:5.509ex;" alt="{\displaystyle {\frac {a}{\sin A}}={\frac {b}{\sin B}}={\frac {c}{\sin C}}=2R}" /></span></dd></dl> <p>on <i>R</i> és el radi de la <a href="/wiki/Circumfer%C3%A8ncia_circumscrita" title="Circumferència circumscrita">circumferència circumscrita</a>. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fitxer: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>Una <a href="/wiki/Corba_de_Lissajous" title="Corba de Lissajous">corba de Lissajous</a>, una figura formada amb funcions basades en la trigonometria.</figcaption></figure> <p>Es pot demostrar a base de dividir el triangle en dos triangles rectangles i fent servir la definició del sinus. El teorema del sinus és útil per a calcular les longituds dels costats desconeguts d'un triangle si es coneixen dos angles i un constat. Aquesta és una situació que succeeix normalment en <a href="/wiki/Triangulaci%C3%B3" title="Triangulació">triangulació</a>, una tècnica per a determinar distàncies desconegudes a base de mesurar dos angles i una distància accessible entre els dos angles. </p> <div class="mw-heading mw-heading3"><h3 id="Teorema_del_cosinus">Teorema del cosinus</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funci%C3%B3_trigonom%C3%A8trica&action=edit&section=17" title="Modifica la secció: Teorema del cosinus"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r30997230" /><div role="note" class="hatnote navigation-not-searchable">Article principal: <a href="/wiki/Teorema_del_cosinus" title="Teorema del cosinus">Teorema del cosinus</a></div> <p>El teorema del cosinus és una extensió del teorema de Pitàgores: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos C\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c5f4795efade2b94b3e89df23c25488cd9ea6de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:24.605ex; height:2.843ex;" alt="{\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos C\,}" /></span></dd></dl> <p>També conegut com: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos 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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos C={\frac {a^{2}+b^{2}-c^{2}}{2ab}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90d013f8ae80e83f4fe4a843989cb3bab8bfc5e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:21.277ex; height:5.843ex;" alt="{\displaystyle \cos C={\frac {a^{2}+b^{2}-c^{2}}{2ab}}}" /></span></dd></dl> <p>En aquesta fórmula l'angle C és oposat al costat c. Aquest teorema es pot demostrar dividint el triangle en dos triangles rectangles i emprant el <a href="/wiki/Teorema_de_Pit%C3%A0gores" title="Teorema de Pitàgores">teorema de Pitàgores</a>. </p><p>El teorema del cosinus es fa servir principalment per determinar un costat d'un triangle si es coneixen els altres dos costats i un dels angles, tot i que de vegades hi pot haver dues solucions positives. També es pot fer servir per a trobar el cosinus d'un angle (i en conseqüència l'angle mateix) si es coneixen tots tres costats. </p> <div class="mw-heading mw-heading3"><h3 id="Teorema_de_la_tangent">Teorema de la tangent</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funci%C3%B3_trigonom%C3%A8trica&action=edit&section=18" title="Modifica la secció: Teorema de la tangent"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r30997230" /><div role="note" class="hatnote navigation-not-searchable">Article principal: <a href="/wiki/Teorema_de_la_tangent" title="Teorema de la tangent">Teorema de la tangent</a></div> <p>El teorema de la tangent estableix que: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {a+b}{a-b}}={\frac {\tan[{\frac {1}{2}}(A+B)]}{\tan[{\frac {1}{2}}(A-B)]}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>+</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mrow> <mrow> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>−</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a+b}{a-b}}={\frac {\tan[{\frac {1}{2}}(A+B)]}{\tan[{\frac {1}{2}}(A-B)]}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc3a9eabede6ceceb176a58754cf6557b46f7c3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:24.307ex; height:7.843ex;" alt="{\displaystyle {\frac {a+b}{a-b}}={\frac {\tan[{\frac {1}{2}}(A+B)]}{\tan[{\frac {1}{2}}(A-B)]}}}" /></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Funcions_periòdiques"><span id="Funcions_peri.C3.B2diques"></span>Funcions periòdiques</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funci%C3%B3_trigonom%C3%A8trica&action=edit&section=19" title="Modifica la secció: Funcions periòdiques"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fitxer:Synthesis_square.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0a/Synthesis_square.gif/220px-Synthesis_square.gif" decoding="async" width="220" height="110" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0a/Synthesis_square.gif/330px-Synthesis_square.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0a/Synthesis_square.gif/440px-Synthesis_square.gif 2x" data-file-width="500" data-file-height="250" /></a><figcaption>Animació d'una síntesi adaptativa d'una <a href="/wiki/Ona_quadrada" title="Ona quadrada">ona quadrada</a> amb un nombre creixent d'harmònics.</figcaption></figure> <p>En física les funcions sinus i cosinus, per exemple, es fan servir per a descriure el <a href="/wiki/Moviment_harm%C3%B2nic_simple" title="Moviment harmònic simple">moviment harmònic simple</a>, el qual és un model de fenòmens naturals, com ara el moviment d'una massa fixada en una molla i, per angles petits, el moviment pendular d'una massa penjada d'una corda. Les funcions sinus i cosinus són les projeccions unidimensionals del <a href="/wiki/Moviment_circular_uniforme" class="mw-redirect" title="Moviment circular uniforme">moviment circular uniforme</a>.