{"id":65752,"date":"2022-04-06T10:56:45","date_gmt":"2022-04-06T13:56:45","guid":{"rendered":"https:\/\/vestibulares.estrategia.com\/portal\/?p=65752"},"modified":"2024-03-08T11:39:38","modified_gmt":"2024-03-08T14:39:38","slug":"funcoes-trigonometricas","status":"publish","type":"post","link":"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/funcoes-trigonometricas\/","title":{"rendered":"Fun\u00e7\u00f5es trigonom\u00e9tricas: f\u00f3rmulas, gr\u00e1ficos e aplica\u00e7\u00f5es"},"content":{"rendered":"<p class=\"wp-block-paragraph\">As <span id=\"urn:local-annotation-645938\" class=\"textannotation disambiguated\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/funcoes-trigonometricas\">fun&ccedil;&otilde;es trigonom&eacute;tricas<\/span> surgem a partir do estudo dos &acirc;ngulos, que s&atilde;o medidos em graus para a <span id=\"urn:enhancement-4712f875-db5e-4d90-9c8a-c3e9d235a966\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/geometria\">geometria<\/span> e s&atilde;o quantificadas em radianos na <span id=\"urn:enhancement-98981ab1-8cff-46e6-8c59-7291a80d767e\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/algebra\">&aacute;lgebra<\/span>, que &eacute; uma rela&ccedil;&atilde;o <span id=\"urn:enhancement-48c2b907-fb41-4858-a7aa-69fceaf07e13\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/matematica\">matem&aacute;tica<\/span> para realizar c&aacute;lculos em <span id=\"urn:enhancement-52c2d45e-7887-4f20-881a-14e3d0ed628d\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/trigonometria\">trigonometria<\/span>.<p class=\"wp-block-paragraph\">Para al&eacute;m de uma simples opera&ccedil;&atilde;o do vestibular, as fun&ccedil;&otilde;es trigonom&eacute;tricas s&atilde;o importantes para entender o comportamento de ondas, as possibilidades m&aacute;ximas e m&iacute;nimas de uma grandeza, entre outras informa&ccedil;&otilde;es que podem ser acessadas com essa ferramenta.<\/p><p class=\"wp-block-paragraph\">Com o objetivo de apresentar e explicar o tema, o Estrat&eacute;gia Vestibulares preparou este artigo. Nele voc&ecirc; encontrar&aacute; um resumo com as principais propriedades das fun&ccedil;&otilde;es seno, <span id=\"urn:enhancement-0e7a1ee9-dbc4-4258-9ece-b99fbbb41607\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cosseno\">cosseno<\/span> e <span id=\"urn:enhancement-594464c2-68f9-4b9d-a38e-f579cbc015f8\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/tangente\">tangente<\/span>, como dom&iacute;nio, imagem e gr&aacute;ficos. Al&eacute;m disso, poder&aacute; acompanhar uma resolu&ccedil;&atilde;o de exerc&iacute;cios sobre o tema. Acompanhe agora!<\/p><div id=\"ez-toc-container\" class=\"ez-toc-v2_0_86 counter-hierarchy ez-toc-counter ez-toc-transparent ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\"><p class=\"ez-toc-title\" style=\"cursor:inherit\">Navegue pelo conte\u00fado<\/p>\n<\/div><nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/funcoes-trigonometricas\/#O-que-sao-funcoes-trigonometricas\" >O que s&atilde;o fun&ccedil;&otilde;es trigonom&eacute;tricas?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/funcoes-trigonometricas\/#Funcoes-trigonometricas-seno\" >Fun&ccedil;&otilde;es trigonom&eacute;tricas: seno&nbsp;<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/funcoes-trigonometricas\/#Funcoes-trigonometricas-cosseno\" >Fun&ccedil;&otilde;es trigonom&eacute;tricas:  cosseno&nbsp;<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/funcoes-trigonometricas\/#Funcoes-trigonometricas-tangente\" >Fun&ccedil;&otilde;es trigonom&eacute;tricas: tangente&nbsp;<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/funcoes-trigonometricas\/#Questoes-sobre-funcoes-trigonometricas\" >Quest&otilde;es sobre fun&ccedil;&otilde;es trigonom&eacute;tricas<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/funcoes-trigonometricas\/#Aprenda-mais-matematica-com-a-Coruja\" >Aprenda mais matem&aacute;tica com a Coruja!