{"id":67196,"date":"2022-04-30T12:00:00","date_gmt":"2022-04-30T15:00:00","guid":{"rendered":"https:\/\/vestibulares.estrategia.com\/portal\/?p=67196"},"modified":"2024-04-17T10:14:03","modified_gmt":"2024-04-17T13:14:03","slug":"operacoes-com-arcos","status":"publish","type":"post","link":"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/operacoes-com-arcos\/","title":{"rendered":"Opera\u00e7\u00f5es com arcos: o que s\u00e3o, ciclo trigonom\u00e9trico e como fazer"},"content":{"rendered":"<p class=\"wp-block-paragraph\">Os <span id=\"urn:enhancement-37731414-8bf1-4290-ad3f-07c154cbd990\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/arcos\">arcos<\/span> de uma <span id=\"urn:enhancement-e260f091-4716-4470-ac39-fd18e1b8d276\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/circunferencia\">circunfer&ecirc;ncia<\/span> s&atilde;o definidos por dois ou mais <span id=\"urn:enhancement-c6fc5b36-3a34-4e8e-bd1e-9bd5ab6d49d2\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/pontos\">pontos<\/span> que dividem o tra&ccedil;ado circular em diferentes &acirc;ngulos. Nos vestibulares, &eacute; frequente a cobran&ccedil;a de quest&otilde;es de opera&ccedil;&otilde;es com <span id=\"urn:enhancement-01e95c9a-0496-4cff-9539-fb7e594f6448\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/arcos\">arcos<\/span>, de maneira contextualizada como os giros de um ponteiro no rel&oacute;gio, a <span id=\"urn:enhancement-3bdb9fb0-35c6-4042-84a1-c7c79c2f1f96\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/rotacao\">rota&ccedil;&atilde;o<\/span> de engrenagens ou rodas de carro.<p class=\"wp-block-paragraph\">Aprenda, a partir de agora, as defini&ccedil;&otilde;es sobre opera&ccedil;&otilde;es com <span id=\"urn:enhancement-b6c45dd4-8463-41af-85b8-f435785f5c8a\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/arcos\">arcos<\/span>, com exemplos de c&aacute;lculos, al&eacute;m de acompanhar a resolu&ccedil;&atilde;o de quest&otilde;es de prova sobre esse assunto. Vamos l&aacute;?!<\/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\/operacoes-com-arcos\/#O-que-sao-operacoes-com-arcos\" >O que s&atilde;o opera&ccedil;&otilde;es com arcos?<\/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\/operacoes-com-arcos\/#Soma-de-arcos\" >Soma de arcos&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\/operacoes-com-arcos\/#Subtracao-de-arcos\" >Subtra&ccedil;&atilde;o de arcos&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\/operacoes-com-arcos\/#Formulas-para-operacoes-com-arcos-duplos\" >F&oacute;rmulas para opera&ccedil;&otilde;es com arcos duplos<\/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\/operacoes-com-arcos\/#Exemplo-de-operacoes-com-arcos-duplos\" >Exemplo de opera&ccedil;&otilde;es com arcos duplos<\/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\/operacoes-com-arcos\/#Estude-com-o-Estrategia-Vestibulares\" >Estude com o Estrat&eacute;gia Vestibulares<\/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\/operacoes-com-arcos\/#Veja-tambem\" >Veja tamb&eacute;m:<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\" id=\"h-o-que-sao-operacoes-com-arcos\"><span class=\"ez-toc-section\" id=\"O-que-sao-operacoes-com-arcos\"><\/span>O que s&atilde;o opera&ccedil;&otilde;es com arcos?