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p><p>Les funcions trigonomètriques també han demostrat ser útils en l'estudi de les <a href="/wiki/Funci%C3%B3_peri%C3%B2dica" title="Funció periòdica">funcions periòdiques</a> en general. Aquestes funcions tenen gràfiques i ones característiquess, útils per a modelitzar fenòmens recurrents com ara les <a href="/wiki/Ona" title="Ona">ones</a> de llum o sonores. Cada senyal es pot escriure com un sumatori (normalment infinit) de funcions sinus i cosinus de diferents freqüències; aquesta és la idea en què es basa l'<a href="/wiki/An%C3%A0lisi_de_Fourier" title="Anàlisi de Fourier">anàlisi de Fourier</a>, on les funcions trigonomètriques es fan servir per a resoldre una gran varietat de problemes de condicions de contorn en equacions diferencials amb derivades parcials. Per exemple, l'<a href="/wiki/Ona_quadrada" title="Ona quadrada">ona quadrada</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\left(x\right)=\left\{{\begin{matrix}-{\frac {1}{2}}\pi \quad x\in \left(-\pi ,0\right)\\{\frac {1}{2}}\pi \quad x\in \left(0,\pi \right)\\\end{matrix}}\right.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>π</mi> <mspace width="1em"></mspace> <mi>x</mi> <mo>∈</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mi>π</mi> <mo>,</mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>π</mi> <mspace width="1em"></mspace> <mi>x</mi> <mo>∈</mo> <mrow> <mo>(</mo> <mrow> <mn>0</mn> <mo>,</mo> <mi>π</mi> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\left(x\right)=\left\{{\begin{matrix}-{\frac {1}{2}}\pi \quad x\in \left(-\pi ,0\right)\\{\frac {1}{2}}\pi \quad x\in \left(0,\pi \right)\\\end{matrix}}\right.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3773fbea3f3038ef2f27e2941ac481e4a61efe3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:28.965ex; height:7.509ex;" alt="{\displaystyle f\left(x\right)=\left\{{\begin{matrix}-{\frac {1}{2}}\pi \quad x\in \left(-\pi ,0\right)\\{\frac {1}{2}}\pi \quad x\in \left(0,\pi \right)\\\end{matrix}}\right.}" /></span></dd></dl> <p>es pot escriure com una <a href="/wiki/S%C3%A8rie_de_Fourier" title="Sèrie de Fourier">sèrie de Fourier</a>.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=2\sum _{n=1}^{\infty }{\sin {\left((2n-1)x\right)} \over (2n-1)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <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> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=2\sum _{n=1}^{\infty }{\sin {\left((2n-1)x\right)} \over (2n-1)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df36bd9398518f1529fd005a35a623cdf83c1ee3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:28.395ex; height:6.843ex;" alt="{\displaystyle f(x)=2\sum _{n=1}^{\infty }{\sin {\left((2n-1)x\right)} \over (2n-1)}}" /></span>.</dd></dl> <p>A l'animació de la dreta es pot veure que amb només una quants termes s'aconsegueix una aproximació força bona. </p> <div class="mw-heading mw-heading2"><h2 id="Vegeu_també"><span id="Vegeu_tamb.C3.A9"></span>Vegeu també</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funci%C3%B3_trigonom%C3%A8trica&action=edit&section=20" title="Modifica la secció: Vegeu també"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Construcci%C3%B3_de_les_taules_trigonom%C3%A8triques" title="Construcció de les taules trigonomètriques">Construcció de les taules trigonomètriques</a></li> <li><a href="/wiki/Funcions_hiperb%C3%B2liques" class="mw-redirect" title="Funcions hiperbòliques">Funcions hiperbòliques</a></li> <li><a href="/wiki/Teorema_de_Pit%C3%A0gores" title="Teorema de Pitàgores">Teorema de Pitàgores</a></li> <li><a href="/wiki/Llista_d%27identitats_trigonom%C3%A8triques" title="Llista d'identitats trigonomètriques">Llista d'identitats trigonomètriques</a></li> <li><a href="/wiki/Demostraci%C3%B3_de_les_identitats_trigonom%C3%A8triques" title="Demostració de les identitats trigonomètriques">Demostració de les identitats trigonomètriques</a></li> <li><a href="/wiki/F%C3%B3rmula_d%27Euler" title="Fórmula d'Euler">Fórmula d'Euler</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funci%C3%B3_trigonom%C3%A8trica&action=edit&section=21" title="Modifica la secció: Notes"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist" style="list-style-type: decimal;"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-oconnor1996-1"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-oconnor1996_1-0">1,0</a></sup> <sup><a href="#cite_ref-oconnor1996_1-1">1,1</a></sup> <sup><a href="#cite_ref-oconnor1996_1-2">1,2</a></sup></span> <span class="reference-text">O'Connor (1996).</span> </li> <li id="cite_note-boyer1991-2"><span class="mw-cite-backlink"><a href="#cite_ref-boyer1991_2-0">↑</a></span> <span class="reference-text">Boyer, pàg. 158–168.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.britannica.com/EBchecked/topic/2127/Abul-Wafa">Abūʾl-Wafā</a> Article sobre Abūʾl-Wafā a l'Enciclopèdia Britànica.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://books.google.cat/books?id=R-s0G5ojNhUC&pg=PR23&lpg=PR23&dq=%22Llibre+de+Geometria%22+%22Abraham+Bar+Hiia%22&source=web&ots=2FHA8bLTmE&sig=DuEcD3HkdwMIjrpvwf8XcCH3fOM&hl=ca&sa=X&oi=book_result&resnum=4&ct=result#PPA82,M1">Llibre de Geometria</a><sup class="noprint Inline-Template"><span title="" style="white-space: nowrap;"><i>[<a href="/wiki/Viquip%C3%A8dia:Enlla%C3%A7os_externs#Manteniment_d'enllaços_externs" title="Viquipèdia:Enllaços externs">Enllaç no actiu</a>]</i></span></sup>, Abraham Bar Hiia (Savasorda), Biblioteca Hebraico-Catalana, <a href="/wiki/Especial:Fonts_bibliogr%C3%A0fiques/978-84-9859-106-4" title="Especial:Fonts bibliogràfiques/978-84-9859-106-4">ISBN 978-84-9859-106-4</a> pàgina 82</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><a href="#cite_ref-5">↑</a></span> <span class="reference-text">Vegeu Maor (1998), capítol 3, referent a etimologia.