<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/funcoes-trigonometricas\/#Veja-tambem\" >Veja tamb&eacute;m:<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\" id=\"h-o-que-sao-funcoes-trigonometricas\"><span class=\"ez-toc-section\" id=\"O-que-sao-funcoes-trigonometricas\"><\/span>O que s&atilde;o fun&ccedil;&otilde;es trigonom&eacute;tricas?<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\">As fun&ccedil;&otilde;es trigonom&eacute;tricas s&atilde;o aquelas que admitem &acirc;ngulos como parte do c&aacute;lculo. Na <span id=\"urn:enhancement-80cbbb1b-49e9-492b-9ac3-4400fe6bd258\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/algebra\">&aacute;lgebra<\/span>, geralmente, esses &acirc;ngulos est&atilde;o convertidos na unidade dos radianos, que s&atilde;o obtidos a partir de um <span id=\"urn:enhancement-ee5f1528-0c9b-4427-b788-5f0fe5a92458\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/circulo\">c&iacute;rculo<\/span> padr&atilde;o, como voc&ecirc; pode observar na imagem abaixo:<\/p><div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/WLKdX0oF6oY4JeTXX3hsCLMMznZyA3Cq3PDScTKGvbNlEyOKy2U0mg1en-G6-0ZtzIf-6QWttTmRPw3kIvAsvRM91huFIJd9F-4NJE2WTV6wS7r6Nq--9yyX838S2QWii8Vxd21w\" alt=\"Fun&ccedil;&otilde;es trigonom&eacute;tricas\"><figcaption class=\"wp-element-caption\">Imagem: Adapta&ccedil;&atilde;o\/Wikimedia<\/figcaption><\/figure>\n<\/div><p class=\"wp-block-paragraph\">Perceba que existe uma correspond&ecirc;ncia entre os &acirc;ngulos do <span id=\"urn:enhancement-edc166e3-94d3-4749-bdc7-9460589d1f66\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/circulo\">c&iacute;rculo<\/span> com os radianos. Ent&atilde;o, voc&ecirc; manter em mente que 180&ordm; = ? radianos. A partir dessa correspond&ecirc;ncia, qualquer &acirc;ngulo pode ser convertido em radianos, se utilizada a <span id=\"urn:enhancement-f2936898-92d4-4afd-9f88-3737486ab334\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/regra-de-tres\">regra de tr&ecirc;s<\/span>.<\/p><p class=\"has-text-align-center wp-block-paragraph\">180 &ordm; &mdash;&mdash;&ndash; ? radianos<br>60&ordm;&nbsp; &nbsp; &ndash;&mdash;&mdash;- x&nbsp;<br>x.180=60.?<br>x.3=?<br>x=?\/3, ou seja, 60&ordm;=?\/3 radianos<\/p><p class=\"wp-block-paragraph\">A partir de agora, observe cada uma das linhas, horizontal e vertical, que comp&otilde;em o <span id=\"urn:enhancement-27cedd3e-88f8-4b65-a2ce-80e50b288a1e\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/ciclo-trigonometrico-2\">c&iacute;rculo trigonom&eacute;trico<\/span>. Entende-se que seu <span id=\"urn:enhancement-dc2b3818-c4a5-4d51-9fcd-2a170b19ff14\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/raio\">raio<\/span> =1 e, a partir disso:<\/p><ul class=\"wp-block-list\">\n<li>O seno &eacute; representado pelo tamanho do lado vertical de um tri&acirc;ngulo ret&acirc;ngulo cujo &acirc;ngulo esteja marcado no c&iacute;rculo;<\/li>\n\n\n\n<li>O <span id=\"urn:enhancement-64d77799-81cf-4e66-881e-3d67090703aa\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cosseno\">cosseno<\/span> &eacute; o lado horizontal do tri&acirc;ngulo ret&acirc;ngulo do &acirc;ngulo em quest&atilde;o; e&nbsp;<\/li>\n\n\n\n<li>A <span id=\"urn:enhancement-da8068a3-09a6-42a4-920e-bf679293a435\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/tangente\">tangente<\/span> do &acirc;ngulo &eacute; dada pela proje&ccedil;&atilde;o da reta que marca o &acirc;ngulo em 90&ordm; com o di&acirc;metro horizontal.