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class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Conteúdo" 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">Conteúdo</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">mover para a barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">ocultar</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">Início</div> </a> </li> <li id="toc-Círculo_trigonométrico" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Círculo_trigonométrico"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Círculo trigonométrico</span> </div> </a> <button aria-controls="toc-Círculo_trigonométrico-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>Alternar a subsecção Círculo trigonométrico</span> </button> <ul id="toc-Círculo_trigonométrico-sublist" class="vector-toc-list"> <li id="toc-Seno" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Seno"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Seno</span> </div> </a> <ul id="toc-Seno-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cosseno" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cosseno"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Cosseno</span> </div> </a> <ul id="toc-Cosseno-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tangente" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tangente"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Tangente</span> </div> </a> <ul id="toc-Tangente-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Relações" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Relações"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Relações</span> </div> </a> <ul id="toc-Relações-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Teorema_de_Pitágoras" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Teorema_de_Pitágoras"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Teorema de Pitágoras</span> </div> </a> <ul id="toc-Teorema_de_Pitágoras-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Aplicações_da_trigonometria" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Aplicações_da_trigonometria"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Aplicações da trigonometria</span> </div> </a> <ul id="toc-Aplicações_da_trigonometria-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Identidades_trigonométricas" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Identidades_trigonométricas"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Identidades trigonométricas</span> </div> </a> <button aria-controls="toc-Identidades_trigonométricas-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>Alternar a subsecção Identidades trigonométricas</span> </button> <ul id="toc-Identidades_trigonométricas-sublist" class="vector-toc-list"> <li id="toc-Identidades_triangulares" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Identidades_triangulares"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Identidades triangulares</span> </div> </a> <ul id="toc-Identidades_triangulares-sublist" class="vector-toc-list"> <li id="toc-Lei_dos_senos" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Lei_dos_senos"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1.1</span> <span>Lei dos senos</span> </div> </a> <ul id="toc-Lei_dos_senos-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lei_dos_cossenos" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Lei_dos_cossenos"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1.2</span> <span>Lei dos cossenos</span> </div> </a> <ul id="toc-Lei_dos_cossenos-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lei_das_tangentes" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Lei_das_tangentes"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1.3</span> <span>Lei das tangentes</span> </div> </a> <ul id="toc-Lei_das_tangentes-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Como_saber_o_ângulo_interno_de_um_triângulo_retângulo" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Como_saber_o_ângulo_interno_de_um_triângulo_retângulo"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Como saber o ângulo interno de um triângulo retângulo</span> </div> </a> <ul id="toc-Como_saber_o_ângulo_interno_de_um_triângulo_retângulo-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Referências" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Referências"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Referências</span> </div> </a> <ul id="toc-Referências-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliografia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliografia"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Bibliografia</span> </div> </a> <ul id="toc-Bibliografia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ligações_externas" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ligações_externas"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Ligações externas</span> </div> </a> <ul id="toc-Ligações_externas-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="Conteúdo" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Alternar o índice" > <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">Alternar o índice</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">Trigonometria</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="Ir para um artigo noutra língua. Disponível em 139 línguas" > <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-139" 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">139 línguas</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Driehoeksmeting" title="Driehoeksmeting — africanês" lang="af" hreflang="af" data-title="Driehoeksmeting" data-language-autonym="Afrikaans" data-language-local-name="africanês" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Trigonometrie" title="Trigonometrie — alemão suíço" lang="gsw" hreflang="gsw" data-title="Trigonometrie" data-language-autonym="Alemannisch" data-language-local-name="alemão suíço" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%89%B5%E1%88%AA%E1%8C%8E%E1%8A%96%E1%88%9C%E1%89%B5%E1%88%AA" title="ትሪጎኖሜትሪ — amárico" lang="am" hreflang="am" data-title="ትሪጎኖሜትሪ" data-language-autonym="አማርኛ" data-language-local-name="amárico" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Trigonometr%C3%ADa" title="Trigonometría — aragonês" lang="an" hreflang="an" data-title="Trigonometría" data-language-autonym="Aragonés" data-language-local-name="aragonês" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-anp mw-list-item"><a href="https://anp.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%A4%BF" title="त्रिकोणमिति — angika" lang="anp" hreflang="anp" data-title="त्रिकोणमिति" data-language-autonym="अंगिका" data-language-local-name="angika" class="interlanguage-link-target"><span>अंगिका</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AD%D8%B3%D8%A7%D8%A8_%D8%A7%D9%84%D9%85%D8%AB%D9%84%D8%AB%D8%A7%D8%AA" title="حساب المثلثات — árabe" lang="ar" hreflang="ar" data-title="حساب المثلثات" data-language-autonym="العربية" data-language-local-name="árabe" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-arz mw-list-item"><a href="https://arz.wikipedia.org/wiki/%D8%AD%D8%B3%D8%A7%D8%A8_%D8%A7%D9%84%D9%85%D8%AB%D9%84%D8%AB%D8%A7%D8%AA" title="حساب المثلثات — Egyptian Arabic" lang="arz" hreflang="arz" data-title="حساب المثلثات" data-language-autonym="مصرى" data-language-local-name="Egyptian Arabic" class="interlanguage-link-target"><span>مصرى</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%A4%E0%A7%8D%E0%A7%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" title="ত্ৰিকোণমিতি — assamês" lang="as" hreflang="as" data-title="ত্ৰিকোণমিতি" data-language-autonym="অসমীয়া" data-language-local-name="assamês" class="interlanguage-link-target"><span>অসমীয়া</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Trigonometr%C3%ADa" title="Trigonometría — asturiano" lang="ast" hreflang="ast" data-title="Trigonometría" data-language-autonym="Asturianu" data-language-local-name="asturiano" 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/Triqonometriya" title="Triqonometriya — azerbaijano" lang="az" hreflang="az" data-title="Triqonometriya" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaijano" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D8%A7%D9%88%DA%86%E2%80%8C%D8%A8%D9%88%D8%AC%D8%A7%D9%82_%D8%A8%DB%8C%D9%84%DB%8C%D9%85%DB%8C" title="اوچ‌بوجاق بیلیمی — South Azerbaijani" lang="azb" hreflang="azb" data-title="اوچ‌بوجاق بیلیمی" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target"><span>تۆرکجه</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%D1%8F" title="Тригонометрия — bashkir" lang="ba" hreflang="ba" data-title="Тригонометрия" data-language-autonym="Башҡортса" data-language-local-name="bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-bat-smg mw-list-item"><a href="https://bat-smg.wikipedia.org/wiki/Tr%C4%97guonuometr%C4%97j%C4%97" title="Trėguonuometrėjė — Samogitian" lang="sgs" hreflang="sgs" data-title="Trėguonuometrėjė" data-language-autonym="Žemaitėška" data-language-local-name="Samogitian" class="interlanguage-link-target"><span>Žemaitėška</span></a></li><li class="interlanguage-link interwiki-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Trigonometriya" title="Trigonometriya — Central Bikol" lang="bcl" hreflang="bcl" data-title="Trigonometriya" data-language-autonym="Bikol Central" data-language-local-name="Central Bikol" class="interlanguage-link-target"><span>Bikol Central</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%8F" title="Трыганаметрыя — bielorrusso" lang="be" hreflang="be" data-title="Трыганаметрыя" data-language-autonym="Беларуская" data-language-local-name="bielorrusso" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%A2%D1%80%D1%8B%D0%B3%D0%B0%D0%BD%D0%B0%D0%BC%D1%8D%D1%82%D1%80%D1%8B%D1%8F" title="Трыганамэтрыя — Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Трыганамэтрыя" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</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%8F" title="Тригонометрия — búlgaro" lang="bg" hreflang="bg" data-title="Тригонометрия" data-language-autonym="Български" data-language-local-name="búlgaro" 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" title="ত্রিকোণমিতি — bengalês" lang="bn" hreflang="bn" data-title="ত্রিকোণমিতি" data-language-autonym="বাংলা" data-language-local-name="bengalês" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Trigonometriezh" title="Trigonometriezh — bretão" lang="br" hreflang="br" data-title="Trigonometriezh" data-language-autonym="Brezhoneg" data-language-local-name="bretão" class="interlanguage-link-target"><span>Brezhoneg</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Trigonometrija" title="Trigonometrija — bósnio" lang="bs" hreflang="bs" data-title="Trigonometrija" data-language-autonym="Bosanski" data-language-local-name="bósnio" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Trigonometria" title="Trigonometria — catalão" lang="ca" hreflang="ca" data-title="Trigonometria" data-language-autonym="Català" data-language-local-name="catalão" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%B3%DB%8E%DA%AF%DB%86%D8%B4%DB%95%D8%B2%D8%A7%D9%86%DB%8C" title="سێگۆشەزانی — curdo central" lang="ckb" hreflang="ckb" data-title="سێگۆشەزانی" data-language-autonym="کوردی" data-language-local-name="curdo central" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-co mw-list-item"><a href="https://co.wikipedia.org/wiki/Trigunumitria" title="Trigunumitria — córsico" lang="co" hreflang="co" data-title="Trigunumitria" data-language-autonym="Corsu" data-language-local-name="córsico" class="interlanguage-link-target"><span>Corsu</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Trigonometrie" title="Trigonometrie — checo" lang="cs" hreflang="cs" data-title="Trigonometrie" data-language-autonym="Čeština" data-language-local-name="checo" 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" title="Тригонометри — chuvash" lang="cv" hreflang="cv" data-title="Тригонометри" data-language-autonym="Чӑвашла" data-language-local-name="chuvash" 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/Trigonometreg" title="Trigonometreg — galês" lang="cy" hreflang="cy" data-title="Trigonometreg" data-language-autonym="Cymraeg" data-language-local-name="galês" 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/Trigonometri" title="Trigonometri — dinamarquês" lang="da" hreflang="da" data-title="Trigonometri" data-language-autonym="Dansk" data-language-local-name="dinamarquês" 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/Trigonometrie" title="Trigonometrie — alemão" lang="de" hreflang="de" data-title="Trigonometrie" data-language-autonym="Deutsch" data-language-local-name="alemão" 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%AF%CE%B1" title="Τριγωνομετρία — grego" lang="el" hreflang="el" data-title="Τριγωνομετρία" data-language-autonym="Ελληνικά" data-language-local-name="grego" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-eml