</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><a href="#cite_ref-6">↑</a></span> <span class="reference-text"><span class="citation" style="font-style:normal" id="CITEREFSozio2005"><span style="font-variant: small-caps;">Sozio</span>, Gerardo «<a rel="nofollow" class="external text" href="https://web.archive.org/web/20120526095021/http://www.parabola.unsw.edu.au/vol41_no1/vol41_no1_3.pdf">Trigonometry: Chords, Arcs and Angles</a>» (en <style data-mw-deduplicate="TemplateStyles:r33711417">.mw-parser-output .languageicon{font-size:0.95em;color:#555;background-color:inherit}@media screen{html.skin-theme-clientpref-night .mw-parser-output .languageicon{background-color:inherit;color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .languageicon{background-color:inherit;color:white}}</style><span class="languageicon" title="en anglès">(anglès)</span>). <i>Parabola</i>. University of New South Wales [Sidney (Austràlia)], Vol. 41, Num. 1, 2005, pàg. 10. Arxivat de l'<a rel="nofollow" class="external text" href="http://www.parabola.unsw.edu.au/vol41_no1/vol41_no1_3.pdf">original</a> el 2012-05-26 [Consulta: 29 desembre 2013].</span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><a href="#cite_ref-7">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.menacorde.com/cursMat/IniciacioMatematiques//s7/2_2_1.html">Iniciació a les matemàtiques per a l'enginyeria</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160313043836/http://menacorde.com/cursmat/iniciaciomatematiques/s7/2_2_1.html">Arxivat</a> 2016-03-13 a <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. Pàgina de la UOC on es defineixen les funcions trigonomètriques basant-se en el triangle rectangle]</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><a href="#cite_ref-8">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://books.google.cat/books?id=im-x_vl6gnsC&pg=PA444&lpg=PA444&dq=%22funci%C3%B3+sinus%22&source=bl&ots=i2BvNfjrWG&sig=lASjbN1WZ0iy-iHO2e0CPyO6AUQ&hl=ca&ei=mJbdSYWUKKHUjAeJ_rCnDg&sa=X&oi=book_result&ct=result&resnum=8#PPA437,M1">Càlculus</a> A la pàgina 437 hi ha la definició del sinus i el cosinus basant-se en la circumferència goniomètrica.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><a href="#cite_ref-9">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20070902195701/http://marx.upf.es/lavisit/MostrarTutorial.jsp?codiTutorial=senyals">Senyals Periòdics.</a> Portal d'Aprenentatge Obert de l'Escola d'Enginyeria de Telecomunicació, Tutorial: Els senyals, Núria Garcia Garcia, Processament del senyal, Característiques bàsiques dels senyals</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><a href="#cite_ref-10">↑</a></span> <span class="reference-text">Vegeu Lars Ahlfors, pàgines 43–44.</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><a href="#cite_ref-11">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://books.google.cat/books?id=ha1ad_abDKwC&pg=PA239&lpg=PA239&dq=%22funci%C3%B3+sinus%22&source=bl&ots=eM2z8URV5r&sig=-kN-OcLzz3UTPKV6JbAytDJWD_k&hl=ca&ei=3qPdSfilOMKgjAfK-4GnDg&sa=X&oi=book_result&ct=result&resnum=5#PPA201,M1">Definició de les funcions trigonomètriques basant-se en sèries</a></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><a href="#cite_ref-12">↑</a></span> <span class="reference-text">Abramowitz; Weisstein.</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><a href="#cite_ref-13">↑</a></span> <span class="reference-text">Needham, pàg. ix.</span> </li> <li id="cite_note-mathworld_double_angle-14"><span class="mw-cite-backlink"><a href="#cite_ref-mathworld_double_angle_14-0">↑</a></span> <span class="reference-text"><span class="citació mathworld" id="Referència-Mathworld-Double-Angle_Formulas"><a href="/wiki/Eric_W._Weisstein" class="mw-redirect" title="Eric W. Weisstein"><span style="font-variant:small-caps; font-variant-caps: small-caps;">Weisstein</span>, Eric W.</a>, <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Double-AngleFormulas.html">«Double-Angle Formulas»</a> a <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a> (en anglès).</span></span> </li> <li id="cite_note-mathworld_multiple_angle-15"><span class="mw-cite-backlink"><a href="#cite_ref-mathworld_multiple_angle_15-0">↑</a></span> <span class="reference-text"><span class="citació mathworld" id="Referència-Mathworld-Multiple-Angle_Formulas"><a href="/wiki/Eric_W._Weisstein" class="mw-redirect" title="Eric W. Weisstein"><span style="font-variant:small-caps; font-variant-caps: small-caps;">Weisstein</span>, Eric W.</a>, <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Multiple-AngleFormulas.html">«Multiple-Angle Formulas»</a> a <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a> (en anglès).</span></span> </li> <li id="cite_note-mathworld_half_angle-16"><span class="mw-cite-backlink"><a href="#cite_ref-mathworld_half_angle_16-0">↑</a></span> <span class="reference-text"><span class="citació mathworld" id="Referència-Mathworld-Half-Angle_Formulas"><a href="/wiki/Eric_W._Weisstein" class="mw-redirect" title="Eric W. Weisstein"><span style="font-variant:small-caps; font-variant-caps: small-caps;">Weisstein</span>, Eric W.</a>, <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Half-AngleFormulas.html">«Half-Angle Formulas»</a> a <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a> (en anglès).</span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><a href="#cite_ref-17">↑</a></span> <span class="reference-text">Kantabutra.</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><a href="#cite_ref-18">↑</a></span> <span class="reference-text">Ara bé, fer això i mantenir la precisió no és trivial, i es poden fer servir mètodes com les taules de precisió de Gal, la reducció de Cody i Waite, i els algorismes de reducció de Payne i Hanek.