&nbsp;<\/li>\n<\/ul><div class=\"wp-block-image\">\n<figure class=\"aligncenter is-resized\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/WD8qFo9h8X5vvofbZdrq1ZQj39DGBHqcFCUOxijslRdztHfinJe4nGBtVEmZu2SrNmul1M84RWnmt8n_qG5_fx2YylCS4nNwmnxdddTBF7NmqkGui3JRCQrrfgohCDkNh9U8vCp-\" alt=\"Fun&ccedil;&otilde;es trigonom&eacute;tricas - seno, cosseno e tangente\" style=\"width:562px;height:545px\"><figcaption class=\"wp-element-caption\"> Imagem: Adapta&ccedil;&atilde;o\/Wikimedia <\/figcaption><\/figure>\n<\/div><p class=\"wp-block-paragraph\">Podemos observar que, por padr&atilde;o, o valor das <span id=\"urn:enhancement-eefe4c8d-a3b4-4f09-834c-fb2dadcd5670\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/retas\">retas<\/span> no <span id=\"urn:enhancement-914d7ff9-fb1b-4120-ab83-69a6c20afd35\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/ciclo-trigonometrico-2\">c&iacute;rculo trigonom&eacute;trico<\/span> varia de -1 a 1. De <span id=\"urn:enhancement-8b06b4d5-3a4d-4cef-96a1-0dbaded53618\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/geometria\">forma<\/span> que se assemelha a colocar a reta real com origem no centro do <span id=\"urn:enhancement-bab8190c-936c-494d-9bab-663633b1ab5a\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/circulo\">c&iacute;rculo<\/span> e seguir a contagem a partir disso. Por essa raz&atilde;o, existem &acirc;ngulos com seno negativo ou positivo, o que tamb&eacute;m acontece nas fun&ccedil;&otilde;es <span id=\"urn:enhancement-4b3f1bb8-7853-40ed-9a11-055a2fe43e58\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cosseno\">cosseno<\/span> e <span id=\"urn:enhancement-05f7885c-d751-4827-845d-3f75b19a6791\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/tangente\">tangente<\/span>.&nbsp;<\/p><p class=\"wp-block-paragraph\">Alguns valores de seno, <span id=\"urn:enhancement-6841b54f-447a-4e6c-ba72-ad637bd49132\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cosseno\">cosseno<\/span> e <span id=\"urn:enhancement-89af7a25-8d3e-4517-abf0-a326f0952991\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/tangente\">tangente<\/span> aparecem com muita frequ&ecirc;ncia nos vestibulares, por isso, foram criadas algumas regras que facilitam decor&aacute;-los.&nbsp;<\/p><p class=\"wp-block-paragraph\">Primeiramente, &eacute; necess&aacute;rio desenhar uma tabela de lado 4&times;4, como a que &eacute; mostrada a seguir. Depois, vamos completar todos os numeradores, de cima para baixo, da esquerda para a direita nas duas primeiras colunas com a sequ&ecirc;ncia 1,2,3,3,2,1.<\/p><figure class=\"wp-block-table\"><table><tbody><tr><td><\/td><td class=\"has-text-align-center\" data-align=\"center\"><\/td><td class=\"has-text-align-center\" data-align=\"center\"><\/td><td><\/td><\/tr><tr><td><\/td><td class=\"has-text-align-center\" data-align=\"center\"><strong>1<\/strong><\/td><td class=\"has-text-align-center\" data-align=\"center\"><strong>3<\/strong><\/td><td><\/td><\/tr><tr><td><\/td><td class=\"has-text-align-center\" data-align=\"center\"><strong>2<\/strong><\/td><td class=\"has-text-align-center\" data-align=\"center\"><strong>2<\/strong><\/td><td><\/td><\/tr><tr><td><\/td><td class=\"has-text-align-center\" data-align=\"center\"><strong>3<\/strong><\/td><td class=\"has-text-align-center\" data-align=\"center\"><strong>1<\/strong><\/td><td><\/td><\/tr><\/tbody><\/table><\/figure><p class=\"wp-block-paragraph\">Ap&oacute;s isso, adicionamos 2 como denominador em todos esse n&uacute;meros. E, agora, colocamos tamb&eacute;m ra&iacute;zes em todos numeradores de valor 2 ou 3.