mw-list-item"><a href="https://eml.wikipedia.org/wiki/Trigonometr%C3%AE" title="Trigonometrî — Emiliano-Romagnolo" lang="egl" hreflang="egl" data-title="Trigonometrî" data-language-autonym="Emiliàn e rumagnòl" data-language-local-name="Emiliano-Romagnolo" class="interlanguage-link-target"><span>Emiliàn e rumagnòl</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Trigonometry" title="Trigonometry — inglês" lang="en" hreflang="en" data-title="Trigonometry" data-language-autonym="English" data-language-local-name="inglês" 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/Trigonometrio" title="Trigonometrio — esperanto" lang="eo" hreflang="eo" data-title="Trigonometrio" 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/Trigonometr%C3%ADa" title="Trigonometría — espanhol" lang="es" hreflang="es" data-title="Trigonometría" data-language-autonym="Español" data-language-local-name="espanhol" 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/Trigonomeetria" title="Trigonomeetria — estónio" lang="et" hreflang="et" data-title="Trigonomeetria" data-language-autonym="Eesti" data-language-local-name="estónio" 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/Trigonometria" title="Trigonometria — basco" lang="eu" hreflang="eu" data-title="Trigonometria" data-language-autonym="Euskara" data-language-local-name="basco" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-ext mw-list-item"><a href="https://ext.wikipedia.org/wiki/Trigonometria" title="Trigonometria — Extremaduran" lang="ext" hreflang="ext" data-title="Trigonometria" data-language-autonym="Estremeñu" data-language-local-name="Extremaduran" class="interlanguage-link-target"><span>Estremeñu</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%AB%D9%84%D8%AB%D8%A7%D8%AA" title="مثلثات — persa" lang="fa" hreflang="fa" data-title="مثلثات" 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/Trigonometria" title="Trigonometria — finlandês" lang="fi" hreflang="fi" data-title="Trigonometria" data-language-autonym="Suomi" data-language-local-name="finlandês" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fiu-vro mw-list-item"><a href="https://fiu-vro.wikipedia.org/wiki/Trigonomeetri%C3%A4" title="Trigonomeetriä — Võro" lang="vro" hreflang="vro" data-title="Trigonomeetriä" data-language-autonym="Võro" data-language-local-name="Võro" class="interlanguage-link-target"><span>Võro</span></a></li><li class="interlanguage-link interwiki-fo mw-list-item"><a href="https://fo.wikipedia.org/wiki/Trigonometri" title="Trigonometri — feroês" lang="fo" hreflang="fo" data-title="Trigonometri" data-language-autonym="Føroyskt" data-language-local-name="feroês" class="interlanguage-link-target"><span>Føroyskt</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Trigonom%C3%A9trie" title="Trigonométrie — francês" lang="fr" hreflang="fr" data-title="Trigonométrie" data-language-autonym="Français" data-language-local-name="francês" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Trigonometrii" title="Trigonometrii — frísio setentrional" lang="frr" hreflang="frr" data-title="Trigonometrii" data-language-autonym="Nordfriisk" data-language-local-name="frísio setentrional" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Triant%C3%A1nacht" title="Triantánacht — irlandês" lang="ga" hreflang="ga" data-title="Triantánacht" data-language-autonym="Gaeilge" data-language-local-name="irlandês" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E4%B8%89%E8%A7%92%E5%AD%B8" title="三角學 — gan" lang="gan" hreflang="gan" data-title="三角學" data-language-autonym="贛語" data-language-local-name="gan" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Trigonom%C3%A9tri" title="Trigonométri — Guianan Creole" lang="gcr" hreflang="gcr" data-title="Trigonométri" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Trigonometr%C3%ADa" title="Trigonometría — galego" lang="gl" hreflang="gl" data-title="Trigonometría" data-language-autonym="Galego" data-language-local-name="galego" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gu mw-list-item"><a href="https://gu.wikipedia.org/wiki/%E0%AA%A4%E0%AB%8D%E0%AA%B0%E0%AA%BF%E0%AA%95%E0%AB%8B%E0%AA%A3%E0%AA%AE%E0%AA%BF%E0%AA%A4%E0%AA%BF" title="ત્રિકોણમિતિ — guzerate" lang="gu" hreflang="gu" data-title="ત્રિકોણમિતિ" data-language-autonym="ગુજરાતી" data-language-local-name="guzerate" class="interlanguage-link-target"><span>ગુજરાતી</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%98%D7%A8%D7%99%D7%92%D7%95%D7%A0%D7%95%D7%9E%D7%98%D7%A8%D7%99%D7%94" title="טריגונומטריה — hebraico" lang="he" hreflang="he" data-title="טריגונומטריה" data-language-autonym="עברית" data-language-local-name="hebraico" 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%A4%BF" title="त्रिकोणमिति — hindi" lang="hi" hreflang="hi" data-title="त्रिकोणमिति" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Trigonometry" title="Trigonometry — Fiji Hindi" lang="hif" hreflang="hif" data-title="Trigonometry" data-language-autonym="Fiji Hindi" data-language-local-name="Fiji Hindi" class="interlanguage-link-target"><span>Fiji Hindi</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Trigonometrija" title="Trigonometrija — croata" lang="hr" hreflang="hr" data-title="Trigonometrija" data-language-autonym="Hrvatski" data-language-local-name="croata" 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/Trigonometria" title="Trigonometria — húngaro" lang="hu" hreflang="hu" data-title="Trigonometria" data-language-autonym="Magyar" data-language-local-name="húngaro" 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%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Եռանկյունաչափություն — arménio" lang="hy" hreflang="hy" data-title="Եռանկյունաչափություն" data-language-autonym="Հայերեն" data-language-local-name="arménio" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Trigonometria" title="Trigonometria — interlíngua" lang="ia" hreflang="ia" data-title="Trigonometria" data-language-autonym="Interlingua" data-language-local-name="interlíngua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-iba mw-list-item"><a href="https://iba.wikipedia.org/wiki/Trigonometri" title="Trigonometri — iban" lang="iba" hreflang="iba" data-title="Trigonometri" data-language-autonym="Jaku Iban" data-language-local-name="iban" class="interlanguage-link-target"><span>Jaku Iban</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Trigonometri" title="Trigonometri — indonésio" lang="id" hreflang="id" data-title="Trigonometri" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonésio" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ilo mw-list-item"><a href="https://ilo.wikipedia.org/wiki/Trigonometria" title="Trigonometria — ilocano" lang="ilo" hreflang="ilo" data-title="Trigonometria" data-language-autonym="Ilokano" data-language-local-name="ilocano" class="interlanguage-link-target"><span>Ilokano</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Trigonometrio" title="Trigonometrio — ido" lang="io" hreflang="io" data-title="Trigonometrio" 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/Hornafr%C3%A6%C3%B0i" title="Hornafræði — islandês" lang="is" hreflang="is" data-title="Hornafræði" data-language-autonym="Íslenska" data-language-local-name="islandês" 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/Trigonometria" title="Trigonometria — italiano" lang="it" hreflang="it" data-title="Trigonometria" data-language-autonym="Italiano" data-language-local-name="italiano" 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%E6%B3%95" title="三角法 — japonês" lang="ja" hreflang="ja" data-title="三角法" data-language-autonym="日本語" data-language-local-name="japonês" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Chriganamichri" title="Chriganamichri — Jamaican Creole English" lang="jam" hreflang="jam" data-title="Chriganamichri" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Trigonom%C3%A8tri" title="Trigonomètri — javanês" lang="jv" hreflang="jv" data-title="Trigonomètri" data-language-autonym="Jawa" data-language-local-name="javanês" class="interlanguage-link-target"><span>Jawa</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%90" title="ტრიგონომეტრია — georgiano" lang="ka" hreflang="ka" data-title="ტრიგონომეტრია" data-language-autonym="ქართული" data-language-local-name="georgiano" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kaa mw-list-item"><a href="https://kaa.wikipedia.org/wiki/Trigonometriya" title="Trigonometriya — kara-kalpak" lang="kaa" hreflang="kaa" data-title="Trigonometriya" data-language-autonym="Qaraqalpaqsha" data-language-local-name="kara-kalpak" class="interlanguage-link-target"><span>Qaraqalpaqsha</span></a></li><li class="interlanguage-link interwiki-kbp mw-list-item"><a href="https://kbp.wikipedia.org/wiki/K%C9%94l%C9%94m%C9%A9%C5%8B_naadozo_t%CA%8A_pilinzi_maz%CA%8A%CA%8A" title="Kɔlɔmɩŋ naadozo tʊ pilinzi mazʊʊ — Kabiye" lang="kbp" hreflang="kbp" data-title="Kɔlɔmɩŋ naadozo tʊ pilinzi mazʊʊ" data-language-autonym="Kabɩyɛ" data-language-local-name="Kabiye" class="interlanguage-link-target"><span>Kabɩyɛ</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.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%8F" title="Тригонометрия — cazaque" lang="kk" hreflang="kk" data-title="Тригонометрия" data-language-autonym="Қазақша" data-language-local-name="cazaque" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%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="ត្រីកោណមាត្រ — khmer" lang="km" hreflang="km" data-title="ត្រីកោណមាត្រ" 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%EB%B2%95" title="삼각법 — coreano" lang="ko" hreflang="ko" data-title="삼각법" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/S%C3%AAgo%C5%9Fenas%C3%AE" title="Sêgoşenasî — curdo" lang="ku" hreflang="ku" data-title="Sêgoşenasî" data-language-autonym="Kurdî" data-language-local-name="curdo" class="interlanguage-link-target"><span>Kurdî</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.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%8F" title="Тригонометрия — quirguiz" lang="ky" hreflang="ky" data-title="Тригонометрия" data-language-autonym="Кыргызча" data-language-local-name="quirguiz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Trigonometria" title="Trigonometria — latim" lang="la" hreflang="la" data-title="Trigonometria" data-language-autonym="Latina" data-language-local-name="latim" 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/Trigonometria" title="Trigonometria — Lingua Franca Nova" lang="lfn" hreflang="lfn" data-title="Trigonometria" 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-lg mw-list-item"><a href="https://lg.wikipedia.org/wiki/Essomampuyisatu_(Trigonometry)" title="Essomampuyisatu (Trigonometry) — ganda" lang="lg" hreflang="lg" data-title="Essomampuyisatu (Trigonometry)" data-language-autonym="Luganda" data-language-local-name="ganda" class="interlanguage-link-target"><span>Luganda</span></a></li><li class="interlanguage-link interwiki-li mw-list-item"><a href="https://li.wikipedia.org/wiki/Goniometrie" title="Goniometrie — limburguês" lang="li" hreflang="li" data-title="Goniometrie" data-language-autonym="Limburgs" data-language-local-name="limburguês" class="interlanguage-link-target"><span>Limburgs</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Trigonometria" title="Trigonometria — lombardo" lang="lmo" hreflang="lmo" data-title="Trigonometria" data-language-autonym="Lombard" data-language-local-name="lombardo" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lo mw-list-item"><a href="https://lo.wikipedia.org/wiki/%E0%BB%84%E0%BA%95%E0%BA%A1%E0%BA%B8%E0%BA%A1" title="ໄຕມຸມ — laosiano" lang="lo" hreflang="lo" data-title="ໄຕມຸມ" data-language-autonym="ລາວ" data-language-local-name="laosiano" class="interlanguage-link-target"><span>ລາວ</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Trigonometrija" title="Trigonometrija — lituano" lang="lt" hreflang="lt" data-title="Trigonometrija" data-language-autonym="Lietuvių" data-language-local-name="lituano" 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/Trigonometrija" title="Trigonometrija — letão" lang="lv" hreflang="lv" data-title="Trigonometrija" data-language-autonym="Latviešu" data-language-local-name="letão" 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%98%D0%B0" title="Тригонометрија — macedónio" lang="mk" hreflang="mk" data-title="Тригонометрија" data-language-autonym="Македонски" data-language-local-name="macedónio" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%A4%E0%B5%8D%E0%B4%B0%E0%B4%BF%E0%B4%95%E0%B5%8B%E0%B4%A3%E0%B4%AE%E0%B4%BF%E0%B4%A4%E0%B4%BF" title="ത്രികോണമിതി — malaiala" lang="ml" hreflang="ml" data-title="ത്രികോണമിതി" data-language-autonym="മലയാളം" data-language-local-name="malaiala" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.