</span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><a href="#cite_ref-19">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://doi.acm.org/10.1145/321941.321944">R. P. Brent, "Fast Multiple-Precision Evaluation of Elementary Functions", J. ACM <b>23</b>, 242 (1976).</a></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><a href="#cite_ref-20">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.udg.edu/Portals/88/Santalo/confer%C3%A8ncies/Un_passeig_per_la_hist%C3%B2ria_del_sistema_m%C3%A8tric.pdf">Un passeig per la història del sistema mètric decimal</a>, Anton Aubanell Pou, Càtedra "Lluís Santaló" de la Universitat de Girona.</span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><a href="#cite_ref-21">↑</a></span> <span class="reference-text"><sup title="Cal donar format a aquesta referència." class="noprint"><span class="noprint" style="color:blue;"><i>[<a href="/wiki/VP:REF" class="mw-redirect" title="VP:REF">enllaç sense format</a>]</i></span></sup> <a rel="nofollow" class="external free" href="http://cns.upf.edu/laura/teaching/ffi/teoria_osc.pdf">http://cns.upf.edu/laura/teaching/ffi/teoria_osc.pdf</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20090920141902/http://cns.upf.edu/laura/teaching/ffi/teoria_osc.pdf">Arxivat</a> 2009-09-20 a <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. Fonaments Físics de la Informàtica] Capítol 2 oscil·lacions</span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><a href="#cite_ref-22">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://dmi.uib.es/~catalina/docencia/edp/fourier.pdf">Notes sobre sèries i transformada de Fourier</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100402042244/http://dmi.uib.es/~catalina/docencia/edp/fourier.pdf">Arxivat</a> 2010-04-02 a <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>., Catalina Sbert, 2005, pègina 2</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Referències"><span id="Refer.C3.A8ncies"></span>Referències</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funci%C3%B3_trigonom%C3%A8trica&action=edit&section=22" title="Modifica la secció: Referències"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <div style="font-size:95%"> <ul><li>Lars Ahlfors. <i>Complex Analysis: an introduction to the theory of analytic functions of one complex variable</i>, segona edició, McGraw-Hill Book Company, Nova York, 1966.</li> <li>Abramowitz, Milton; Irene A. Stegun. <i>Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables</i>, Dover, Nova York. (1964). <a href="/wiki/Especial:Fonts_bibliogr%C3%A0fiques/0-486-61272-4" title="Especial:Fonts bibliogràfiques/0-486-61272-4">ISBN 0-486-61272-4</a>.</li> <li><a href="/wiki/Carl_Boyer" class="mw-redirect" title="Carl Boyer">Boyer, Carl B.</a>. <i>A History of Mathematics</i>, John Wiley & Sons, Inc., segona edició. (1991). <a href="/wiki/Especial:Fonts_bibliogr%C3%A0fiques/0-471-54397-7" title="Especial:Fonts bibliogràfiques/0-471-54397-7">ISBN 0-471-54397-7</a>.</li> <li>Joseph, George G. <i>The Crest of the Peacock: Non-European Roots of Mathematics</i>, segona edició Penguin Books, Londres. (2000). <a href="/wiki/Especial:Fonts_bibliogr%C3%A0fiques/0-691-00659-8" title="Especial:Fonts bibliogràfiques/0-691-00659-8">ISBN 0-691-00659-8</a>.</li> <li>Kantabutra, Vitit. "On hardware for computing exponential and trigonometric functions," <i>IEEE Trans. Computers</i> <b>45</b> (3), 328–339 (1996).</li> <li>Maor, Eli. <i><a rel="nofollow" class="external text" href="http://www.pupress.princeton.edu/books/maor/">Trigonometric Delights</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20060414195120/http://www.pupress.princeton.edu/books/maor/">Arxivat</a> 2006-04-14 a <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>.</i>, Princeton Univ. Press. (1998). Reimpressió (25 febrer de 2002): <a href="/wiki/Especial:Fonts_bibliogr%C3%A0fiques/0-691-09541-8" title="Especial:Fonts bibliogràfiques/0-691-09541-8">ISBN 0-691-09541-8</a>.</li> <li>Needham, Tristan. <a rel="nofollow" class="external text" href="http://www.usfca.edu/vca/PDF/vca-preface.pdf">"Preface"</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20040602145226/http://www.usfca.edu/vca/PDF/vca-preface.pdf">Arxivat</a> 2004-06-02 a <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>." a <i><a rel="nofollow" class="external text" href="http://www.usfca.edu/vca/">Visual Complex Analysis</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20080607065324/http://www.usfca.edu/vca/">Arxivat</a> 2008-06-07 a <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>.</i>. Oxford University Press, (1999). <a href="/wiki/Especial:Fonts_bibliogr%C3%A0fiques/0-19-853446-9" title="Especial:Fonts bibliogràfiques/0-19-853446-9">ISBN 0-19-853446-9</a>.</li> <li>O'Connor, J.J.; E.F. Robertson. <a rel="nofollow" class="external text" href="http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Trigonometric_functions.html">"Trigonometric functions"</a> <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">Arxivat</a> 2013-01-20 a <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>., <i>Arxiu d'història de les matemàtiques a MacTutor</i>. (1996).</li> <li>O'Connor, J.J.; 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>, <i>Arxiu d'història de les matemàtiques a MacTutor</i>. (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">Arxivat</a> 2006-05-05 a <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>.. <i>Arxiu d'història de les matemàtiques a MacTutor</i>. (2002).</li> <li>Weisstein, Eric W. <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Tangent.html">"Tangent"</a> a <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>, accés el <a href="/wiki/21_de_gener" title="21 de gener">21 de gener</a> de <a href="/wiki/2006" title="2006">2006</a>.