<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\">Opera&ccedil;&otilde;es com <span id=\"urn:enhancement-c6e6edf7-5a6d-4942-8d4d-10ef075f9b79\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/arcos\">arcos<\/span> s&atilde;o ferramentas da <span id=\"urn:enhancement-3fff990f-624b-4213-84a4-a62e01d3905f\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/trigonometria\">trigonometria<\/span> propostas para as situa&ccedil;&otilde;es em que os &acirc;ngulos s&atilde;o somados ou subtra&iacute;dos entre si. &Eacute; poss&iacute;vel relacionar esses c&aacute;lculos tanto com a <span id=\"urn:enhancement-1f4e78ed-7977-4721-8e31-83e92dbb5a79\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/geometria-euclidiana\">geometria plana<\/span>, como com as<a href=\"https:\/\/vestibulares.estrategia.com\/portal\/matematica\/funcoes-trigonometricas\/\" target=\"_blank\"> fun&ccedil;&otilde;es trigonom&eacute;tricas<\/a>: seno, cosseno e tangente.<\/p><p class=\"wp-block-paragraph\">Dada a rela&ccedil;&atilde;o entre <a href=\"https:\/\/vestibulares.estrategia.com\/portal\/matematica\/arcos-e-angulos\/\" target=\"_blank\">arcos e &acirc;ngulos<\/a>, sabe-se que todo <span id=\"urn:enhancement-01141fe4-3344-4eaa-b353-bcd48331a87b\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/angulo\">&acirc;ngulo<\/span> ter&aacute; um arco correspondente no <span id=\"urn:enhancement-af163092-ca4f-48f8-ac56-10df25d5258e\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/ciclo-trigonometrico-2\">ciclo trigonom&eacute;trico<\/span>. De forma que, para o <span id=\"urn:enhancement-824f07d2-d4a1-4eb4-8c65-0e07055db9ea\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/angulo\">&acirc;ngulo<\/span> &alpha; = 45&ordm;, podem ser delimitados <span id=\"urn:enhancement-1304092f-8da5-4fce-b736-b8c7dc0c6a46\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/dois-pontos\">dois pontos<\/span> que conferem o arco 45&ordm;.&nbsp;<\/p><p class=\"wp-block-paragraph\">Lembre-se que um <span id=\"urn:enhancement-93bd0ab9-5da4-42a9-92b7-3c5b7ebf356f\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/circulo\">c&iacute;rculo<\/span> possui, ao todo, 360&ordm;. De forma que, sempre que um arco &gamma; &eacute; estabelecido dentro dele, haver&aacute; um outro <span id=\"urn:enhancement-169d9821-4980-471c-b254-f23a94604664\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/angulo\">&acirc;ngulo<\/span> &Omega; formado. De forma que&nbsp; &gamma; +&nbsp; &Omega; = 360&ordm;.<\/p><div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/onu16vffEL_yC6-ZKWh2Zw9SRQJ54F0nYEWCc_zG9nuqoI40TUEmbH4ZK-bZQv18Hz08CIlTwFrJOZeyLybtIdJfVrHRVrFT3YI_a5ySolZ81nb7VE7BqjpznhbVHLXGccuegqg_7w6Ag16F7opSBW0\" alt=\"exemplos de arcos de &acirc;ngulos\"><figcaption class=\"wp-element-caption\">Imagem: Adapta&ccedil;&atilde;o\/Wikimedia<\/figcaption><\/figure>\n<\/div><h3 class=\"wp-block-heading\" id=\"h-tabela-trigonometrica-para-operacoes-com-arcos-e-angulos\">Tabela trigonom&eacute;trica para opera&ccedil;&otilde;es com arcos e &acirc;ngulos<\/h3><p class=\"wp-block-paragraph\">A <a href=\"https:\/\/www.youtube.com\/watch?v=rpDtjvuJDjM&amp;pp=ygUdZXN0cmF0ZWdpYSB2ZXN0IHRyaWdvbm9tZXRyaWE%3D\" target=\"_blank\">trigonometria <\/a>apresenta muitos valores irracionais e todos entre 1 e 0. Por isso, &eacute; complexo decorar as raz&otilde;es trigonom&eacute;tricas de todos os &acirc;ngulos. Assim, existe uma tabela trigonom&eacute;trica que agrupa todos os &acirc;ngulos not&aacute;veis e recorrentes em exerc&iacute;cios de ensino m&eacute;dio e vestibulares.