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" title="त्रिकोणमिती — marata" lang="mr" hreflang="mr" data-title="त्रिकोणमिती" data-language-autonym="मराठी" data-language-local-name="marata" 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/Trigonometri" title="Trigonometri — malaio" lang="ms" hreflang="ms" data-title="Trigonometri" data-language-autonym="Bahasa Melayu" data-language-local-name="malaio" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%90%E1%80%BC%E1%80%AE%E1%80%82%E1%80%AD%E1%80%AF%E1%80%94%E1%80%AD%E1%80%AF%E1%80%99%E1%80%B1%E1%80%90%E1%80%BC%E1%80%AE" title="တြီဂိုနိုမေတြီ — birmanês" lang="my" hreflang="my" data-title="တြီဂိုနိုမေတြီ" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="birmanês" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Trigonometrie" title="Trigonometrie — baixo-alemão" lang="nds" hreflang="nds" data-title="Trigonometrie" data-language-autonym="Plattdüütsch" data-language-local-name="baixo-alemão" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.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%A4%BF" title="त्रिकोणमिति — nepalês" lang="ne" hreflang="ne" data-title="त्रिकोणमिति" data-language-autonym="नेपाली" data-language-local-name="nepalês" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-new mw-list-item"><a href="https://new.wikipedia.org/wiki/%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%97%E0%A5%8B%E0%A4%A8%E0%A5%8B%E0%A4%AE%E0%A5%87%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF" title="त्रिगोनोमेत्रि — newari" lang="new" hreflang="new" data-title="त्रिगोनोमेत्रि" data-language-autonym="नेपाल भाषा" data-language-local-name="newari" class="interlanguage-link-target"><span>नेपाल भाषा</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Goniometrie" title="Goniometrie — neerlandês" lang="nl" hreflang="nl" data-title="Goniometrie" data-language-autonym="Nederlands" data-language-local-name="neerlandês" 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/Trigonometri" title="Trigonometri — norueguês nynorsk" lang="nn" hreflang="nn" data-title="Trigonometri" data-language-autonym="Norsk nynorsk" data-language-local-name="norueguês 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/Trigonometri" title="Trigonometri — norueguês bokmål" lang="nb" hreflang="nb" data-title="Trigonometri" data-language-autonym="Norsk bokmål" data-language-local-name="norueguês 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/Trigonometria" title="Trigonometria — occitano" lang="oc" hreflang="oc" data-title="Trigonometria" data-language-autonym="Occitan" data-language-local-name="occitano" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Rogkofa" title="Rogkofa — oromo" lang="om" hreflang="om" data-title="Rogkofa" data-language-autonym="Oromoo" data-language-local-name="oromo" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-or mw-list-item"><a href="https://or.wikipedia.org/wiki/%E0%AC%A4%E0%AD%8D%E0%AC%B0%E0%AC%BF%E0%AC%95%E0%AD%8B%E0%AC%A3%E0%AC%AE%E0%AC%BF%E0%AC%A4%E0%AC%BF" title="ତ୍ରିକୋଣମିତି — oriá" lang="or" hreflang="or" data-title="ତ୍ରିକୋଣମିତି" data-language-autonym="ଓଡ଼ିଆ" data-language-local-name="oriá" class="interlanguage-link-target"><span>ଓଡ଼ିଆ</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%A4%E0%A8%BF%E0%A8%95%E0%A9%8B%E0%A8%A3%E0%A8%AE%E0%A8%BF%E0%A8%A4%E0%A9%80" title="ਤਿਕੋਣਮਿਤੀ — panjabi" lang="pa" hreflang="pa" data-title="ਤਿਕੋਣਮਿਤੀ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="panjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Trygonometria" title="Trygonometria — polaco" lang="pl" hreflang="pl" data-title="Trygonometria" data-language-autonym="Polski" data-language-local-name="polaco" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Trigonometr%C3%ACa" title="Trigonometrìa — Piedmontese" lang="pms" hreflang="pms" data-title="Trigonometrìa" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D9%B9%D8%B1%DB%8C%DA%AF%D9%86%D9%88%D9%85%DB%8C%D9%B9%D8%B1%DB%8C" title="ٹریگنومیٹری — Western Punjabi" lang="pnb" hreflang="pnb" data-title="ٹریگنومیٹری" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/Wamp%27artupuykama" title="Wamp'artupuykama — quíchua" lang="qu" hreflang="qu" data-title="Wamp'artupuykama" data-language-autonym="Runa Simi" data-language-local-name="quíchua" class="interlanguage-link-target"><span>Runa Simi</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Trigonometrie" title="Trigonometrie — romeno" lang="ro" hreflang="ro" data-title="Trigonometrie" data-language-autonym="Română" data-language-local-name="romeno" 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%8F" title="Тригонометрия — russo" lang="ru" hreflang="ru" data-title="Тригонометрия" data-language-autonym="Русский" data-language-local-name="russo" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-rue mw-list-item"><a href="https://rue.wikipedia.org/wiki/%D0%A2%D1%80%D1%96%D2%91%D0%BE%D0%BD%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D1%96%D1%8F" title="Тріґонометрія — Rusyn" lang="rue" hreflang="rue" data-title="Тріґонометрія" data-language-autonym="Русиньскый" data-language-local-name="Rusyn" class="interlanguage-link-target"><span>Русиньскый</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Trigunomitr%C3%ACa" title="Trigunomitrìa — siciliano" lang="scn" hreflang="scn" data-title="Trigunomitrìa" data-language-autonym="Sicilianu" data-language-local-name="siciliano" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Trigonometry" title="Trigonometry — scots" lang="sco" hreflang="sco" data-title="Trigonometry" data-language-autonym="Scots" data-language-local-name="scots" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Trigonometrija" title="Trigonometrija — servo-croata" lang="sh" hreflang="sh" data-title="Trigonometrija" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="servo-croata" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-shi mw-list-item"><a href="https://shi.wikipedia.org/wiki/Askti%C9%A3mr" title="Asktiɣmr — tachelhit" lang="shi" hreflang="shi" data-title="Asktiɣmr" data-language-autonym="Taclḥit" data-language-local-name="tachelhit" class="interlanguage-link-target"><span>Taclḥit</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%BA" title="ත්‍රිකෝණමිතිය — cingalês" lang="si" hreflang="si" data-title="ත්‍රිකෝණමිතිය" data-language-autonym="සිංහල" data-language-local-name="cingalês" 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/Trigonometry" title="Trigonometry — Simple English" lang="en-simple" hreflang="en-simple" data-title="Trigonometry" 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/Trigonometria" title="Trigonometria — eslovaco" lang="sk" hreflang="sk" data-title="Trigonometria" data-language-autonym="Slovenčina" data-language-local-name="eslovaco" 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/Trigonometrija" title="Trigonometrija — esloveno" lang="sl" hreflang="sl" data-title="Trigonometrija" data-language-autonym="Slovenščina" data-language-local-name="esloveno" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Pimagonyonhatu" title="Pimagonyonhatu — shona" lang="sn" hreflang="sn" data-title="Pimagonyonhatu" data-language-autonym="ChiShona" data-language-local-name="shona" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Tirignoometeri" title="Tirignoometeri — somali" lang="so" hreflang="so" data-title="Tirignoometeri" data-language-autonym="Soomaaliga" data-language-local-name="somali" class="interlanguage-link-target"><span>Soomaaliga</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Trigonometria" title="Trigonometria — albanês" lang="sq" hreflang="sq" data-title="Trigonometria" data-language-autonym="Shqip" data-language-local-name="albanês" 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%D0%B0" title="Тригонометрија — sérvio" lang="sr" hreflang="sr" data-title="Тригонометрија" data-language-autonym="Српски / srpski" data-language-local-name="sérvio" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-stq mw-list-item"><a href="https://stq.wikipedia.org/wiki/Trigonometrie" title="Trigonometrie — Saterland Frisian" lang="stq" hreflang="stq" data-title="Trigonometrie" data-language-autonym="Seeltersk" data-language-local-name="Saterland Frisian" class="interlanguage-link-target"><span>Seeltersk</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Trigonometri" title="Trigonometri — sueco" lang="sv" hreflang="sv" data-title="Trigonometri" data-language-autonym="Svenska" data-language-local-name="sueco" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Trigonometria" title="Trigonometria — suaíli" lang="sw" hreflang="sw" data-title="Trigonometria" data-language-autonym="Kiswahili" data-language-local-name="suaíli" class="interlanguage-link-target"><span>Kiswahili</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" title="முக்கோணவியல் — tâmil" lang="ta" hreflang="ta" data-title="முக்கோணவியல்" data-language-autonym="தமிழ்" data-language-local-name="tâmil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%A4%E0%B1%8D%E0%B0%B0%E0%B0%BF%E0%B0%95%E0%B1%8B%E0%B0%A3%E0%B0%AE%E0%B0%BF%E0%B0%A4%E0%B0%BF" title="త్రికోణమితి — telugu" lang="te" hreflang="te" data-title="త్రికోణమితి" data-language-autonym="తెలుగు" data-language-local-name="telugu" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.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%8F" title="Тригонометрия — tajique" lang="tg" hreflang="tg" data-title="Тригонометрия" data-language-autonym="Тоҷикӣ" data-language-local-name="tajique" 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%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="ตรีโกณมิติ — tailandês" lang="th" hreflang="th" data-title="ตรีโกณมิติ" data-language-autonym="ไทย" data-language-local-name="tailandês" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tk mw-list-item"><a href="https://tk.wikipedia.org/wiki/Trigonometri%C3%BDa" title="Trigonometriýa — turcomano" lang="tk" hreflang="tk" data-title="Trigonometriýa" data-language-autonym="Türkmençe" data-language-local-name="turcomano" class="interlanguage-link-target"><span>Türkmençe</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Trigonometriya" title="Trigonometriya — tagalo" lang="tl" hreflang="tl" data-title="Trigonometriya" data-language-autonym="Tagalog" data-language-local-name="tagalo" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Trigonometri" title="Trigonometri — turco" lang="tr" hreflang="tr" data-title="Trigonometri" data-language-autonym="Türkçe" data-language-local-name="turco" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.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%8F" title="Тригонометрия — tatar" lang="tt" hreflang="tt" data-title="Тригонометрия" data-language-autonym="Татарча / tatarça" data-language-local-name="tatar" class="interlanguage-link-target"><span>Татарча / tatarça</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%D1%96%D1%8F" title="Тригонометрія — ucraniano" lang="uk" hreflang="uk" data-title="Тригонометрія" data-language-autonym="Українська" data-language-local-name="ucraniano" 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" title="مثلثیات — urdu" lang="ur" hreflang="ur" data-title="مثلثیات" data-language-autonym="اردو" data-language-local-name="urdu" 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/Trigonometriya" title="Trigonometriya — usbeque" lang="uz" hreflang="uz" data-title="Trigonometriya" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="usbeque" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://vec.wikipedia.org/wiki/Trigonometria" title="Trigonometria — Venetian" lang="vec" hreflang="vec" data-title="Trigonometria" data-language-autonym="Vèneto" data-language-local-name="Venetian" class="interlanguage-link-target"><span>Vèneto</span></a></li><li class="interlanguage-link interwiki-vep mw-list-item"><a href="https://vep.wikipedia.org/wiki/Trigonometrii" title="Trigonometrii — Veps" lang="vep" hreflang="vep" data-title="Trigonometrii" data-language-autonym="Vepsän kel’" data-language-local-name="Veps" class="interlanguage-link-target"><span>Vepsän kel’</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/L%C6%B0%E1%BB%A3ng_gi%C3%A1c" title="Lượng giác — vietnamita" lang="vi" hreflang="vi" data-title="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-wa mw-list-item"><a href="https://wa.wikipedia.org/wiki/Trigonometreye" title="Trigonometreye — valão" lang="wa" hreflang="wa" data-title="Trigonometreye" data-language-autonym="Walon" data-language-local-name="valão" class="interlanguage-link-target"><span>Walon</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Trigonometriya" title="Trigonometriya — waray" lang="war" hreflang="war" data-title="Trigonometriya" data-language-autonym="Winaray" data-language-local-name="waray" class="interlanguage-link-target"><span>Winaray</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%AD%A6" title="三角学 — wu" lang="wuu" hreflang="wuu" data-title="三角学" data-language-autonym="吴语" data-language-local-name="wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-xmf mw-list-item"><a href="https://xmf.