</li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Enllaços_externs"><span id="Enlla.C3.A7os_externs"></span>Enllaços externs</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funci%C3%B3_trigonom%C3%A8trica&action=edit&section=23" title="Modifica la secció: Enllaços externs"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r33663753">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:#f9f9f9;display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}.mw-parser-output .side-box-center{clear:both;margin:auto}}</style><div class="side-box metadata side-box-right plainlinks"> <div class="side-box-flex"> <div class="side-box-image"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/40px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/60px-Commons-logo.svg.png 1.5x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">A <span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/P%C3%A0gina_principal?uselang=ca">Wikimedia Commons</a></span> hi ha contingut multimèdia relatiu a: <i><b><a href="https://commons.wikimedia.org/wiki/Category:Trigonometric_functions" class="extiw" title="commons:Category:Trigonometric functions">Funció trigonomètrica</a></b></i></div></div> </div> <ul><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> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r33711417" /><span class="languageicon" title="En anglès">(anglès)</span></li></ul> <p><br /> </p> <div role="navigation" class="navbox" aria-labelledby="Trigonometria" style="padding:3px"><table class="nowraplinks collapsible collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><div class="plainlinks hlist navbar mini"><ul><li class="nv-view"><span typeof="mw:File"><a href="/wiki/Plantilla:Trigonometria" title="Plantilla:Trigonometria"><img alt="Vegeu aquesta plantilla" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/Commons-emblem-notice.svg/20px-Commons-emblem-notice.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/Commons-emblem-notice.svg/40px-Commons-emblem-notice.svg.png 1.5x" data-file-width="48" data-file-height="48" /></a></span></li></ul></div><div id="Trigonometria" style="font-size:114%;margin:0 4em"><a href="/wiki/Trigonometria" title="Trigonometria">Trigonometria</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a class="mw-selflink selflink">Funcions<br />trigonomètriques</a></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/Sinus" title="Sinus">Sinus</a> <i>(sin)</i><span style="font-weight:bold;"> ·</span> <a href="/wiki/Cosinus" class="mw-redirect" title="Cosinus">Cosinus</a> <i>(cos)</i><span style="font-weight:bold;"> ·</span> <a href="/wiki/Tangent_(trigonometria)" class="mw-redirect" title="Tangent (trigonometria)">Tangent</a> <i>(tan)</i><span style="font-weight:bold;"> ·</span> <a href="/wiki/Cotangent" class="mw-redirect" title="Cotangent">Cotangent</a> <i>(cot)</i><span style="font-weight:bold;"> ·</span> <a href="/wiki/Secant" class="mw-redirect" title="Secant">Secant</a> <i>(sec)</i><span style="font-weight:bold;"> ·</span> <a href="/wiki/Cosecant" class="mw-redirect" title="Cosecant">Cosecant</a> <i>(csc)</i><span style="font-weight:bold;"> ·</span> <a href="/wiki/Versinus" title="Versinus">Versinus</a> <i>(versin)</i><span style="font-weight:bold;"> ·</span> <a href="/wiki/Coversinus" class="mw-redirect" title="Coversinus">Coversinus</a> <i>(coversin)</i><span style="font-weight:bold;"> ·</span> <a href="/wiki/Semiversinus" class="mw-redirect" title="Semiversinus">Semiversinus</a> <i>(semiversin)</i><span style="font-weight:bold;"> ·</span> <a href="/w/index.php?title=Vercosinus&action=edit&redlink=1" class="new" title="Vercosinus (encara no existeix)">Vercosinus</a> <i>(vercos)</i><span style="font-weight:bold;"> ·</span> <a href="/wiki/Exsecant" title="Exsecant">Exsecant</a> <i>(exsec)</i><span style="font-weight:bold;"> ·</span> <a href="/wiki/Excosecant" class="mw-redirect" title="Excosecant">Excosecant</a> <i>(excsc)</i></div></td><td class="navbox-image" rowspan="5" style="width:1px;padding:0px 0px 0px 2px"><div><span typeof="mw:File"><a href="/wiki/Fitxer: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></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Inverses_de_les_funcions_trigonom%C3%A8triques" title="Inverses de les funcions trigonomètriques">Funcions<br />trigonomètriques<br />inverses</a></th><td class="navbox-list navbox-even" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/Arcsinus" class="mw-redirect" title="Arcsinus">Arcsinus</a> <i>(arcsin)</i><span style="font-weight:bold;"> ·</span> <a href="/wiki/Arccosinus" class="mw-redirect" title="Arccosinus">Arccosinus</a> <i>(arccos)</i><span style="font-weight:bold;"> ·</span> <a href="/wiki/Arctangent" class="mw-redirect" title="Arctangent">Arctangent</a> <i>(arctan)</i><span style="font-weight:bold;"> ·</span> <a href="/wiki/Arccotangent" class="mw-redirect" title="Arccotangent">Arccotangent</a> <i>(arccotan)</i><span style="font-weight:bold;"> ·</span> <a href="/wiki/Arcsecant" class="mw-redirect" title="Arcsecant">Arcsecant</a> <i>(arcsec)</i><span style="font-weight:bold;"> ·</span> <a href="/wiki/Arccosecant" class="mw-redirect" title="Arccosecant">Arccosecant</a> <i>(arccosec)</i></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Teoremes</th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/w/index.php?title=Teorema_de_la_corda&action=edit&redlink=1" class="new" title="Teorema de la corda (encara no existeix)">Teorema de la corda</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Teorema_del_sinus" title="Teorema del sinus">Teorema del sinus</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Teorema_del_cosinus" title="Teorema del cosinus">Teorema del