&nbsp;<\/p><figure class=\"wp-block-table\"><table><tbody><tr><td><\/td><td class=\"has-text-align-center\" data-align=\"center\"><\/td><td class=\"has-text-align-center\" data-align=\"center\"><\/td><td><\/td><\/tr><tr><td><\/td><td class=\"has-text-align-center\" data-align=\"center\">1<strong>\/2<\/strong><\/td><td class=\"has-text-align-center\" data-align=\"center\"><strong>&radic;<\/strong>3<strong>\/2<\/strong><\/td><td><\/td><\/tr><tr><td><\/td><td class=\"has-text-align-center\" data-align=\"center\"><strong>&radic;<\/strong>2<strong>\/2<\/strong><\/td><td class=\"has-text-align-center\" data-align=\"center\"><strong>&radic;<\/strong>2<strong>\/2<\/strong><\/td><td><\/td><\/tr><tr><td><\/td><td class=\"has-text-align-center\" data-align=\"center\"><strong>&radic;<\/strong>3<strong>\/2<\/strong><\/td><td class=\"has-text-align-center\" data-align=\"center\">1<strong>\/2<\/strong><\/td><td><\/td><\/tr><\/tbody><\/table><\/figure><p class=\"wp-block-paragraph\">Por fim, completa-se a &uacute;ltima coluna da tabela com a sequ&ecirc;ncia de &radic;3\/3, 1, &radic;3.<\/p><figure class=\"wp-block-table\"><table><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\"><\/td><td class=\"has-text-align-center\" data-align=\"center\"><\/td><td class=\"has-text-align-center\" data-align=\"center\"><\/td><td class=\"has-text-align-center\" data-align=\"center\"><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><\/td><td class=\"has-text-align-center\" data-align=\"center\">1\/2<\/td><td class=\"has-text-align-center\" data-align=\"center\">&radic;3\/2<\/td><td class=\"has-text-align-center\" data-align=\"center\"><strong>&radic;3\/3<\/strong><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><\/td><td class=\"has-text-align-center\" data-align=\"center\">&radic;2\/2<\/td><td class=\"has-text-align-center\" data-align=\"center\">&radic;2\/2<\/td><td class=\"has-text-align-center\" data-align=\"center\"><strong>1<\/strong><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><\/td><td class=\"has-text-align-center\" data-align=\"center\">&radic;3\/2<\/td><td class=\"has-text-align-center\" data-align=\"center\">1\/2<\/td><td class=\"has-text-align-center\" data-align=\"center\"><strong>&radic;3<\/strong><\/td><\/tr><\/tbody><\/table><\/figure><p class=\"wp-block-paragraph\">Em ordem de cima para baixo temos os &acirc;ngulos 30&ordm;,45&ordm; e 60&ordm;, descritos com seus senos, cossenos e tangentes nas colunas (da esquerda para direita), veja:<\/p><figure class=\"wp-block-table\"><table><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\"><\/td><td class=\"has-text-align-center\" data-align=\"center\"><strong><span id=\"urn:enhancement-782728b0-db97-4711-9a31-289abca0d8f0\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/sen\">sen<\/span><\/strong><\/td><td class=\"has-text-align-center\" data-align=\"center\"><strong><span id=\"urn:enhancement-fc481071-de10-4bc1-833e-6a8c7889a0dc\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cosseno\">cos<\/span><\/strong><\/td><td class=\"has-text-align-center\" data-align=\"center\"><strong>tg<\/strong><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><strong>30&ordm;<\/strong><\/td><td class=\"has-text-align-center\" data-align=\"center\">1\/2<\/td><td class=\"has-text-align-center\" data-align=\"center\">&radic;3\/2<\/td><td