<\/p><div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/7nqjK98gxsiz2tMg8MpMFUXLtMFrfDKfHiigf39dJbAPAgcitFrHLbTj-mX-vZkn1wZ6nzRi2SfsaBDmVxgBsYqUvtbgfvuoYn_OWOq1XIREzeQPS7rtUbP4l1uo8oJUA3Lpqf8fvoOAa0iMybwmnRs\" alt=\"tabela trigonom&eacute;trica\"><figcaption class=\"wp-element-caption\">Imagem: Reprodu&ccedil;&atilde;o\/Wikimedia<\/figcaption><\/figure>\n<\/div><p class=\"wp-block-paragraph\">Como diferencial no momento da prova, memorizar o valor dessas raz&otilde;es trigonom&eacute;tricas pode ser &uacute;til para ganhar tempo e resolver as quest&otilde;es com um pouco mais de velocidade.&nbsp;<\/p><h3 class=\"wp-block-heading\" id=\"h-exemplo-do-uso-de-arcos-e-angulos\">Exemplo do uso de arcos e &acirc;ngulos<\/h3><p class=\"wp-block-paragraph\">Associa&ccedil;&otilde;es de <span id=\"urn:enhancement-8b8ada47-7fd8-41ff-bd14-b9b189dd02e8\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/arcos\">arcos<\/span>, &acirc;ngulos e <span id=\"urn:enhancement-f2cf40a3-84eb-4053-9f62-4e0bb5ebb399\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/geometria-euclidiana\">geometria plana<\/span>,&nbsp; al&eacute;m de serem importantes no estudo trigonom&eacute;trico, tamb&eacute;m podem ser levadas em considera&ccedil;&atilde;o quanto ao comprimento <span id=\"urn:enhancement-631d98e6-f259-4335-b2d4-79c19053d6f1\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/circunferencia\">circunfer&ecirc;ncia<\/span> que est&aacute; contido em um dado arco. Acompanhe o exemplo:<\/p><p class=\"wp-block-paragraph\">Se um pneu de carro tem 60 cm de di&acirc;metro e &eacute; rolado no ch&atilde;o em um arco X que tem tg X = &radic;3, qual a dist&acirc;ncia entre o <span id=\"urn:enhancement-e6e6bcff-aa53-459b-a6ea-09948d28906e\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/pontos\">ponto<\/span> de in&iacute;cio do rolamento e o <span id=\"urn:enhancement-fd0d5541-4e05-4e42-a556-8e993b1da153\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/pontos\">ponto<\/span> de parada?<\/p><div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/8q-cGVxmpDoH-GJS6vHFCRnesT5uBKuIPOE_rYJiIeiEEwfLIGTfh430ekwmh1bJeCUeKfVMwaoe7NJMqgowX-Rcm23lXlsrcDFTAMh8Pxaey8TXVMDQNyixsCnMC9eJIdXoS0QwgByRqdyrNeslAA\" alt=\"exerc&iacute;cios de opera&ccedil;&otilde;es com arco\"><figcaption class=\"wp-element-caption\">Imagem: Reprodu&ccedil;&atilde;o\/Wikimedia<\/figcaption><\/figure>\n<\/div><p class=\"wp-block-paragraph\">A imagem acima demonstra, hipoteticamente, que o arco percorrido pelo pneu tem um <span id=\"urn:enhancement-fb8b1d72-3f6c-4d28-b9db-7dda52f8a69c\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/angulo\">&acirc;ngulo<\/span> X, em que tg X = &radic;3. Conforme a tabela trigonom&eacute;trica apresentada acima, o <span