cosinus</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Teorema_de_la_tangent" title="Teorema de la tangent">Teorema de la tangent</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Teorema_de_l%27Huilier" title="Teorema de l'Huilier">Teorema de l'Huilier</a><span style="font-weight:bold;"> ·</span> <a href="/w/index.php?title=Teorema_de_Neper&action=edit&redlink=1" class="new" title="Teorema de Neper (encara no existeix)">Teorema de Neper</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Teorema_de_Pit%C3%A0gores" title="Teorema de Pitàgores">Teorema de Pitàgores</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Teorema_de_Tales" title="Teorema de Tales">Teorema de Tales</a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Fòrmules</th><td class="navbox-list navbox-even" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/F%C3%B3rmula_de_De_Moivre" title="Fórmula de De Moivre">Fórmula de De Moivre</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/F%C3%B3rmula_d%27Euler" title="Fórmula d'Euler">Fórmula d'Euler</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/F%C3%B3rmula_d%27Her%C3%B3" title="Fórmula d'Heró">Fórmula d'Heró</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/F%C3%B3rmula_de_Mollweide" title="Fórmula de Mollweide">Fórmula de Mollweide</a><span style="font-weight:bold;"> ·</span> <a href="/w/index.php?title=F%C3%B3rmules_de_prostaferesi&action=edit&redlink=1" class="new" title="Fórmules de prostaferesi (encara no existeix)">Fórmules de prostaferesi</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/F%C3%B3rmula_de_la_tangent_de_l%27angle_meitat" title="Fórmula de la tangent de l'angle meitat">Fórmula de la tangent de l'angle meitat</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/F%C3%B3rmules_de_Vincenty" title="Fórmules de Vincenty">Fórmules de Vincenty</a><span style="font-weight:bold;"> ·</span> <a href="/w/index.php?title=F%C3%B3rmula_de_Werner&action=edit&redlink=1" class="new" title="Fórmula de Werner (encara no existeix)">Fórmula de Werner</a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><i>Vegeu també</i></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/Hist%C3%B2ria_de_les_funcions_trigonom%C3%A8triques" title="Història de les funcions trigonomètriques">Història</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Llista_d%27identitats_trigonom%C3%A8triques" title="Llista d'identitats trigonomètriques">Identitats</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Demostraci%C3%B3_de_les_identitats_trigonom%C3%A8triques" title="Demostració de les identitats trigonomètriques">Demostracions</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Construcci%C3%B3_de_les_taules_trigonom%C3%A8triques" title="Construcció de les taules trigonomètriques">Taules</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Constants_trigonom%C3%A8triques_exactes" title="Constants trigonomètriques exactes">Constants exactes</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/CORDIC" title="CORDIC">CORDIC</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Derivaci%C3%B3_de_les_funcions_trigonom%C3%A8triques" title="Derivació de les funcions trigonomètriques">Derivades</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Primitives_de_funcions_trigonom%C3%A8triques" title="Primitives de funcions trigonomètriques">Integrals</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Llista_d%27integrals_d%27inverses_de_funcions_trigonom%C3%A8triques" title="Llista d'integrals d'inverses de funcions trigonomètriques">Integrals de les funcions inverses</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Trigonometria_esf%C3%A8rica" title="Trigonometria esfèrica">Trigonometria esfèrica</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Circumfer%C3%A8ncia_goniom%C3%A8trica" title="Circumferència goniomètrica">Circumferència goniomètrica</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Resoluci%C3%B3_de_triangles" title="Resolució de triangles">Resolució de triangles</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Sinusoide" title="Sinusoide">Sinusoide</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Funci%C3%B3_peri%C3%B2dica" title="Funció periòdica">Funció periòdica</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/S%C3%A8rie_de_Fourier" title="Sèrie de Fourier">Sèrie de Fourier</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Triangulaci%C3%B3" title="Triangulació">Triangulació</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Funci%C3%B3_hiperb%C3%B2lica" title="Funció hiperbòlica">Funció hiperbòlica</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Funci%C3%B3_Gudermanniana" class="mw-redirect" title="Funció Gudermanniana">Funció Gudermanniana</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Radian" title="Radian">Radian</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Projecci%C3%B3_azimutal_estereogr%C3%A0fica" title="Projecció azimutal estereogràfica">Projecció azimutal estereogràfica</a></div></td></tr></tbody></table></div> <div role="navigation" class="navbox" aria-label="Navbox" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Control_d%27autoritats" title="Control d'autoritats">Registres d'autoritat</a></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/Biblioth%C3%A8que_nationale_de_France" class="mw-redirect" title="Bibliothèque nationale de France">BNF</a> (<a rel="nofollow" class="external text" href="https://catalogue.bnf.fr/ark:/12148/cb12168469v">1</a>)</li> <li><a href="/wiki/Gemeinsame_Normdatei" title="Gemeinsame Normdatei">GND</a> (<a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4186137-1">1</a>)</li> <li><a href="/wiki/LCCN" class="mw-redirect" title="LCCN">LCCN</a> (<a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/sh85137518">1</a>)</li> <li><a