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%90" title="ტრიგონომეტრია — Mingrelian" lang="xmf" hreflang="xmf" data-title="ტრიგონომეტრია" data-language-autonym="მარგალური" data-language-local-name="Mingrelian" class="interlanguage-link-target"><span>მარგალური</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%98%D7%A8%D7%99%D7%92%D7%90%D7%A0%D7%90%D7%9E%D7%A2%D7%98%D7%A8%D7%99%D7%A2" title="טריגאנאמעטריע — iídiche" lang="yi" hreflang="yi" data-title="טריגאנאמעטריע" data-language-autonym="ייִדיש" data-language-local-name="iídiche" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-yo mw-list-item"><a href="https://yo.wikipedia.org/wiki/Trigonom%E1%BA%B9%CC%81tr%C3%AC" title="Trigonomẹ́trì — ioruba" lang="yo" hreflang="yo" data-title="Trigonomẹ́trì" data-language-autonym="Yorùbá" data-language-local-name="ioruba" class="interlanguage-link-target"><span>Yorùbá</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E4%B8%89%E8%A7%92%E5%AD%A6" title="三角学 — chinês" lang="zh" hreflang="zh" data-title="三角学" data-language-autonym="中文" data-language-local-name="chinês" class="interlanguage-link-target"><span>中文</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-hoat" title="Saⁿ-kak-hoat — min nan" lang="nan" hreflang="nan" data-title="Saⁿ-kak-hoat" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="min nan" 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%AD%B8" title="三角學 — cantonês" lang="yue" hreflang="yue" data-title="三角學" data-language-autonym="粵語" data-language-local-name="cantonês" 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/Q8084#sitelinks-wikipedia" title="Editar hiperligações interlínguas" class="wbc-editpage">Editar hiperligações</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="Espaços nominais"> <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/Trigonometria" title="Ver a página de conteúdo [c]" accesskey="c"><span>Artigo</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Discuss%C3%A3o:Trigonometria" rel="discussion" title="Discussão sobre o conteúdo da página [t]" accesskey="t"><span>Discussão</span></a></li> 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href="/wiki/Especial:P%C3%A1ginas_afluentes/Trigonometria" title="Lista de todas as páginas que contêm hiperligações para esta [j]" accesskey="j"><span>Páginas afluentes</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Especial:Altera%C3%A7%C3%B5es_relacionadas/Trigonometria" rel="nofollow" title="Mudanças recentes nas páginas para as quais esta contém hiperligações [k]" accesskey="k"><span>Alterações relacionadas</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Wikipedia:Carregar_ficheiro" title="Carregar ficheiros [u]" accesskey="u"><span>Carregar ficheiro</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Especial:P%C3%A1ginas_especiais" title="Lista de páginas especiais [q]" accesskey="q"><span>Páginas especiais</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Trigonometria&oldid=67548647" title="Hiperligação permanente para esta revisão desta página"><span>Hiperligação permanente</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Trigonometria&action=info" title="Mais informações sobre esta página"><span>Informações da página</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Especial:Citar&page=Trigonometria&id=67548647&wpFormIdentifier=titleform" title="Informação sobre como citar esta página"><span>Citar esta página</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Especial:UrlShortener&url=https%3A%2F%2Fpt.wikipedia.org%2Fwiki%2FTrigonometria"><span>Obter URL encurtado</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Especial:QrCode&url=https%3A%2F%2Fpt.wikipedia.org%2Fwiki%2FTrigonometria"><span>Descarregar código 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"> Imprimir/exportar </div> <div 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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">Aspeto</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">mover para a barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">ocultar</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Origem: Wikipédia, a enciclopédia livre.</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="pt" dir="ltr"><p><b>Trigonometria</b> (do <a href="/wiki/L%C3%ADngua_grega" title="Língua grega">grego</a> <i>trigōnon</i> "<a href="/wiki/Tri%C3%A2ngulo" title="Triângulo">triângulo</a>" + <i>metron</i> "medida") é um ramo da <a href="/wiki/Matem%C3%A1tica" title="Matemática">matemática</a> que estuda as relações entre os comprimentos de 2 lados de um <a href="/wiki/Tri%C3%A2ngulo_ret%C3%A2ngulo" title="Triângulo retângulo">triângulo retângulo</a> (triângulo onde um dos <a href="/wiki/%C3%82ngulo" title="Ângulo">ângulos</a> mede 90 <a href="/wiki/Grau_(geometria)" title="Grau (geometria)">graus</a>), para diferentes valores de um dos seus ângulos agudos. A abordagem da trigonometria penetra outros campos da <a href="/wiki/Geometria" title="Geometria">geometria</a>, como o estudo de <a href="/wiki/Esfera" title="Esfera">esferas</a> usando a <a href="/wiki/Trigonometria_esf%C3%A9rica" title="Trigonometria esférica">trigonometria esférica</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span>[</span>1<span>]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span>[</span>2<span>]</span></a></sup> </p><p>A trigonometria tem aplicações importantes em vários ramos, tanto como na <a href="/wiki/Matem%C3%A1tica_pura" title="Matemática pura">matemática pura</a>, quanto na <a href="/wiki/Matem%C3%A1tica_aplicada" title="Matemática aplicada">matemática aplicada</a> e, consequentemente, nas <a href="/wiki/Ci%C3%AAncias_naturais" title="Ciências naturais">ciências naturais</a>. </p><p>Foram os <a href="/wiki/Babil%C3%B3nios" class="mw-redirect" title="Babilónios">babilónios</a> os primeiros a usá-la.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span>[</span>3<span>]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Círculo_trigonométrico"><span id="C.C3.ADrculo_trigonom.C3.A9trico"></span>Círculo trigonométrico</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Trigonometria&veaction=edit&section=1" title="Editar secção: Círculo trigonométrico" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Trigonometria&action=edit&section=1" title="Editar código-fonte da secção: Círculo trigonométrico"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="hatnote"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Magnifying_glass_01.svg/17px-Magnifying_glass_01.svg.png" decoding="async" width="17" height="17" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Magnifying_glass_01.svg/26px-Magnifying_glass_01.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Magnifying_glass_01.svg/34px-Magnifying_glass_01.svg.png 2x" data-file-width="663" data-file-height="659" /></span></span>Ver artigo principal: <a href="/wiki/C%C3%ADrculo_trigonom%C3%A9trico" class="mw-redirect" title="Círculo trigonométrico">Círculo trigonométrico</a></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Circulo_Trigonometrico_tangente.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Circulo_Trigonometrico_tangente.png/290px-Circulo_Trigonometrico_tangente.png" decoding="async" width="290" height="222" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Circulo_Trigonometrico_tangente.png/435px-Circulo_Trigonometrico_tangente.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Circulo_Trigonometrico_tangente.png/580px-Circulo_Trigonometrico_tangente.png 2x" data-file-width="585" data-file-height="447" /></a><figcaption>Círculo trigonométrico</figcaption></figure> <p>A Trigonometria (trigono: triângulo e metria: medidas) é o ramo da <a href="/wiki/Matem%C3%A1tica" title="Matemática">Matemática</a> que estuda a <i>proporção</i>, fixa, entre os comprimentos dos lados de um <i>triângulo retângulo</i>, para os diversos valores de um dos seus <a href="/wiki/%C3%82ngulo_agudo" class="mw-redirect" title="Ângulo agudo">ângulos agudos</a>. (Entre estes ângulos, os de 30º, 45º e 60º são denominados <a href="/w/index.php?title=%C3%82ngulo_not%C3%A1vel&action=edit&redlink=1" class="new" title="Ângulo notável (página não existe)">ângulos notáveis</a>.) As proporções entre os 3 lados dos triângulos retângulos são denominadas de seno, cosseno, tangente, cotangente, entre várias outras, dependendo dos lados considerados na proporção. </p><p>Já o Círculo Trigonométrico é um recurso criado para facilitar a visualização destas proporções entre os lados dos triângulos retângulos. Ele consiste em uma <a href="/wiki/Circunfer%C3%AAncia" title="Circunferência">circunferência</a> orientada de raio unitário, centrada na origem dos 2 <a href="/wiki/Eixos" class="mw-redirect mw-disambig" title="Eixos">eixos</a> de um <a href="/wiki/Plano_cartesiano" class="mw-redirect" title="Plano cartesiano">plano cartesiano</a> ortogonal, ou seja, um plano definido por duas retas perpendiculares entre si, ambas com o valor 0 (zero) no ponto onde elas se cortam. Existem dois sentidos de marcação dos arcos no círculo: o sentido positivo, chamado de <a href="/wiki/Anti-hor%C3%A1rio" class="mw-redirect" title="Anti-horário">anti-horário</a>, que se dá a partir da origem dos arcos até o lado terminal do ângulo correspondente ao arco; e o sentido negativo, ou <a href="/wiki/Sentido_hor%C3%A1rio" class="mw-redirect" title="Sentido horário">horário</a>, que se dá no sentido contrário ao anterior. </p> <div class="mw-heading mw-heading3"><h3 id="Seno">Seno</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Trigonometria&veaction=edit&section=2" title="Editar secção: Seno" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Trigonometria&action=edit&section=2" title="Editar código-fonte da secção: Seno"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="hatnote"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Magnifying_glass_01.svg/17px-Magnifying_glass_01.svg.png" decoding="async" width="17" height="17" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Magnifying_glass_01.svg/26px-Magnifying_glass_01.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Magnifying_glass_01.svg/34px-Magnifying_glass_01.svg.png 2x" data-file-width="663" data-file-height="659" /></span></span>Ver artigo principal: <a href="/wiki/Seno" title="Seno">Seno</a></div> <p>Dado um triângulo retângulo, o seno de um dos seus 2 ângulos agudos é a razão entre o comprimento do cateto oposto a este ângulo e o comprimento da <a href="/wiki/Hipotenusa" title="Hipotenusa">hipotenusa</a>, calculada, como toda razão, pela <a href="/wiki/Divis%C3%A3o" title="Divisão">divisão</a> de um valor pelo outro, a referência da razão. </p><p>No círculo trigonométrico, o seno de um ângulo qualquer pode ser visualizado na projeção do seu <a href="/wiki/Raio_(geometria)" title="Raio (geometria)">raio</a> (por definição igual a 1) sobre o eixo vertical. </p><p>Como o seno é esta projeção e o raio do círculo trigonométrico é igual a 1, segue que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x\in \mathbb {R} ,-1\leq \operatorname {sen} (x)\leq 1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>≤</mo> <mi>sen</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x\in \mathbb {R} ,-1\leq \operatorname {sen} (x)\leq 1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af7076c4c70d5e2da6b44ddd26d3916d293d5915" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.532ex; height:2.843ex;" alt="{\displaystyle \forall x\in \mathbb {R} ,-1\leq \operatorname {sen} (x)\leq 1,}"></span> ou seja, a imagem do seno é o intervalo fechado <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [-1,1].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [-1,1].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ca7e11ebbb9225cdba1609c129e46d0ec5101a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.108ex; height:2.843ex;" alt="{\displaystyle [-1,1].}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Cosseno">Cosseno</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Trigonometria&veaction=edit&section=3" title="Editar secção: Cosseno" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Trigonometria&action=edit&section=3" title="Editar código-fonte da secção: Cosseno"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="hatnote"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Magnifying_glass_01.svg/17px-Magnifying_glass_01.svg.png" decoding="async" width="17" height="17" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Magnifying_glass_01.svg/26px-Magnifying_glass_01.