id=\"urn:enhancement-892ca089-5166-4b11-83ef-6fac867923b3\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/angulo\">&acirc;ngulo<\/span> que apresenta esse valor de tangente &eacute; X = 60&ordm;. Para continuar os c&aacute;lculos e alcan&ccedil;ar as informa&ccedil;&otilde;es requeridas no enunciado, &eacute; importante conhecer a f&oacute;rmula que determina o comprimento de uma <a href=\"https:\/\/vestibulares.estrategia.com\/portal\/matematica\/circulo-e-circunferencia\/\">circunfer&ecirc;ncia<\/a>:<\/p><p class=\"has-text-align-center wp-block-paragraph\">C = 2.&pi;.r<\/p><p class=\"wp-block-paragraph\">Como o raio &eacute; dado por 2.r = di&acirc;metro, ent&atilde;o:<\/p><p class=\"has-text-align-center wp-block-paragraph\">r = di&acirc;metro \/2<br>r = 60 cm \/2<br><strong>r = 30 cm&nbsp;<\/strong><\/p><p class=\"wp-block-paragraph\">Adicionamos, agora, o valor do raio na f&oacute;rmula da <span id=\"urn:enhancement-52abbd6f-39e4-4df8-9290-93a8de0c91f8\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/circunferencia\">circunfer&ecirc;ncia<\/span>:<\/p><p class=\"has-text-align-center wp-block-paragraph\">C = 2.&pi;.30<br>C = 60&pi; cm<\/p><p class=\"wp-block-paragraph\">Levando em considera&ccedil;&atilde;o que o comprimento da <span id=\"urn:enhancement-109a172f-2bbb-4b58-8330-84c3334bbf20\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/circunferencia\">circunfer&ecirc;ncia<\/span> se refere ao <span id=\"urn:enhancement-19ec7840-2d6d-4c26-a4a8-03d53979cc96\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/angulo\">&acirc;ngulo<\/span> total do <span id=\"urn:enhancement-61b89c35-f329-4c25-8416-fe95de19a5c3\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/circulo\">c&iacute;rculo<\/span>, que &eacute; de 360&ordm;, podemos continuar as contas com <a href=\"https:\/\/vestibulares.estrategia.com\/portal\/matematica\/regra-de-tres\/\">regra de tr&ecirc;s<\/a>:<\/p><p class=\"has-text-align-center wp-block-paragraph\">360&ordm; &mdash; &mdash;&nbsp; 60&pi; cm<br>60&ordm;&nbsp; &mdash; &mdash; x cm<\/p><p class=\"has-text-align-center wp-block-paragraph\">60 . 60 = 360 . x<br>3600 = 360 x<\/p><p class=\"has-text-align-center wp-block-paragraph\">x = 3600\/360<\/p><p class=\"has-text-align-center wp-block-paragraph\"><strong>x = 10 cm<\/strong><\/p><p class=\"wp-block-paragraph\">Perceba como a rela&ccedil;&atilde;o entre o arco, o <span id=\"urn:enhancement-143e429f-6873-4119-a406-735ca9e0331a\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/angulo\">&acirc;ngulo<\/span> e a <span id=\"urn:enhancement-a2a0dab5-f9f5-49f9-a0f9-8cf6a3197e5e\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/circunferencia\">circunfer&ecirc;ncia<\/span> foram essenciais para encontrar o percurso do pneu quando ele &eacute; virado em 60&ordm;.<\/p><p class=\"wp-block-paragraph\">Agora que j&aacute; vimos um exemplo da aplica&ccedil;&atilde;o dos <span id=\"urn:enhancement-60457522-2326-47f0-83db-398053ad02e3\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/arcos\">arcos<\/span> de &acirc;ngulos na <span id=\"urn:enhancement-42ffc5dd-4f8a-4bf2-bf27-53d89b8dec14\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/geometria-euclidiana\">geometria plana<\/span> e conhecemos a tabela trigonom&eacute;trica para consulta de valores, vamos nos aprofundar na compreens&atilde;o dos <span id=\"urn:enhancement-726e28f6-c0bc-4586-b0f9-7d6cd0980fe3\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/arcos\">arcos<\/span> somados ou subtra&iacute;dos no panorama trigonom&eacute;trico.