href="/wiki/National_Diet_Library" class="mw-redirect" title="National Diet Library">NDL</a> (<a rel="nofollow" class="external text" href="https://id.ndl.go.jp/auth/ndlna/00570156">1</a>)</li> <li><a href="/wiki/Biblioteca_Nacional_de_la_Rep%C3%BAblica_Txeca" title="Biblioteca Nacional de la República Txeca">NKC</a> (<a rel="nofollow" class="external text" href="https://aleph.nkp.cz/F/?func=find-c&local_base=aut&ccl_term=ica=ph923034&CON_LNG=ENG">1</a>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Bases d'informació</th><td class="navbox-list navbox-even" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/Encyclop%C3%A6dia_Britannica_Online" title="Encyclopædia Britannica Online">Britannica</a> (<a rel="nofollow" class="external text" href="https://www.britannica.com/topic/trigonometric-function">1</a>)</li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://auth.wikimedia.org/loginwiki/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Obtingut de «<a dir="ltr" href="https://ca.wikipedia.org/w/index.php?title=Funció_trigonomètrica&oldid=34633281#Tangent">https://ca.wikipedia.org/w/index.php?title=Funció_trigonomètrica&oldid=34633281#Tangent</a>»</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Especial:Categorias" title="Especial:Categorias">Categories</a>: <ul><li><a href="/wiki/Categoria:Trigonometria" title="Categoria:Trigonometria">Trigonometria</a></li><li><a href="/wiki/Categoria:Funcions" title="Categoria:Funcions">Funcions</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Categories ocultes: <ul><li><a href="/wiki/Categoria:Articles_amb_enlla%C3%A7os_externs_no_actius" title="Categoria:Articles amb enllaços externs no actius">Articles amb enllaços externs no actius</a></li><li><a href="/wiki/Categoria:Articles_amb_la_plantilla_Webarchive_amb_enlla%C3%A7_wayback" title="Categoria:Articles amb la plantilla Webarchive amb enllaç wayback">Articles amb la plantilla Webarchive amb enllaç wayback</a></li><li><a href="/wiki/Categoria:Articles_amb_refer%C3%A8ncies_sense_format" title="Categoria:Articles amb referències sense format">Articles amb referències sense format</a></li><li><a href="/wiki/Categoria:P%C3%A0gines_amb_enlla%C3%A7_commonscat_des_de_Wikidata" title="Categoria:Pàgines amb enllaç commonscat des de Wikidata">Pàgines amb enllaç commonscat des de Wikidata</a></li><li><a href="/wiki/Categoria:Llista_d%27articles_de_qualitat" title="Categoria:Llista d'articles de qualitat">Llista d'articles de qualitat</a></li><li><a href="/wiki/Categoria:Control_d%27autoritats" title="Categoria:Control d'autoritats">Control d'autoritats</a></li><li><a href="/wiki/Categoria:Articles_de_qualitat_de_matem%C3%A0tiques" title="Categoria:Articles de qualitat de matemàtiques">Articles de qualitat de matemàtiques</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> La pàgina va ser modificada per darrera vegada el 18 feb 2025 a les 12:41.</li> <li id="footer-info-copyright">El text està disponible sota la <a href="/wiki/Viquip%C3%A8dia:Text_de_la_llic%C3%A8ncia_de_Creative_Commons_Reconeixement-Compartir_Igual_4.0_No_adaptada" title="Viquipèdia:Text de la llicència de Creative Commons Reconeixement-Compartir Igual 4.0 No adaptada"> Llicència de Creative Commons Reconeixement i Compartir-Igual</a>; es poden aplicar termes addicionals. Vegeu les <a class="external text" href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use/ca">Condicions d'ús</a>. Wikipedia® (Viquipèdia™) és una <a href="/wiki/Marca_comercial" title="Marca comercial">marca registrada</a> de <a rel="nofollow" class="external text" href="https://www.wikimediafoundation.org">Wikimedia Foundation, Inc</a>.<br /></li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Política de privadesa</a></li> <li id="footer-places-about"><a href="/wiki/Viquip%C3%A8dia:Quant_a_la_Viquip%C3%A8dia">Quant al projecte Viquipèdia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Viquip%C3%A8dia:Av%C3%ADs_d%27exempci%C3%B3_de_responsabilitat">Descàrrec de responsabilitat</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Codi de conducta</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Desenvolupadors</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/ca.wikipedia.org">Estadístiques</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Declaració de cookies</a></li> <li id="footer-places-mobileview"><a href="//ca.m.wikipedia.org/w/index.php?title=Funci%C3%B3_trigonom%C3%A8trica&mobileaction=toggle_view_mobile#Tangent" class="noprint stopMobileRedirectToggle">Versió per a mòbils</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://www.wikimedia.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><picture><source media="(min-width: 500px)" srcset="/static/images/footer/wikimedia-button.svg" width="84" height="29"><img src="/static/images/footer/wikimedia.svg" width="25" height="25" alt="Wikimedia Foundation" lang="en" loading="lazy"></picture></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><picture><source media="(min-width: 500px)" srcset="/w/resources/assets/poweredby_mediawiki.svg" width="88" height="31"><img src="/w/resources/assets/mediawiki_compact.svg" alt="Powered by MediaWiki" lang="en" width="25" height="25" loading="lazy"></picture></a></li> </ul> </footer> </div> </div> </div> <div class="vector-header-container vector-sticky-header-container"> <div id="vector-sticky-header" class="vector-sticky-header"> <div class="vector-sticky-header-start"> <div class="vector-sticky-header-icon-start vector-button-flush-left vector-button-flush-right" aria-hidden="true"> <button class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-sticky-header-search-toggle" tabindex="-1" data-event-name="ui.vector-sticky-search-form.icon"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Cerca</span> </button> </div> <div