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Magnifying_glass_01.svg/34px-Magnifying_glass_01.svg.png 2x" data-file-width="663" data-file-height="659" /></span></span>Ver artigo principal: <a href="/wiki/Cosseno" title="Cosseno">Cosseno</a></div> <p>Dado um triângulo retângulo, o cosseno de um dos seus 2 ângulos agudos é a razão entre o comprimento do cateto adjacente a este ângulo e o comprimento da <a href="/wiki/Hipotenusa" title="Hipotenusa">hipotenusa</a>, calculada, como toda razão, pela divisão de um valor pelo outro, a referência da razão. </p><p>No círculo trigonométrico, o cosseno de um ângulo qualquer pode ser visualizado na projeção do seu raio (por definição igual a 1) sobre o eixo horizontal. </p><p>Como o cosseno é esta projeção, e o raio do círculo trigonométrico é igual a 1, segue que, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x\in \mathbb {R} ,-1\leq \operatorname {cos} (x)\leq 1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>≤</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x\in \mathbb {R} ,-1\leq \operatorname {cos} (x)\leq 1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6ee1cf7d471aed7ea4e8b76eb5498f6213a0346" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.402ex; height:2.843ex;" alt="{\displaystyle \forall x\in \mathbb {R} ,-1\leq \operatorname {cos} (x)\leq 1,}"></span> ou seja, a imagem do cosseno é o intervalo fechado <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [-1,1].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [-1,1].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ca7e11ebbb9225cdba1609c129e46d0ec5101a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.108ex; height:2.843ex;" alt="{\displaystyle [-1,1].}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Tangente">Tangente</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Trigonometria&veaction=edit&section=4" title="Editar secção: Tangente" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Trigonometria&action=edit&section=4" title="Editar código-fonte da secção: Tangente"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="hatnote"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Magnifying_glass_01.svg/17px-Magnifying_glass_01.svg.png" decoding="async" width="17" height="17" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Magnifying_glass_01.svg/26px-Magnifying_glass_01.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Magnifying_glass_01.svg/34px-Magnifying_glass_01.svg.png 2x" data-file-width="663" data-file-height="659" /></span></span>Ver artigo principal: <a href="/wiki/Tangente" title="Tangente">Tangente</a></div> <p>Dado um triângulo retângulo, a tangente de um dos seus 2 ângulos agudos é a razão entre o comprimento do cateto oposto a este ângulo e o comprimento do cateto adjacente a ele, calculada, como toda razão, pela divisão de um valor pelo outro, a referência da razão. </p><p>No círculo trigonométrico, o valor da tangente de um ângulo qualquer pode ser visualizado na reta <a href="/wiki/Vertical" class="mw-redirect" title="Vertical">vertical</a> que tangencia este círculo no ponto em que ele corta o eixo horizontal do lado direito. Nesta reta tangente ao círculo trigonométrico, o valor da tangente trigonométrica de qualquer ângulo é representado pelo segmento que vai do ponto em que ela corta o eixo horizontal até o ponto em que ela corta a reta que contém o raio do círculo trigonométrico para o ângulo considerado. Para avaliar este valor, deve-se compará-lo com o raio do círculo trigonométrico que, por definição, é igual a 1, de preferência quando este raio se encontra sobre a parte superior do eixo ortogonal vertical. Observe que, enquanto o seno e o cosseno são sempre menores do que o raio do círculo trigonométrico e, portanto, menores do que 1, a tangente trigonométrica pode ser tanto menor quanto maior do que 1. </p> <div class="mw-heading mw-heading2"><h2 id="Relações"><span id="Rela.C3.A7.C3.B5es"></span>Relações</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Trigonometria&veaction=edit&section=5" title="Editar secção: Relações" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Trigonometria&action=edit&section=5" title="Editar código-fonte da secção: Relações"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dois <a href="/wiki/Tri%C3%A2ngulo" title="Triângulo">triângulos</a> são ditos <i>semelhantes</i> se um pode ser obtido pela expansão uniforme do outro. Este é o caso se, e somente se, seus <a href="/wiki/%C3%82ngulo" title="Ângulo">ângulos</a> correspondentes são iguais. O fato crucial sobre triângulos semelhantes é que <i>os <a href="/wiki/Comprimento" title="Comprimento">comprimentos</a> de seus lados são <a href="/wiki/Proporcionalidade" title="Proporcionalidade">proporcionais</a></i>. Isto é, se o maior lado de um triângulo é duas vezes maior que o lado do triângulo similar, então o menor lado será também duas vezes maior que o menor lado do outro triângulo, e o comprimento do lado médio será duas vezes o valor do lado correspondente do outro triângulo. Assim, a <a href="/wiki/Raz%C3%A3o_(matem%C3%A1tica)" title="Razão (matemática)">razão</a> do maior lado e menor lado do primeiro triângulo será a mesma razão do maior lado e o menor lado do outro triângulo. </p><p>Usando estes fatos, definem-se as <a href="/wiki/Fun%C3%A7%C3%A3o_matem%C3%A1tica" class="mw-redirect" title="Função matemática">funções</a> trigonométricas, começando pelos <a href="/wiki/Tri%C3%A2ngulo_ret%C3%A2ngulo" title="Triângulo retângulo">triângulos retângulos</a> (triângulos com um <a href="/wiki/%C3%82ngulo_reto" title="Ângulo reto">ângulo reto</a> 90 <a href="/wiki/Grau_(geometria)" title="Grau (geometria)">graus</a> ou <a href="/wiki/Pi" title="Pi">π</a>/2 <a href="/wiki/Radiano" title="Radiano">radianos</a>). O maior lado em um triângulo qualquer é sempre o lado oposto ao maior ângulo e devido a soma dos ângulos de um triângulo ser 180 graus ou π radianos, o maior ângulo em um triângulo retângulo é o ângulo reto. O maior lado nesse triângulo, consequentemente, é o lado oposto ao ângulo reto, chamado de hipotenusa e os demais lados são chamados de <a href="/wiki/Cateto" title="Cateto">catetos</a>. </p><p>Dois triângulos retângulos que compartilham um segundo ângulo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> são necessariamente similares, e a proporção (ou razão) entre o comprimento do lado oposto a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> e o comprimento da hipotenusa será, portanto, a mesma nos dois triângulos. Este valor será um <a href="/wiki/N%C3%BAmero" title="Número">número</a> entre <a href="/wiki/Zero" class="mw-redirect" title="Zero">0</a> e <a href="/wiki/Um" title="Um">1</a> que depende apenas 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 \alpha .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/794850adc0db51d11a6d8cfa857538183424909c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.134ex; height:1.676ex;" alt="{\displaystyle \alpha .}"></span> Este número é chamado de <a href="/wiki/Seno" title="Seno">seno</a><sup id="cite_ref-ALGOS_4-0" class="reference"><a href="#cite_note-ALGOS-4"><span>[</span>4<span>]</span></a></sup> de A e é escrito como <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sen} \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sen</mi> <mo>⁡</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {sen} \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c0dd3174920ae321326506888e45c038f77e960" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.116ex; height:1.676ex;" alt="{\displaystyle \operatorname {sen} \alpha }"></span> Similarmente, pode-se definir : </p> <ul><li>o <a href="/wiki/Cosseno" title="Cosseno">cosseno</a> 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 \alpha :}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha :}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3dc633c1cfa93fe91f681273fcebb66d206e66bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.78ex; height:1.676ex;" alt="{\displaystyle \alpha :}"></span> é a proporção do comprimento do cateto adjacente ao ângulo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> em relação ao comprimento da hipotenusa;</li> <li>a <a href="/wiki/Tangente" title="Tangente">tangente</a> trigonométrica 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 \alpha :}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha :}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3dc633c1cfa93fe91f681273fcebb66d206e66bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.78ex; height:1.676ex;" alt="{\displaystyle \alpha :}"></span> é a proporção do comprimento do cateto oposto ao ângulo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> em relação ao comprimento do cateto adjacente;</li> <li>a <a href="/wiki/Cotangente" title="Cotangente">cotangente</a> 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 \alpha :}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha :}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3dc633c1cfa93fe91f681273fcebb66d206e66bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.78ex; height:1.676ex;" alt="{\displaystyle \alpha :}"></span> é a proporção do comprimento do cateto adjacente ao ângulo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> em relação ao comprimento do cateto oposto - é o inverso da tangente;</li> <li>a <a href="/wiki/Secante_(trigonometria)" title="Secante (trigonometria)">secante</a> trigonométrica 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 \alpha :}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha :}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3dc633c1cfa93fe91f681273fcebb66d206e66bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.78ex; height:1.676ex;" alt="{\displaystyle \alpha :}"></span> é a proporção do comprimento da hipotenusa em relação ao comprimento do cateto adjacente ao ângulo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> - é o inverso do cosseno;</li> <li>a <a href="/wiki/Cossecante" title="Cossecante">cossecante</a> 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 \alpha :}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha :}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3dc633c1cfa93fe91f681273fcebb66d206e66bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.78ex; height:1.676ex;" alt="{\displaystyle \alpha :}"></span> é a proporção do comprimento da hipotenusa em relação ao comprimento do cateto oposto ao ângulo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> - é o inverso do seno.</li></ul> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Tri%C3%A2ngulo_Ret%C3%A2ngulo.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/Tri%C3%A2ngulo_Ret%C3%A2ngulo.jpg/220px-Tri%C3%A2ngulo_Ret%C3%A2ngulo.jpg" decoding="async" width="220" height="156" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/Tri%C3%A2ngulo_Ret%C3%A2ngulo.jpg/330px-Tri%C3%A2ngulo_Ret%C3%A2ngulo.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/62/Tri%C3%A2ngulo_Ret%C3%A2ngulo.jpg/440px-Tri%C3%A2ngulo_Ret%C3%A2ngulo.jpg 2x" data-file-width="880" data-file-height="622" /></a><figcaption>Triângulo Retângulo</figcaption></figure><p>Seja o triângulo ABC, retângulo em A, têm-se, pelas definições de <a href="/wiki/Seno" title="Seno">seno</a> e <a href="/wiki/Cosseno" title="Cosseno">cosseno</a>: </p><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:C%C3%ADrculo_trigonom%C3%A9trico.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/C%C3%ADrculo_trigonom%C3%A9trico.svg/220px-C%C3%ADrculo_trigonom%C3%A9trico.svg.png" decoding="async" width="220" height="205" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/C%C3%ADrculo_trigonom%C3%A9trico.svg/330px-C%C3%ADrculo_trigonom%C3%A9trico.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/71/C%C3%ADrculo_trigonom%C3%A9trico.svg/440px-C%C3%ADrculo_trigonom%C3%A9trico.svg.png 2x" data-file-width="696" data-file-height="648" /></a><figcaption>Círculo trigonométrico, com a posição das funções seno, cosseno, tangente e cotangente explicitadas</figcaption></figure> <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 \operatorname {sen} \alpha ={{\text{cateto oposto}} \over {\text{hipotenusa}}}={{\text{c}} \over {\text{a}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sen</mi> <mo>⁡</mo> <mi>α</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mtext>cateto oposto</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>hipotenusa</mtext> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>a</mtext> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {sen} \alpha ={{\text{cateto oposto}} \over {\text{hipotenusa}}}={{\text{c}} \over {\text{a}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2573a1c3e693c612bb7fb28de16dfb27ee2eba8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.528ex; height:5.676ex;" alt="{\displaystyle \operatorname {sen} \alpha ={{\text{cateto oposto}} \over {\text{hipotenusa}}}={{\text{c}} \over {\text{a}}}}"></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 \qquad \cos \alpha ={{\text{cateto adjacente}} \over {\text{hipotenusa}}}={{\text{b}} \over {\text{a}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="2em" /> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mtext>cateto adjacente</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>hipotenusa</mtext> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mtext>b</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>a</mtext> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \qquad \cos \alpha ={{\text{cateto adjacente}} \over {\text{hipotenusa}}}={{\text{b}} \over {\text{a}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6e5f827e08aaba77e52b518dd28ee5d6248a0eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:35.196ex; height:5.843ex;" alt="{\displaystyle \qquad \cos \alpha ={{\text{cateto adjacente}} \over {\text{hipotenusa}}}={{\text{b}} \over {\text{a}}}}"></span></dd> <dd></dd></dl> <p>Estas são as mais importantes <a href="/wiki/Fun%C3%A7%C3%A3o_trigonom%C3%A9trica" title="Função trigonométrica">funções trigonométricas</a>; outras funções podem ser definidas tomando as razões dos outros lados de um triângulo retângulo, mas podem ser expressas em termos de <a href="/wiki/Seno" title="Seno">seno</a> e <a href="/wiki/Cosseno" title="Cosseno">cosseno</a>. São elas a <a href="/wiki/Tangente" title="Tangente">tangente</a>, <a href="/wiki/Secante_(trigonometria)" title="Secante (trigonometria)">secante</a>, <a href="/wiki/Cotangente" title="Cotangente">cotangente</a>, e <a href="/wiki/Cossecante" title="Cossecante">cossecante</a>. </p><p>No triângulo ABC acima, a tangente 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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> pode ser calculada através da razão entre o cateto oposto e o adjacente, como se observa em sua definição. Porém ela também pode ser obtida pela <a href="/wiki/Raz%C3%A3o_(matem%C3%A1tica)" title="Razão (matemática)">razão</a> entre seno e cosseno, da seguinte forma: </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 \alpha ={{\text{cateto oposto}} \over {\text{cateto adjacente}}}={{\text{a}}.