<\/p><h2 class=\"wp-block-heading\" id=\"h-soma-de-arcos-nbsp\"><span class=\"ez-toc-section\" id=\"Soma-de-arcos\"><\/span>Soma de arcos&nbsp;<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\">Como em qualquer <span id=\"urn:enhancement-cca9576e-1fda-4d36-b9f6-72c496e78e4b\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/adicao\">soma<\/span>, quando dois <span id=\"urn:enhancement-2db46af3-5916-478d-afbc-f268dca5a5cd\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/arcos\">arcos<\/span> s&atilde;o adicionados entre si, um novo valor de <span id=\"urn:enhancement-b20cfd04-a898-4cb4-9613-90c34a7cce3d\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/angulo\">&acirc;ngulo<\/span> &eacute; encontrado. Por exemplo, se somarmos 30&ordm; e 60&ordm; encontramos o <span id=\"urn:enhancement-f2a63fd7-555a-4797-98cf-427b83465fb4\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/angulo\">&acirc;ngulo<\/span> de 90&ordm;.<\/p><p class=\"wp-block-paragraph\">Por mais que essas afirma&ccedil;&otilde;es pare&ccedil;am &oacute;bvias, &eacute; importante tomar cuidado: a <span id=\"urn:enhancement-5df082b9-d3e8-4ce5-8402-dc12f54c89b4\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/adicao\">soma<\/span> das <span id=\"urn:enhancement-f73d17c4-8b5e-4d60-949c-bf1eee70ec6a\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/funcoes-trigonometricas\">fun&ccedil;&otilde;es trigonom&eacute;tricas<\/span> dos <span id=\"urn:enhancement-10426549-a6a3-4259-b315-35ca432cbda0\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/arcos\">arcos<\/span> n&atilde;o seguem o mesmo padr&atilde;o. Por exemplo, se somarmos o sen 30&ordm; e o sen 60&ordm; n&atilde;o encontramos o valor do sen 90&ordm;, observe:&nbsp;<\/p><p class=\"has-text-align-center wp-block-paragraph\">sen 30&ordm; = &frac12;<br>sen 60&ordm; = &radic;3\/2<\/p><p class=\"has-text-align-center wp-block-paragraph\">sen 30&ordm; + sen 60&ordm; = (1+&radic;3)\/2&nbsp;&nbsp;<br>sen 90&ordm; = 1<strong><br><\/strong><br><strong>sen 30&ordm; + sen 60&ordm; &ne; sen 90&ordm;<\/strong><\/p><p class=\"wp-block-paragraph\">Situa&ccedil;&atilde;o semelhante acontecer&aacute; com o cosseno e a tangente dos &acirc;ngulos, seja na <span id=\"urn:enhancement-b5915d3e-149b-4459-9897-598fec53a93a\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/adicao\">soma<\/span> de <span id=\"urn:enhancement-9c5176d7-8e7f-4d31-b7f5-a1cd8ce8680e\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/arcos\">arcos<\/span> ou em sua subtra&ccedil;&atilde;o. Isso acontece porque a rela&ccedil;&atilde;o entre as <span id=\"urn:enhancement-729676bd-701b-4955-a6bb-ce1797e892fa\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/funcoes-trigonometricas\">fun&ccedil;&otilde;es trigonom&eacute;tricas<\/span> n&atilde;o acontece de maneira linear: ela &eacute; dada pela proje&ccedil;&atilde;o dos &acirc;ngulos no <span id=\"urn:enhancement-d35e3662-85d9-41ca-9fc1-4df2f8568bab\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/ciclo-trigonometrico-2\">ciclo trigonom&eacute;trico<\/span>.