class=\"has-text-align-center\" data-align=\"center\">&radic;3\/3<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><strong>45&ordm;<\/strong><\/td><td class=\"has-text-align-center\" data-align=\"center\">&radic;2\/2<\/td><td class=\"has-text-align-center\" data-align=\"center\">&radic;2\/2<\/td><td class=\"has-text-align-center\" data-align=\"center\">1<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><strong>60&ordm;<\/strong><\/td><td class=\"has-text-align-center\" data-align=\"center\">&radic;3\/2<\/td><td class=\"has-text-align-center\" data-align=\"center\">1\/2<\/td><td class=\"has-text-align-center\" data-align=\"center\">&radic;3<\/td><\/tr><\/tbody><\/table><\/figure><p class=\"wp-block-paragraph\">A melhor <span id=\"urn:enhancement-467553cd-303f-488f-bdd3-b85dfdaed37b\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/geometria\">forma<\/span> de decorar esse processo &eacute; com a par&oacute;dia de jingle bells:&nbsp;<\/p><p class=\"has-text-align-center wp-block-paragraph\">&ldquo;um, dois, tr&ecirc;s, tr&ecirc;s, dois, um, tudo sobre dois.<br>A raiz vai no tr&ecirc;s e tamb&eacute;m no dois.<br>A <span id=\"urn:enhancement-67dc1d95-d668-4f39-ba98-bc8981a3174e\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/tangente\">tangente<\/span> &eacute; diferente vejam s&oacute; voc&ecirc;s:<br>Raiz de tr&ecirc;s sobre tr&ecirc;s, um, raiz de tr&ecirc;s.&rdquo;<\/p><h2 class=\"wp-block-heading\" id=\"h-funcoes-trigonometricas-seno-nbsp\"><span class=\"ez-toc-section\" id=\"Funcoes-trigonometricas-seno\"><\/span> Fun&ccedil;&otilde;es trigonom&eacute;tricas: seno&nbsp;<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\">Como j&aacute; foi dito na <span id=\"urn:enhancement-12c65729-0eae-4642-8115-ac8059159125\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/sen\">fun&ccedil;&atilde;o seno<\/span>, &eacute; considerado o valor do lado vertical do tri&acirc;ngulo ret&acirc;ngulo, no <span id=\"urn:enhancement-1ca1f1fe-ef93-412d-9c81-0c2b66564079\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/ciclo-trigonometrico-2\">c&iacute;rculo trigonom&eacute;trico<\/span>. Assim, quando um &acirc;ngulo de projeta para cima, seu seno &eacute; positivo. Entretanto, se ele est&aacute; marcado na parte de baixo do <span id=\"urn:enhancement-92921f2f-b27a-4952-b8d0-0d5090661480\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/circulo\">c&iacute;rculo<\/span>, seu seno ser&aacute; negativo, como mostra o diagrama abaixo.<\/p><div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/vK4n7J_pCJiTCPVDlyHtDGCaF21KI1fCyZ8nYJ4HRfIFj7MwrI9fOB1SIaBNc74t9yLWIT-Eb7yO7stWAbkA1ko9cgPmivvfXwUpT_gFC-TEZJoOdcV2gFz51MJ4082LmTe-ji5T\" alt=\"Fun&ccedil;&otilde;es trigonom&eacute;tricas - seno\"><\/figure>\n<\/div><p class=\"wp-block-paragraph\">A lei de forma&ccedil;&atilde;o dessa fun&ccedil;&atilde;o &eacute; dada por f(x) = <span id=\"urn:enhancement-3568b2d2-ac3e-4cf9-8d4b-927f34365466\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/sen\">sen<\/span>(x). A partir disso, podem ser feitas algumas considera&ccedil;&otilde;es, em termos de quais intervalos reais abrangem os valores de x e de f(x), acompanhe nos pr&oacute;ximos t&oacute;picos.