\operatorname {sen} \alpha \over {\text{a}}.\cos \alpha }={\operatorname {sen} \alpha \over \cos \alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mi>α</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mtext>cateto oposto</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>cateto adjacente</mtext> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>a</mtext> </mrow> <mo>.</mo> <mi>sen</mi> <mo>⁡</mo> <mi>α</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>a</mtext> </mrow> <mo>.</mo> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sen</mi> <mo>⁡</mo> <mi>α</mi> </mrow> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan \alpha ={{\text{cateto oposto}} \over {\text{cateto adjacente}}}={{\text{a}}.\operatorname {sen} \alpha \over {\text{a}}.\cos \alpha }={\operatorname {sen} \alpha \over \cos \alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54bcbf4d666a20815796b1667e459b91c5f23fa2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:45.87ex; height:5.676ex;" alt="{\displaystyle \tan \alpha ={{\text{cateto oposto}} \over {\text{cateto adjacente}}}={{\text{a}}.\operatorname {sen} \alpha \over {\text{a}}.\cos \alpha }={\operatorname {sen} \alpha \over \cos \alpha }}"></span></dd></dl> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Secante_e_Cossecante_no_C%C3%ADrculo_Trigonom%C3%A9trico.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Secante_e_Cossecante_no_C%C3%ADrculo_Trigonom%C3%A9trico.jpg/220px-Secante_e_Cossecante_no_C%C3%ADrculo_Trigonom%C3%A9trico.jpg" decoding="async" width="220" height="191" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Secante_e_Cossecante_no_C%C3%ADrculo_Trigonom%C3%A9trico.jpg/330px-Secante_e_Cossecante_no_C%C3%ADrculo_Trigonom%C3%A9trico.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/93/Secante_e_Cossecante_no_C%C3%ADrculo_Trigonom%C3%A9trico.jpg/440px-Secante_e_Cossecante_no_C%C3%ADrculo_Trigonom%C3%A9trico.jpg 2x" data-file-width="880" data-file-height="765" /></a><figcaption>Secante e Cossecante no círculo trigonométrico unitário</figcaption></figure><p>Da mesma forma que a <a href="/wiki/Tangente" title="Tangente">tangente</a>, a <a href="/wiki/Cotangente" title="Cotangente">cotangente</a> 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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> pode ser definida como uma <a href="/wiki/Raz%C3%A3o_(matem%C3%A1tica)" title="Razão (matemática)">razão</a> entre catetos, nesse caso como a razão entre os catetos adjacente e oposto. Portanto, a cotangente pode ser expressa através da razão entre <a href="/wiki/Cosseno" title="Cosseno">cosseno</a> e <a href="/wiki/Seno" title="Seno">seno</a> como também sendo o <a href="/wiki/Inverso_multiplicativo" title="Inverso multiplicativo">inverso</a> da tangente. </p><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 \cot \alpha ={{\text{cateto adjacente}} \over {\text{cateto oposto}}}={{\text{a}}.\cos \alpha \over {\text{a}}.\operatorname {sen} \alpha }={\cos \alpha \over \operatorname {sen} \alpha }={1 \over \tan \alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cot</mi> <mo>⁡</mo> <mi>α</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mtext>cateto adjacente</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>cateto oposto</mtext> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>a</mtext> </mrow> <mo>.</mo> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>a</mtext> </mrow> <mo>.</mo> <mi>sen</mi> <mo>⁡</mo> <mi>α</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> </mrow> <mrow> <mi>sen</mi> <mo>⁡</mo> <mi>α</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>tan</mi> <mo>⁡</mo> <mi>α</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cot \alpha ={{\text{cateto adjacente}} \over {\text{cateto oposto}}}={{\text{a}}.\cos \alpha \over {\text{a}}.\operatorname {sen} \alpha }={\cos \alpha \over \operatorname {sen} \alpha }={1 \over \tan \alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f61115885b54ce0125fa7ad99273e6baed50e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:54.778ex; height:5.843ex;" alt="{\displaystyle \cot \alpha ={{\text{cateto adjacente}} \over {\text{cateto oposto}}}={{\text{a}}.\cos \alpha \over {\text{a}}.\operatorname {sen} \alpha }={\cos \alpha \over \operatorname {sen} \alpha }={1 \over \tan \alpha }}"></span> </p><p>A <a href="/wiki/Secante_(trigonometria)" title="Secante (trigonometria)">secante</a> e a <a href="/wiki/Cossecante" title="Cossecante">cossecante</a> ficam definidas por serem o <a href="/wiki/Inverso_multiplicativo" title="Inverso multiplicativo">inverso</a> do <a href="/wiki/Cosseno" title="Cosseno">cosseno</a> e do <a href="/wiki/Seno" title="Seno">seno</a>, respectivamente. Portanto, a secante e a cossecante 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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> podem ser expressas da seguinte forma: </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 \qquad \sec \alpha ={1 \over \cos \alpha }={{\text{hipotenusa}} \over {\text{cateto adjacente}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="2em" /> <mi>sec</mi> <mo>⁡</mo> <mi>α</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mtext>hipotenusa</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>cateto adjacente</mtext> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \qquad \sec \alpha ={1 \over \cos \alpha }={{\text{hipotenusa}} \over {\text{cateto adjacente}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e8053d3a0aadb3b672db30156077428a5791e0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:38.759ex; height:5.843ex;" alt="{\displaystyle \qquad \sec \alpha ={1 \over \cos \alpha }={{\text{hipotenusa}} \over {\text{cateto adjacente}}}}"></span></dd> <dd></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 \qquad \csc \alpha ={1 \over \operatorname {sen} \alpha }={{\text{hipotenusa}} \over {\text{cateto oposto}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="2em" /> <mi>csc</mi> <mo>⁡</mo> <mi>α</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>sen</mi> <mo>⁡</mo> <mi>α</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mtext>hipotenusa</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>cateto oposto</mtext> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \qquad \csc \alpha ={1 \over \operatorname {sen} \alpha }={{\text{hipotenusa}} \over {\text{cateto oposto}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51481dc4d46252ec8604ddef5fcffe4d9eb6e623" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:35.866ex; height:5.843ex;" alt="{\displaystyle \qquad \csc \alpha ={1 \over \operatorname {sen} \alpha }={{\text{hipotenusa}} \over {\text{cateto oposto}}}}"></span></dd> <dd></dd></dl> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Sundial_Warsaw.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/91/Sundial_Warsaw.jpg/220px-Sundial_Warsaw.jpg" decoding="async" width="220" height="177" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/91/Sundial_Warsaw.jpg/330px-Sundial_Warsaw.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/91/Sundial_Warsaw.jpg/440px-Sundial_Warsaw.jpg 2x" data-file-width="1007" data-file-height="808" /></a><figcaption>Relógio de sol</figcaption></figure><figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:UnitCircle.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4d/UnitCircle.png/220px-UnitCircle.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4d/UnitCircle.png/330px-UnitCircle.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4d/UnitCircle.png/440px-UnitCircle.png 2x" data-file-width="600" data-file-height="600" /></a><figcaption>O círculo unitário</figcaption></figure><p>Até então, as funções trigonométricas tem sido definidas por ângulos entre 0 e 90 graus (0 e π/2 radianos) apenas. Usando um <a href="/wiki/C%C3%ADrculo_unit%C3%A1rio" title="Círculo unitário">círculo unitário</a>, pode-se estendê-los para todos argumentos <a href="/wiki/Positivo" class="mw-disambig" title="Positivo">positivos</a> e <a href="/wiki/Negativo" class="mw-disambig" title="Negativo">negativos</a> (veja <a href="/wiki/Fun%C3%A7%C3%A3o_trigonom%C3%A9trica" title="Função trigonométrica">função trigonométrica</a>). </p><p>Uma vez que as funções seno e cosseno tenham sido tabuladas (ou computadas por uma <a href="/wiki/Calculadora" title="Calculadora">calculadora</a>), pode-se responder virtualmente todas questões sobre triângulos arbitrários, usando a <a href="/wiki/Lei_dos_senos" title="Lei dos senos">lei dos senos</a> e a <a href="/wiki/Lei_dos_cossenos" title="Lei dos cossenos">lei dos cossenos</a>. Estas leis podem ser usadas para calcular os ângulos restantes e lados de qualquer triângulo bem como dois lados e um ângulo ou dois ângulos e um lado ou três lados conhecidos. </p><p>Alguns matemáticos acreditam que a trigonometria foi originalmente inventada para calcular <a href="/wiki/Rel%C3%B3gio_de_sol" title="Relógio de sol">relógios de sol</a>, um tradicional exercício em antigos livros. Isto é também muito importante para a <a href="/wiki/Agrimensura" title="Agrimensura">agrimensura</a>. </p> <div style="clear:both;"></div> <div class="mw-heading mw-heading2"><h2 id="Teorema_de_Pitágoras"><span id="Teorema_de_Pit.C3.A1goras"></span>Teorema de Pitágoras</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Trigonometria&veaction=edit&section=6" title="Editar secção: Teorema de Pitágoras" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Trigonometria&action=edit&section=6" title="Editar código-fonte da secção: Teorema de Pitágoras"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="hatnote"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Magnifying_glass_01.svg/17px-Magnifying_glass_01.svg.png" decoding="async" width="17" height="17" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Magnifying_glass_01.svg/26px-Magnifying_glass_01.