&nbsp;<\/p><p class=\"wp-block-paragraph\">Veja, na imagem a seguir, que todos esses valores s&atilde;o retirados de tri&acirc;ngulos ret&acirc;ngulos que surgem na <span id=\"urn:enhancement-879c5fcb-b2dc-47a5-84a8-f4aa542ef847\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/circunferencia\">circunfer&ecirc;ncia<\/span> de raio 1.&nbsp;<\/p><div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/O5EED9fqqmW26L2iZ3cldGd06o7dUvtnXp5AvpQ9TakiQjwbM-FPrCUjGUR8LuvSrsYX_Dl2vaN9YQdXfJZwjtgY7-OmH7bHy0BAex5JvebjSunuEpNbLLqIPeKOSWAUE_ia9dfZzijjK4t1tKDnTA\" alt=\"Arcos de &acirc;ngulos: soma de arcos\"><\/figure>\n<\/div><h3 class=\"wp-block-heading\" id=\"h-formula-para-operacoes-com-arcos-soma\">F&oacute;rmula para opera&ccedil;&otilde;es com arcos: soma<\/h3><h4 class=\"wp-block-heading\" id=\"h-cosseno-da-soma\">Cosseno da soma<\/h4><p class=\"wp-block-paragraph\">Com as manipula&ccedil;&otilde;es <span id=\"urn:enhancement-0118538e-1988-42f5-9334-bd265eec801a\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/matematica\">matem&aacute;ticas<\/span> necess&aacute;rias, foram encontradas f&oacute;rmulas que sintetizam o estudo da <span id=\"urn:enhancement-6c2c3b7d-1a5b-489e-b738-2674f3060ee6\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/adicao\">soma<\/span> de <span id=\"urn:enhancement-f3474b8d-c6df-4bca-bad6-19d184987195\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/arcos\">arcos<\/span> e as <span id=\"urn:enhancement-92a8b981-de10-41a9-9979-f0c84173865a\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/funcoes-trigonometricas\">fun&ccedil;&otilde;es trigonom&eacute;tricas<\/span>, como voc&ecirc; acompanhar&aacute; abaixo. Considere dois &acirc;ngulos gen&eacute;ricos dados por a e b.<\/p><p class=\"has-text-align-center wp-block-paragraph\"><strong>cos (a+b) = cos(a).cos(b) &ndash; sen(a).sen(b)<\/strong><\/p><p class=\"wp-block-paragraph\">Para memorizar f&oacute;rmula do seno da <span id=\"urn:enhancement-fcb0e07e-c224-4c76-b230-975b118c3dd3\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/adicao\">soma<\/span> de dois <span id=\"urn:enhancement-e03b164b-5ec5-4299-a9da-3cf9b6b390e6\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/arcos\">arcos<\/span>, &eacute; poss&iacute;vel lembrar da seguinte frase, muito famosa no mundo dos vestibulares:<\/p><p class=\"has-text-align-center wp-block-paragraph\">&ldquo;<strong>Cos<\/strong>seno de<strong> a mais b<\/strong> &eacute;:<br><strong>co<\/strong>&ccedil;a <strong>a<\/strong>, <strong>co<\/strong>&ccedil;a <strong>b<\/strong>;<br><strong>troca o sina<\/strong>l,<br><strong>s<\/strong>em <strong>sab<\/strong>&ecirc;&rdquo;<\/p><p class=\"wp-block-paragraph\">Na sonoriza&ccedil;&atilde;o da estrofe entende-se um indicativo para a constru&ccedil;&atilde;o da f&oacute;rmula, assim:<\/p><p class=\"wp-block-paragraph\">O termo &ldquo;co&ccedil;a&rdquo; faz refer&ecirc;ncia &agrave; fun&ccedil;&atilde;o cosseno e as consoantes esses (s) do &uacute;ltimo verso apontam para a fun&ccedil;&atilde;o seno. A troca de sinais demonstra que os dois termos devem ser subtra&iacute;dos