role="search" class="vector-search-box-vue vector-search-box-show-thumbnail vector-search-box"> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail"> <form action="/w/index.php" id="vector-sticky-search-form" class="cdx-search-input cdx-search-input--has-end-button"> <div class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Cerca a Viquipèdia"> <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Especial:Cerca"> </div> <button class="cdx-button cdx-search-input__end-button">Cerca</button> </form> </div> </div> </div> <div class="vector-sticky-header-context-bar"> <nav aria-label="Contingut" class="vector-toc-landmark"> <div id="vector-sticky-header-toc" class="vector-dropdown mw-portlet mw-portlet-sticky-header-toc vector-sticky-header-toc vector-button-flush-left" > <input type="checkbox" id="vector-sticky-header-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-sticky-header-toc" class="vector-dropdown-checkbox " aria-label="Commuta la taula de continguts." > <label id="vector-sticky-header-toc-label" for="vector-sticky-header-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Commuta la taula de continguts.</span> </label> <div class="vector-dropdown-content"> <div id="vector-sticky-header-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div class="vector-sticky-header-context-bar-primary" aria-hidden="true" ><span class="mw-page-title-main">Funció trigonomètrica</span></div> </div> </div> <div class="vector-sticky-header-end" aria-hidden="true"> <div class="vector-sticky-header-icons"> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-talk-sticky-header" tabindex="-1" data-event-name="talk-sticky-header"><span class="vector-icon mw-ui-icon-speechBubbles mw-ui-icon-wikimedia-speechBubbles"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-subject-sticky-header" tabindex="-1" data-event-name="subject-sticky-header"><span class="vector-icon mw-ui-icon-article mw-ui-icon-wikimedia-article"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-history-sticky-header" tabindex="-1" data-event-name="history-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-history mw-ui-icon-wikimedia-wikimedia-history"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only mw-watchlink" id="ca-watchstar-sticky-header" tabindex="-1" data-event-name="watch-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-star mw-ui-icon-wikimedia-wikimedia-star"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-edit-sticky-header" tabindex="-1" data-event-name="wikitext-edit-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-wikiText mw-ui-icon-wikimedia-wikimedia-wikiText"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-ve-edit-sticky-header" tabindex="-1" data-event-name="ve-edit-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-edit mw-ui-icon-wikimedia-wikimedia-edit"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-viewsource-sticky-header" tabindex="-1" data-event-name="ve-edit-protected-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-editLock mw-ui-icon-wikimedia-wikimedia-editLock"></span> <span></span> </a> </div> <div class="vector-sticky-header-buttons"> <button class="cdx-button cdx-button--weight-quiet mw-interlanguage-selector" id="p-lang-btn-sticky-header" tabindex="-1" data-event-name="ui.dropdown-p-lang-btn-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-language mw-ui-icon-wikimedia-wikimedia-language"></span> <span>74 llengües</span> </button> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive" id="ca-addsection-sticky-header" tabindex="-1" data-event-name="addsection-sticky-header"><span class="vector-icon mw-ui-icon-speechBubbleAdd-progressive mw-ui-icon-wikimedia-speechBubbleAdd-progressive"></span> <span>Afegeix un tema</span> </a> </div> <div class="vector-sticky-header-icon-end"> <div class="vector-user-links"> </div> </div> </div> </div> </div> <div class="mw-portlet mw-portlet-dock-bottom emptyPortlet" id="p-dock-bottom"> <ul> </ul> </div> <script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-65585cc8dc-kjfc7","wgBackendResponseTime":305,"wgPageParseReport":{"limitreport":{"cputime":"0.441","walltime":"0.839","ppvisitednodes":{"value":2509,"limit":1000000},"postexpandincludesize":{"value":38502,"limit":2097152},"templateargumentsize":{"value":2000,"limit":2097152},"expansiondepth":{"value":12,"limit":100},"expensivefunctioncount":{"value":9,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":23858,"limit":5000000},"entityaccesscount":{"value":2,"limit":400},"timingprofile":["100.00% 285.677 1 -total"," 27.18% 77.650 1 Plantilla:Commonscat"," 25.26% 72.167 1 Plantilla:Referències"," 22.64% 64.676 1 Plantilla:Sister"," 21.84% 62.406 1 Plantilla:Caixa_lateral"," 16.10% 45.990 8 Plantilla:Article_principal"," 12.90% 36.846 1 Plantilla:Ref-publicació"," 7.38% 21.070 1 Plantilla:Autoritat"," 7.17% 20.491 1 Plantilla:Trigonometria"," 6.91% 19.734 1 Plantilla:Article_de_qualitat"]},"scribunto":{"limitreport-timeusage":{"value":"0.083","limit":"10.000"},"limitreport-memusage":{"value":2596424,"limit":52428800}},"cachereport":{"origin":"mw-web.eqiad.main-75687f9f4b-kztmk","timestamp":"20250323090438","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Funci\u00f3 trigonom\u00e8trica","url":"https:\/\/ca.wikipedia.org\/wiki\/Funci%C3%B3_trigonom%C3%A8trica#Tangent","sameAs":"http:\/\/www.wikidata.org\/entity\/Q93344","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q93344","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2007-06-05T09:52:42Z","dateModified":"2025-02-18T10:41:12Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/9\/9d\/Circle-trig6.svg","headline":"funci\u00f3 la variable de la qual \u00e9s un angle"}</script> </body> </html>

CINXE.COM

Funció trigonomètrica - Viquipèdia, l'enciclopèdia lliure