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Magnifying_glass_01.svg/34px-Magnifying_glass_01.svg.png 2x" data-file-width="663" data-file-height="659" /></span></span>Ver artigo principal: <a href="/wiki/Teorema_de_Pit%C3%A1goras" title="Teorema de Pitágoras">Teorema de Pitágoras</a></div> <p>O <a href="/wiki/Teorema" title="Teorema">teorema</a> de <a href="/wiki/Pit%C3%A1goras" title="Pitágoras">Pitágoras</a> estabelece que "A soma do <a href="/wiki/Quadrado_(aritm%C3%A9tica)" class="mw-redirect" title="Quadrado (aritmética)">quadrado</a> das medidas dos <a href="/wiki/Cateto" title="Cateto">catetos</a> (lados que formam o ângulo de 90°, neste caso c e <i>b</i>) é igual ao quadrado da medida da <a href="/wiki/Hipotenusa" title="Hipotenusa">hipotenusa</a> (lado oposto ao ângulo de 90°, ou a)". Assim: a² = b² + c² . Um <a href="/wiki/Corol%C3%A1rio" title="Corolário">corolário</a> desse teorema é que se os dois catetos forem de mesmo tamanho, a hipotenusa vale o produto do cateto pela <a href="/wiki/Raiz_quadrada" title="Raiz quadrada">raiz quadrada</a> de 2. </p> <div class="mw-heading mw-heading2"><h2 id="Aplicações_da_trigonometria"><span id="Aplica.C3.A7.C3.B5es_da_trigonometria"></span>Aplicações da trigonometria</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Trigonometria&veaction=edit&section=7" title="Editar secção: Aplicações da trigonometria" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Trigonometria&action=edit&section=7" title="Editar código-fonte da secção: Aplicações da trigonometria"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Existem diversas aplicações da trigonometria e das funções trigonométricas. Por exemplo, a técnica da triangulação é usada em <a href="/wiki/Astronomia" title="Astronomia">astronomia</a> para estimar a <a href="/wiki/Dist%C3%A2ncia" title="Distância">distância</a> das <a href="/wiki/Estrelas_pr%C3%B3ximas" class="mw-redirect" title="Estrelas próximas">estrelas próximas</a>; em <a href="/wiki/Geografia" title="Geografia">geografia</a> para estimar distâncias entre divisas e em <i>sistemas de navegação por <a href="/wiki/Sat%C3%A9lite_artificial" title="Satélite artificial">satélite</a></i>. As funções seno e cosseno são fundamentais para a teoria das <a href="/wiki/Fun%C3%A7%C3%A3o_peri%C3%B3dica" title="Função periódica">funções periódicas</a>, as quais descrevem as ondas sonoras e luminosas. </p><p>Campos que fazem uso da trigonometria ou funções trigonométricas incluem astronomia (especialmente para localização de posições aparentes de objetos celestes, em qual a trigonometria esférica é essencial) e portanto <a href="/wiki/Navega%C3%A7%C3%A3o" title="Navegação">navegação</a> (nos <a href="/wiki/Oceano" title="Oceano">oceanos</a>, em <a href="/wiki/Avi%C3%A3o" title="Avião">aviões</a>, e no <a href="/wiki/Espa%C3%A7o" class="mw-disambig" title="Espaço">espaço</a>), <a href="/wiki/Teoria_musical" title="Teoria musical">teoria musical</a>, <a href="/wiki/Ac%C3%BAstica" title="Acústica">acústica</a>, <a href="/wiki/%C3%93ptica" title="Óptica">óptica</a>, <a href="/wiki/Mercado_financeiro" title="Mercado financeiro">análise de mercado</a>, <a href="/wiki/Eletr%C3%B4nica" title="Eletrônica">eletrônica</a>, <a href="/wiki/Teoria_da_probabilidade" class="mw-redirect" title="Teoria da probabilidade">teoria da probabilidade</a>, <a href="/wiki/Estat%C3%ADstica" title="Estatística">estatística</a>, <a href="/wiki/Biologia" title="Biologia">biologia</a>, <i>equipamentos médicos</i> (por exemplo, <a href="/wiki/Tomografia_Computadorizada" class="mw-redirect" title="Tomografia Computadorizada">Tomografia Computadorizada</a> e <a href="/wiki/Ultrassom" title="Ultrassom">Ultrassom</a>), <a href="/wiki/Farm%C3%A1cia" title="Farmácia">farmácia</a>, <a href="/wiki/Qu%C3%ADmica" title="Química">química</a>, <a href="/wiki/Teoria_dos_n%C3%BAmeros" title="Teoria dos números">teoria dos números</a> (e portanto <a href="/wiki/Criptologia" title="Criptologia">criptologia</a>), <a href="/wiki/Sismologia" title="Sismologia">sismologia</a>, <a href="/wiki/Meteorologia" title="Meteorologia">meteorologia</a>, <a href="/wiki/Oceanografia" title="Oceanografia">oceanografia</a>, muitas das ciências <a href="/wiki/F%C3%ADsica" title="Física">físicas</a>, solos (inspeção e <a href="/wiki/Geodesia" class="mw-redirect" title="Geodesia">geodesia</a>), <a href="/wiki/Arquitetura" title="Arquitetura">arquitetura</a>, <a href="/wiki/Fon%C3%A9tica" title="Fonética">fonética</a>, <a href="/wiki/Economia" title="Economia">economia</a>, <a href="/wiki/Engenharia" title="Engenharia">engenharia</a>, <a href="/wiki/Computa%C3%A7%C3%A3o_gr%C3%A1fica" title="Computação gráfica">gráficos computadorizados</a>, <a href="/wiki/Cartografia" title="Cartografia">cartografia</a>, <a href="/wiki/Cristalografia" title="Cristalografia">cristalografia</a> e <a href="/wiki/Jogo_eletr%C3%B4nico" title="Jogo eletrônico">desenvolvimento de jogos</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Identidades_trigonométricas"><span id="Identidades_trigonom.C3.A9tricas"></span>Identidades trigonométricas</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Trigonometria&veaction=edit&section=8" title="Editar secção: Identidades trigonométricas" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Trigonometria&action=edit&section=8" title="Editar código-fonte da secção: Identidades trigonométricas"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="hatnote"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Magnifying_glass_01.svg/17px-Magnifying_glass_01.svg.png" decoding="async" width="17" height="17" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Magnifying_glass_01.svg/26px-Magnifying_glass_01.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Magnifying_glass_01.svg/34px-Magnifying_glass_01.svg.png 2x" data-file-width="663" data-file-height="659" /></span></span>Ver artigo principal: <a href="/wiki/Identidade_trigonom%C3%A9trica" title="Identidade trigonométrica">Identidade trigonométrica</a></div><p>Algumas <a href="/wiki/Equa%C3%A7%C3%A3o" title="Equação">equações</a> envolvendo <a href="/wiki/Fun%C3%A7%C3%A3o_trigonom%C3%A9trica" title="Função trigonométrica">funções trigonométricas</a> são verdade para todos os <a href="/wiki/%C3%82ngulo" title="Ângulo">ângulos</a> e são conhecidas como "identidades trigonométricas". Muitas expressam relações geométricas importantes. Por exemplo, as identidades Pitagoreanas são uma expressão do <a href="/wiki/Teorema_de_Pit%C3%A1goras" title="Teorema de Pitágoras">Teorema de Pitágoras</a>. Essas identidades podem ser encontradas, junto com suas demonstrações, na página <a href="/wiki/Identidade_trigonom%C3%A9trica" title="Identidade trigonométrica">Identidade Trigonométrica</a>. </p><div class="mw-heading mw-heading3"><h3 id="Identidades_triangulares">Identidades triangulares</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Trigonometria&veaction=edit&section=9" title="Editar secção: Identidades triangulares" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Trigonometria&action=edit&section=9" title="Editar código-fonte da secção: Identidades triangulares"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right mw-image-border" typeof="mw:File"><a href="/wiki/Ficheiro:Triangle55.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/9/92/Triangle55.png" decoding="async" width="300" height="157" class="mw-file-element" data-file-width="300" data-file-height="157" /></a><figcaption></figcaption></figure> <p>As identidades que se seguem referem-se a um triângulo com ângulos <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {A}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {A}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03abe3d30d22b974ebd7293a3a7067fb690b1825" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.487ex; height:3.343ex;" alt="{\displaystyle {\widehat {A}},}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo>^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/679a6a1e396ee7d5fa5856f4bdfa0122a95aa064" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.843ex;" alt="{\displaystyle {\widehat {B}}}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>C</mi> <mo>^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8253b043a217859a990c55e9420fa88e9a53d9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.019ex; width:1.809ex; height:3.009ex;" alt="{\displaystyle {\widehat {C}}}"></span> e lados de comprimentos <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f059f053fcf9f421b7c74362cf3bd5ed024e19d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.877ex; height:2.009ex;" alt="{\displaystyle a,}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae5e8f9eb465084d3a00a24026b80652b74ef58e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.654ex; height:2.009ex;" alt="{\displaystyle c,}"></span> como na figura ao lado. Repare que o lado oposto ao ângulo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo>^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5575024af93e0125d130908ca42fc3c347b800a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.019ex; width:1.859ex; height:3.009ex;" alt="{\displaystyle {\widehat {A}}}"></span> é o de comprimento <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f059f053fcf9f421b7c74362cf3bd5ed024e19d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.877ex; height:2.009ex;" alt="{\displaystyle a,}"></span> o lado oposto ao ângulo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo>^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/679a6a1e396ee7d5fa5856f4bdfa0122a95aa064" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.843ex;" alt="{\displaystyle {\widehat {B}}}"></span> é o de comprimento <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> e o lado oposto ao ângulo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>C</mi> <mo>^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8253b043a217859a990c55e9420fa88e9a53d9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.019ex; width:1.809ex; height:3.009ex;" alt="{\displaystyle {\widehat {C}}}"></span> é o de comprimento <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13b8d90daa52ffa8e5988459b6f10ef4d64ee5da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.654ex; height:1.676ex;" alt="{\displaystyle c.}"></span> </p> <div class="mw-heading mw-heading4"><h4 id="Lei_dos_senos">Lei dos senos</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Trigonometria&veaction=edit&section=10" title="Editar secção: Lei dos senos" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Trigonometria&action=edit&section=10" title="Editar código-fonte da secção: Lei dos senos"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="hatnote"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Magnifying_glass_01.svg/17px-Magnifying_glass_01.svg.png" decoding="async" width="17" height="17" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Magnifying_glass_01.svg/26px-Magnifying_glass_01.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Magnifying_glass_01.svg/34px-Magnifying_glass_01.svg.png 2x" data-file-width="663" data-file-height="659" /></span></span>Ver artigo principal: <a href="/wiki/Lei_dos_senos" title="Lei dos senos">Lei dos senos</a></div> <p>A <b><a href="/wiki/Lei_dos_senos" title="Lei dos senos">lei dos senos</a></b><sup id="cite_ref-INFOESC_5-0" class="reference"><a href="#cite_note-INFOESC-5"><span>[</span>5<span>]</span></a></sup> para um triângulo arbitrário diz: </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 {\operatorname {sen} {\widehat {A}}}{a}}={\frac {\operatorname {sen} {\widehat {B}}}{b}}={\frac {\operatorname {sen} {\widehat {C}}}{c}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sen</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo>^</mo> </mover> </mrow> </mrow> </mrow> <mi>a</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sen</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo>^</mo> </mover> </mrow> </mrow> </mrow> <mi>b</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sen</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>C</mi> <mo>^</mo> </mover> </mrow> </mrow> </mrow> <mi>c</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\operatorname {sen} {\widehat {A}}}{a}}={\frac {\operatorname {sen} {\widehat {B}}}{b}}={\frac {\operatorname {sen} {\widehat {C}}}{c}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ac9b366af5db194ad651f57a49f2464f7ed197c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:25.633ex; height:6.176ex;" alt="{\displaystyle {\frac {\operatorname {sen} {\widehat {A}}}{a}}={\frac {\operatorname {sen} {\widehat {B}}}{b}}={\frac {\operatorname {sen} {\widehat {C}}}{c}},}"></span></dd></dl> <p>ou equivalentemente: </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}{\operatorname {sen} {\widehat {A}}}}={\frac {b}{\operatorname {sen} {\widehat {B}}}}={\frac {c}{\operatorname {sen} {\widehat {C}}}}=2R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mrow> <mi>sen</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo>^</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mrow> <mi>sen</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo>^</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mrow> <mi>sen</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>C</mi> <mo>^</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a}{\operatorname {sen} {\widehat {A}}}}={\frac {b}{\operatorname {sen} {\widehat {B}}}}={\frac {c}{\operatorname {sen} {\widehat {C}}}}=2R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cb63c61c7307a82d8263562a201305ee769ec5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:31.011ex; height:6.176ex;" alt="{\displaystyle {\frac {a}{\operatorname {sen} {\widehat {A}}}}={\frac {b}{\operatorname {sen} {\widehat {B}}}}={\frac {c}{\operatorname {sen} {\widehat {C}}}}=2R}"></span></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Lei_dos_cossenos">Lei dos cossenos</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Trigonometria&veaction=edit&section=11" title="Editar secção: Lei dos cossenos" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Trigonometria&action=edit&section=11" title="Editar código-fonte da secção: Lei dos cossenos"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="hatnote"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Magnifying_glass_01.svg/17px-Magnifying_glass_01.svg.png" decoding="async" width="17" height="17" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Magnifying_glass_01.svg/26px-Magnifying_glass_01.