<\/p><h3 class=\"wp-block-heading\" id=\"h-dominio-e-imagem-da-funcao-seno\">Dom&iacute;nio e imagem da fun&ccedil;&atilde;o seno<\/h3><p class=\"wp-block-paragraph\">O dom&iacute;nio das fun&ccedil;&otilde;es trigonom&eacute;tricas diz respeito ao intervalo de varia&ccedil;&atilde;o do &acirc;ngulo x. No caso de f(x) = <span id=\"urn:enhancement-7e2585ba-5f7e-43fe-8948-445cb178d3fd\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/sen\">sen<\/span>(x), podemos observar que qualquer &acirc;ngulo possuir&aacute; um valor correspondente em radianos. Assim, x pode ser encontrado em todos os conjuntos dos n&uacute;meros reais, de maneira que o dom&iacute;nio de <span id=\"urn:enhancement-fbfd06b0-1a05-4c66-bd5e-dd2bf396cc93\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/sen\">sen<\/span>(x) &eacute; tal que f: R &rarr; R.<\/p><p class=\"wp-block-paragraph\">A imagem (Im) da fun&ccedil;&atilde;o, por sua vez, marca o intervalo em que ela se encontra no eixo y do ciclo trigonom&eacute;trico. No caso da fun&ccedil;&atilde;o seno, observa-se que ela pode variar de -1 a 1, ou seja, Im sen(x) = [-1;1].<\/p><p class=\"wp-block-paragraph\">Essas propriedades podem ser observadas a partir do gr&aacute;fico das fun&ccedil;&otilde;es trigonom&eacute;tricas. Para f(x)=sen(x), teremos a seguinte rela&ccedil;&atilde;o entre &acirc;ngulo em radianos (eixo x) e seu seno (eixo y):<\/p><div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/sXYCdjrvDVE-IOgwYwcqSMubCfDZqLbJ8Ev595mnJMkOi61WQHSduun_OHdcKOA8u9KYrMHFcH-Z3V70xr6pEV04OsBpIPHmAN3D7YqnOUAfIj2wHYVoGEtMhghXum0_CDlHiEjM\" alt=\"Gr&aacute;fico de fun&ccedil;&otilde;es trigonom&eacute;tricas - seno\"><figcaption class=\"wp-element-caption\">Imagem: Reprodu&ccedil;&atilde;o\/Wikimedia<\/figcaption><\/figure>\n<\/div><h2 class=\"wp-block-heading\" id=\"h-funcoes-trigonometricas-cosseno-nbsp\"><span class=\"ez-toc-section\" id=\"Funcoes-trigonometricas-cosseno\"><\/span>Fun&ccedil;&otilde;es trigonom&eacute;tricas:  cosseno&nbsp;<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\">A fun&ccedil;&atilde;o <span id=\"urn:enhancement-adbf0f23-9947-4b6d-aa6d-2654c0dbb010\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cosseno\">cosseno<\/span> &eacute; dada pela equa&ccedil;&atilde;o f(x) = <span id=\"urn:enhancement-86d2757f-afcb-4a2d-a449-913eb6a6ddf7\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cosseno\">cos<\/span>(x). Sua varia&ccedil;&atilde;o no <span id=\"urn:enhancement-db822739-0f9b-40d1-b39d-f889fca7f811\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/ciclo-trigonometrico-2\">ciclo trigonom&eacute;trico<\/span> &eacute; regida pelo di&acirc;metro horizontal, quando os valores est&atilde;o no hemisf&eacute;rio direito, ser&atilde;o positivos. Na por&ccedil;&atilde;o esquerda, negativos:<\/p><div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/xuNGKdLGtWtUl7KrbEff-7w5k9IPXx2xQi2QeUu_J1wjwiJQPWmKadsq0-Q2sAzD_eAfUKbGjYyOAxHbKVvRJYuNJrBER9iHbBtQQnc-B-L2YbQoMB5HMesPE5gjx8Ob_fYq_4yJ\" alt=\"Fun&ccedil;&otilde;es trigonom&eacute;tricas - cosseno\"><\/figure>\n<\/div><h3 class=\"wp-block-heading\" id=\"h-dominio-e-imagem-da-funcao-cosseno\">Dom&iacute;nio e imagem da fun&ccedil;&atilde;o cosseno<\/h3><p class=\"wp-block-paragraph\">Como os &acirc;ngulos, em <span id=\"urn:enhancement-d73bd134-5847-45a5-89de-6ba740941ac4\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cosseno\">cos<\/span>(x), podem variar dentro do conjunto dos n&uacute;meros reais, entende-se que o dom&iacute;nio da fun&ccedil;&atilde;o <span id=\"urn:enhancement-1b0f77b6-4bbb-441f-a9be-cb17d43c51f8\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cosseno\">cosseno<\/span> tamb&eacute;m &eacute; dado por f: R &rarr; R. Semelhantemente, no eixo y do <span id=\"urn:enhancement-99e4705e-c11c-40bc-8c68-137ef4b99f47\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/ciclo-trigonometrico-2\">c&iacute;rculo trigonom&eacute;trico<\/span>, o valor de <span