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Magnifying_glass_01.svg/34px-Magnifying_glass_01.svg.png 2x" data-file-width="663" data-file-height="659" /></span></span>Ver artigo principal: <a href="/wiki/Lei_dos_cossenos" title="Lei dos cossenos">Lei dos cossenos</a></div> <p>A <b><a href="/wiki/Lei_dos_cossenos" title="Lei dos cossenos">lei dos cossenos</a></b> (também conhecida como <a href="/wiki/F%C3%B3rmula" class="mw-disambig" title="Fórmula">fórmula</a> dos cossenos) é uma extensão do <a href="/wiki/Teorema_de_Pit%C3%A1goras" title="Teorema de Pitágoras">teorema de Pitágoras</a> para triângulos arbitrários: </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 {\widehat {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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>C</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos {\widehat {C}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a1b8e956ce49fc5132e556641bbc76bb1957e72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.889ex; height:3.343ex;" alt="{\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos {\widehat {C}},}"></span></dd></dl> <p>ou equivalentemente: </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 {\widehat {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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>C</mi> <mo>^</mo> </mover> </mrow> </mrow> <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 {\widehat {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/65145ec4fdf1aed5f93a1bc1a05b9ad2ff60e937" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:21.301ex; height:5.843ex;" alt="{\displaystyle \cos {\widehat {C}}={\frac {a^{2}+b^{2}-c^{2}}{2ab}}}"></span></dd></dl> <p>o teorema de Pitágoras é um caso particular da Lei dos Cossenos, quando o cosseno de 90°é 0. </p> <div class="mw-heading mw-heading4"><h4 id="Lei_das_tangentes">Lei das tangentes</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Trigonometria&veaction=edit&section=12" title="Editar secção: Lei das tangentes" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Trigonometria&action=edit&section=12" title="Editar código-fonte da secção: Lei das tangentes"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="hatnote"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Magnifying_glass_01.svg/17px-Magnifying_glass_01.svg.png" decoding="async" width="17" height="17" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Magnifying_glass_01.svg/26px-Magnifying_glass_01.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Magnifying_glass_01.svg/34px-Magnifying_glass_01.svg.png 2x" data-file-width="663" data-file-height="659" /></span></span>Ver artigo principal: <a href="/wiki/Lei_das_tangentes" title="Lei das tangentes">Lei das tangentes</a></div> <p>A <b>lei das tangentes</b>: </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 \left[{\tfrac {1}{2}}(A+B)\right]}{\tan \left[{\tfrac {1}{2}}(A-B)\right]}}}"> <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> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>+</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> </mrow> <mrow> <mi>tan</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>−</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a+b}{a-b}}={\frac {\tan \left[{\tfrac {1}{2}}(A+B)\right]}{\tan \left[{\tfrac {1}{2}}(A-B)\right]}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21e586b944db5905c0bc192c18573ecb20e63503" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:24.952ex; height:7.843ex;" alt="{\displaystyle {\frac {a+b}{a-b}}={\frac {\tan \left[{\tfrac {1}{2}}(A+B)\right]}{\tan \left[{\tfrac {1}{2}}(A-B)\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 {\frac {b+c}{b-c}}={\frac {\tan \left[{\tfrac {1}{2}}(B+C)\right]}{\tan \left[{\tfrac {1}{2}}(B-C)\right]}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mo>+</mo> <mi>c</mi> </mrow> <mrow> <mi>b</mi> <mo>−</mo> <mi>c</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>tan</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mi>B</mi> <mo>+</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> </mrow> <mrow> <mi>tan</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mi>B</mi> <mo>−</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {b+c}{b-c}}={\frac {\tan \left[{\tfrac {1}{2}}(B+C)\right]}{\tan \left[{\tfrac {1}{2}}(B-C)\right]}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a56b78d0ecd84140de0ddaf336178524041bd55a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:24.753ex; height:7.843ex;" alt="{\displaystyle {\frac {b+c}{b-c}}={\frac {\tan \left[{\tfrac {1}{2}}(B+C)\right]}{\tan \left[{\tfrac {1}{2}}(B-C)\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 {\frac {a+c}{a-c}}={\frac {\tan \left[{\tfrac {1}{2}}(A+C)\right]}{\tan \left[{\tfrac {1}{2}}(A-C)\right]}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>+</mo> <mi>c</mi> </mrow> <mrow> <mi>a</mi> <mo>−</mo> <mi>c</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>tan</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>+</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> </mrow> <mrow> <mi>tan</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>−</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a+c}{a-c}}={\frac {\tan \left[{\tfrac {1}{2}}(A+C)\right]}{\tan \left[{\tfrac {1}{2}}(A-C)\right]}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a605092f58553f5b67a63b4e22823439384b271" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:24.964ex; height:7.843ex;" alt="{\displaystyle {\frac {a+c}{a-c}}={\frac {\tan \left[{\tfrac {1}{2}}(A+C)\right]}{\tan \left[{\tfrac {1}{2}}(A-C)\right]}}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Como_saber_o_ângulo_interno_de_um_triângulo_retângulo"><span id="Como_saber_o_.C3.A2ngulo_interno_de_um_tri.C3.A2ngulo_ret.C3.A2ngulo"></span>Como saber o ângulo interno de um triângulo retângulo</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Trigonometria&veaction=edit&section=13" title="Editar secção: Como saber o ângulo interno de um triângulo retângulo" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Trigonometria&action=edit&section=13" title="Editar código-fonte da secção: Como saber o ângulo interno de um triângulo retângulo"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Sendo: </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(A)={\frac {\operatorname {sen} (A)}{\cos(A)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sen</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan(A)={\frac {\operatorname {sen} (A)}{\cos(A)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d02cb54dbf51c7771d1cd8ecad7c0209db9a6550" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:17.64ex; height:6.509ex;" alt="{\displaystyle \tan(A)={\frac {\operatorname {sen} (A)}{\cos(A)}}}"></span></dd></dl> <p>em que: </p> <ul><li>Sen(A) é comprimento do cateto oposto e</li> <li>Cos(A) é o comprimento do cateto adjacente.</li></ul> <p>A tangente inversa: </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 ^{-1}(A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan ^{-1}(A)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a86722f8fb208ab3be18ce66add86d3f8dbc1561" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.245ex; height:3.176ex;" alt="{\displaystyle \tan ^{-1}(A)}"></span></dd></dl> <p>ou: </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 \arctan \left({\frac {\operatorname {sen}(A)}{\cos(A)}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sen</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arctan \left({\frac {\operatorname {sen}(A)}{\cos(A)}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9267de383def7be8eedbc04e9ce353b64a47cee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:17.517ex; height:6.509ex;" alt="{\displaystyle \arctan \left({\frac {\operatorname {sen}(A)}{\cos(A)}}\right)}"></span></dd></dl> <p>é o <a href="/wiki/%C3%82ngulo_interno" title="Ângulo interno">ângulo interno</a>. </p> <h2 id="Referências" style="cursor: help;" title="Esta seção foi configurada para não ser editável diretamente. Edite a página toda ou a seção anterior em vez disso."><span id="Refer.C3.AAncias"></span>Referências</h2> <div class="reflist" style="list-style-type: decimal;"><div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Linton, 2004</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><cite class="citation web"><a rel="nofollow" class="external text" href="http://www.brasilescola.com/matematica/trigonometria.htm">«Trigonometria»</a>. Brasil Escola<span class="reference-accessdate">. Consultado em 23 de fevereiro de 2012</span></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3ATrigonometria&rft.btitle=Trigonometria&rft.genre=unknown&rft.pub=Brasil+Escola&rft_id=http%3A%2F%2Fwww.brasilescola.com%2Fmatematica%2Ftrigonometria.htm&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></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="https://zap.aeiou.pt/afinal-os-babilonios-nao-gregos-os-primeiros-estudar-trigonometria-171538">Afinal, foram os babilónios (e não gregos) os primeiros a estudar trigonometria, por Lusa, ZAP.aeiou, 25 Agosto, 2017</a></span> </li> <li id="cite_note-ALGOS-4"><span class="mw-cite-backlink"><a href="#cite_ref-ALGOS_4-0">↑</a></span> <span class="reference-text"><cite class="citation web">Marques, Paulo. <a rel="nofollow" class="external text" href="http://www.algosobre.com.br/matematica/trigonometria-funcoes.html">«Trigonometria, Funções»</a>. algosobre<span class="reference-accessdate">. Consultado em 24 de fevereiro de 2012</span></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3ATrigonometria&rft.au=Marques%2C+Paulo&rft.btitle=Trigonometria%2C+Fun%C3%A7%C3%B5es&rft.genre=unknown&rft.pub=algosobre&rft_id=http%3A%2F%2Fwww.algosobre.com.br%2Fmatematica%2Ftrigonometria-funcoes.html&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-INFOESC-5"><span class="mw-cite-backlink"><a href="#cite_ref-INFOESC_5-0">↑</a></span> <span class="reference-text"><cite class="citation web">Ribeiro, Thyago. <a rel="nofollow" class="external text" href="http://www.infoescola.com/trigonometria/lei-dos-senos-e-dos-cossenos/">«Lei dos Senos e dos Cossenos»</a>. Infoescola<span class="reference-accessdate">. Consultado em 25 de fevereiro de 2012</span></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3ATrigonometria&rft.au=Ribeiro%2C+Thyago&rft.btitle=Lei+dos+Senos+e+dos+Cossenos&rft.genre=unknown&rft.pub=Infoescola&rft_id=http%3A%2F%2Fwww.infoescola.com%2Ftrigonometria%2Flei-dos-senos-e-dos-cossenos%2F&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Bibliografia">Bibliografia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Trigonometria&veaction=edit&section=14" title="Editar secção: Bibliografia" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Trigonometria&action=edit&section=14" title="Editar código-fonte da secção: Bibliografia"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Christopher M. Linton (2004). From Eudoxus to Einstein: A History of Mathematical Astronomy . Cambridge University Press.</li> <li>Weisstein, Eric W. "Trigonometric Addition Formulas". Wolfram <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>.</li> <li>Christopher Mark Linton (2006) "The Trigonometric... and His Live.</li> <li>IEZZI, Gelson. Fundamentos de Matemática Elementar, trigonometria. 8 ed. São Paulo: Atual, 2004.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Ligações_externas"><span id="Liga.C3.A7.C3.B5es_externas"></span>Ligações externas</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Trigonometria&veaction=edit&section=15" title="Editar secção: Ligações externas" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Trigonometria&action=edit&section=15" title="Editar código-fonte da secção: Ligações externas"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.matematica.pucminas.br/profs/web_walter/oficinas/oficina0822005.pdf">Oficina de trigonometria</a>Acessado em 24 de maio de 2008</li> <li><a rel="nofollow" class="external text" href="http://pessoal.sercomtel.com.br/matematica/trigonom/trigonometria.htm">Ensino de Trigonometria</a></li> <li><cite class="citation web"><a rel="nofollow" class="external text" href="http://www.3eck.org/triangle/en/calculator_simple.php">«Triangle calculator»</a> (em inglês)</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3ATrigonometria&rft.btitle=Triangle+calculator&rft.genre=unknown&rft_id=http%3A%2F%2Fwww.3eck.org%2Ftriangle%2Fen%2Fcalculator_simple.php&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><a rel="nofollow" class="external text" href="http://www.mat.ufg.br/cursos/rialma/docentes/jhilario/orientacoes/trigonometria.rtf">Melhoria do Ensino da Trigonometria</a></li> <li><a rel="nofollow" class="external text" href="http://www.ccmn.ufrj.br/curso/trabalhos/pdf/matematica-trabalhos/conceitos_tecnologias_funcoes/helena.pdf">Ensino de Identidades Trigonométricas</a></li></ul> <div role="navigation" class="navbox" aria-labelledby="Áreas_da_matemática" 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