id=\"urn:enhancement-489c95b9-c876-4487-8a0d-2727cccb7d7e\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cosseno\">cos<\/span> (x) se estende entre 1- e 1, ou seja, Im <span id=\"urn:enhancement-0cf2f4e1-66de-4759-8369-fd16065b2754\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cosseno\">cos<\/span>(x) = [-1;1]. Observe no gr&aacute;fico:<\/p><div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/F7_Yu4E-c0G6o8_thnLnL5jZE-v6MSRYgwyiAlxaL-poyJFzE28hBMslQ2T3uab4k3M8o8WnEEXAnedowXg5xMKudA1nC4JG3z5Jn2cvcHluHr6RnVIKcgAziSOAmUQyzGeVQWWC\" alt=\"Gr&aacute;ficos das fun&ccedil;&otilde;es trigonom&eacute;tricas - cosseno\"><\/figure>\n<\/div><p class=\"wp-block-paragraph\">Veja que o cosseno do &acirc;ngulo 0 radianos &eacute; igual a 1, j&aacute; que ele ocupa todo o <span id=\"urn:enhancement-ba5a6190-3e45-42dd-83c3-50cf5ae09eeb\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/raio\">raio<\/span> do ciclo trigonom&eacute;trico. De maneira an&aacute;loga, o cosseno de &frac12; ? (90&ordm;) &eacute; 0, j&aacute; que o posicionamento desse &acirc;ngulo no <span id=\"urn:enhancement-57bda0a5-f60c-4cc1-9dcb-f3beac615627\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/plano-cartesiano\">plano cartesiano<\/span> do <span id=\"urn:enhancement-a14afe99-1419-4472-a3f5-b348345a6d5c\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/circulo\">c&iacute;rculo<\/span> &eacute; igual a (0,1).<\/p><div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/aIf1iA8cONw7xVF4HEWPb-NLH-VLHbJaSzmMvvrM5Q8e4bYfz2yT_dUqChDN7t7BnJXHI05ByLI2h4bBpLwGAyZK313kdJBEFcMiieRN74b6M7ERhzfrDICRdIF4Hc7aqeaojn7R\" alt=\"Fun&ccedil;&otilde;es trigonom&eacute;tricas - cosseno e seno\"><\/figure>\n<\/div><h2 class=\"wp-block-heading\" id=\"h-funcoes-trigonometricas-tangente-nbsp\"><span class=\"ez-toc-section\" id=\"Funcoes-trigonometricas-tangente\"><\/span>Fun&ccedil;&otilde;es trigonom&eacute;tricas: tangente&nbsp;<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\">Como nas outras fun&ccedil;&otilde;es trigonom&eacute;tricas, a lei de forma&ccedil;&atilde;o da fun&ccedil;&atilde;o <span id=\"urn:enhancement-a62e39e0-3600-4657-8a05-ecf4d061a3a6\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/tangente\">tangente<\/span> &eacute; tal que f(x) = tg(x). Apesar de tal semelhan&ccedil;a, essa equa&ccedil;&atilde;o possui muitas particularidades, j&aacute; que &eacute; uma raz&atilde;o entre <span id=\"urn:enhancement-4a8565d3-47ee-417d-bd29-53198334a519\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/sen\">sen<\/span>(x) \/ <span id=\"urn:enhancement-5efc16ca-d4b8-4c9f-8b20-09af49c0741d\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cosseno\">cos<\/span>(x).<\/p><p class=\"wp-block-paragraph\">Como, em uma divis&atilde;o, o denominador n&atilde;o pode ser igual a zero, n&atilde;o existe resposta para f(x) = tg(x) quando o cos(x) = 0. Por essa raz&atilde;o, todos os &acirc;ngulos que tem 0 como cosseno n&atilde;o s&atilde;o aceitos no dom&iacute;nio de tg(x).<\/p><p class=\"wp-block-paragraph\">Para garantir essa afirma&ccedil;&atilde;o, entende-se que, dada um valor inteiro k: Dom(tg) = {x &isin; R&#9474;x &ne; de &pi;\/2 + k&pi;; k &isin; Z}. Isso indica que toda vez que x = &pi;\/2 + k&pi;, a fun&ccedil;&atilde;o tangente n&atilde;o ser&aacute; definida.<\/p><p class=\"wp-block-paragraph\">Em termos de imagem, ela pode variar infinitamente, j&aacute; que a reta que a define n&atilde;o est&aacute; dentro do ciclo trigonom&eacute;trico e sempre &eacute; crescente. Como voc&ecirc; pode observar no gr&aacute;fico abaixo, o valor de tg(x) tende ao infinito:<\/p><div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/jQb_0C8fFnfz6wD2E8aM_BqycRqqXvoMN1tuMh4M4KZqqBzYwHjSruVg4bpcsFHhxYw9WS4FHwp4xL9_bi29aKtR-0WbgFZCN64mqfmtmgmMryIVdI41l13Uor9DZUjF_SlAHqFv\" alt=\"Gr&aacute;fico da fun&ccedil;&atilde;o trigonom&eacute;trica tangente\"><\/figure>\n<\/div><h2 class=\"wp-block-heading\" id=\"h-questoes-sobre-funcoes-trigonometricas\"><span class=\"ez-toc-section\" id=\"Questoes-sobre-funcoes-trigonometricas\"><\/span>Quest&otilde;es sobre fun&ccedil;&otilde;es trigonom&eacute;tricas<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\"><span id=\"urn:enhancement-cc6d0820-8357-49ca-8fba-775b93c4b2d7\" class=\"textannotation disambiguated wl-event\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/enem-64632\">ENEM<\/span> 2017<\/p><p class=\"wp-block-paragraph\">Raios de luz solar est&atilde;o atingindo a superf&iacute;cie de um lago formando um &acirc;ngulo x com a sua superf&iacute;cie, conforme indica a figura.<\/p><p class=\"wp-block-paragraph\">Em determinadas condi&ccedil;&otilde;es, pode-se supor que a intensidade luminosa desses raios, na superf&iacute;cie do lago, seja dada aproximadamente por l(x) = k.sen(x) sendo k uma constante, e supondo-se que x est&aacute; entre 0&ordm; e 90&ordm;.<\/p><div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/uVl1VovPb5TEEJJBfeJn8LQgVKRxaHSywvzxY8XfE25FFzprbKRcnSg8qGLGdOEGJcfmu1wiPJg8DybOnAfJJhl8Zsdb2SElL0HHjWhrgi-JbekVZ-vytXXO-r6tvJXUpNvkyyNh\" alt=\"\"><\/figure>\n<\/div><p class=\"wp-block-paragraph\">Quando x = 30&ordm;, a intensidade luminosa se reduz a qual percentual de seu valor m&aacute;ximo?<\/p><p class=\"wp-block-paragraph\">a) 33%<br>b) 50%<br>c) 57%<br>d) 70%<br>e) 86%<\/p><p class=\"wp-block-paragraph\">Como k &eacute; uma constante, o valor m&aacute;ximo de l(x) &eacute; encontrado com <span id=\"urn:enhancement-3fb22cb8-c4a8-4891-beee-733b8dc9c92d\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/sen\">sen<\/span>(x) &eacute; m&aacute;ximo. Da imagem da <span id=\"urn:enhancement-91e35567-1ef2-48fe-9ce6-a74126910b20\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/sen\">fun&ccedil;&atilde;o seno<\/span>, sabemos o maior valor assumido por <span id=\"urn:enhancement-ff60547c-19aa-4e5a-9f46-7277882c4111\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/sen\">sen<\/span>(x) = 1. Portanto:&nbsp;<\/p><p class=\"wp-block-paragraph\">l (x) m&aacute;ximo = k.1<\/p><p class=\"wp-block-paragraph\">Quando x = 30&ordm;, teremos que <span id=\"urn:enhancement-af1047cf-7312-42cc-9e0c-acea0b257e99\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/sen\">sen<\/span> 30&ordm; = &frac12;, ent&atilde;o:<\/p><p class=\"wp-block-paragraph\">l (30&ordm;)&nbsp; = k.1\/2<\/p><p class=\"wp-block-paragraph\">Ent&atilde;o, entre l(x) m&aacute;ximo e l (30)&ordm; teremos uma perda de intensidade luminosa de 50%, como prediz a alternativa B.&nbsp;<\/p><h2 class=\"wp-block-heading\" id=\"h-aprenda-mais-matematica-com-a-coruja\"><span class=\"ez-toc-section\" id=\"Aprenda-mais-matematica-com-a-Coruja\"><\/span>Aprenda mais matem&aacute;tica com a Coruja!<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\">Voc&ecirc; sabia que tirar suas d&uacute;vidas te ajuda a construir o conhecimento? Ao formular a pergunta e buscar compreender as respostas voc&ecirc; tem a chance de consolidar o conte&uacute;do. Por isso, o Estrat&eacute;gia Vestibulares disponibiliza um f&oacute;rum de d&uacute;vidas com monitores especializados para te ajudar a resolver quest&otilde;es e sanar seus questionamentos. Quer saber mais? 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