quando deseja-se fazer a <span id=\"urn:enhancement-4b6cb666-1392-41bb-bce3-bde53137bf32\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/adicao\">soma<\/span> de <span id=\"urn:enhancement-cfc880c4-0f18-4b10-ab1b-1aef82932b38\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/arcos\">arcos<\/span>. De maneira an&aacute;loga, eles devem ser somados se o objetivo &eacute; subtrair os <span id=\"urn:enhancement-5a2e49b8-058d-4f33-aba4-3555cb3529c8\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/arcos\">arcos<\/span>.<\/p><h4 class=\"wp-block-heading\" id=\"h-seno-da-soma\">Seno da soma<\/h4><p class=\"has-text-align-center wp-block-paragraph\"><strong>sen(a+b) = sen(a).cos(b) + sen (b).cos(a)<\/strong><\/p><p class=\"wp-block-paragraph\">E4-3555cb3529c8&Prime; class=&rdquo;textannotation&rdquo;&gt;ssa f&oacute;rmula &eacute;, geralmente, decorada a partir da par&oacute;dia de um poema famoso da literatura brasileira, escrito por <span id=\"urn:enhancement-2475666d-a8a3-4437-a7fe-44a5e1c2a3a2\" c>lass=&rdquo;textannotation disambiguated wl-person&rdquo; itemid=&rdquo;https:\/\/data.wordlift.io\/wl110249\/entity\/goncalves-dias&rdquo;&gt;Gon&ccedil;alves Dias<\/span>:&nbsp;<\/p><p class=\"has-text-align-center wp-block-paragraph\">&ldquo;<em>Na minha terra tem palmeiras onde 437-a7fe-44a5e1c2a3a2&Prime; class=&rdquo;textannotation&rdquo;&gt;canta o sabi&aacute;,&nbsp;<\/em><br><strong><em>sen<\/em><\/strong><em>o <\/em><strong><em>a<\/em><\/strong><em>, <\/em><strong><em>cos<\/em><\/strong><em>seno <\/em><strong><em>b<\/em><\/strong><em>;<\/em><br><strong>&gt;<em>sen<\/em><\/strong><em>o <\/em><strong><em>b<\/em><\/strong><em>, <\/em><strong><em>cos<\/em><\/strong><em>seno<\/em><strong><em> a<\/em><\/strong><em>.&rdquo;<\/em><\/p><p class=\"wp-block-paragraph\">Com isso, voc&ecirc; se lembrar&aacute; da ordem das opera&ccedil;&otilde;es e percebe que n&atilde;o ser&aacute; necess&aacute;rio trocar o sinal da opera&ccedil;&atilde;o no momento de montagem dos c&aacute;lculos.<\/p><h4 class=\"wp-block-heading\" id=\"h-tangente-da-soma\">Tangente da soma<\/h4><div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/CdRc5-4DWBQUnL_xrGmoSaJqb3SZGCgD8N0JpIsNn7cJg0ektSCztW_K-rqoNlTO_ro88g6EqdR7atrmNrdn37EjTWdZoVunKnuE24QzeSBlZHx7cGr4x1FMH8NNoB4W4tHqYsuELFWuKRktgvGdhg\" alt=\"tangente da soma\"><\/figure>\n<\/div><p class=\"wp-block-paragraph\">Para essa opera&ccedil;&atilde;o, teremos a seguinte forma de memorizar:<\/p><p class=\"has-text-align-center wp-block-paragraph\">&ldquo;Tem gente que <strong>a<\/strong>ma, tem gente que <strong>b<\/strong>eija;<br>H<strong>um<\/strong> (1), tem gente que <strong>a<\/strong>ma e <strong>b<\/strong>eija&rdquo;<\/p><p class=\"wp-block-paragraph\">Note que o som de &ldquo;tem gente&rdquo; se assemelha ao de &ldquo;tangente&rdquo;, assim como as primeiras letras das palavras ama e beija, fazem analogia aos &acirc;ngulos a e b. E, por fim, a <span id=\"urn:enhancement-99b2e532-5948-4b61-a6fa-d6b5ba7c4d16\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/expressoes-algebricas\">express&atilde;o<\/span> &ldquo;hum&rdquo; se refere ao numeral.<\/p><h2 class=\"wp-block-heading\" id=\"h-subtracao-de-arcos-nbsp\"><span class=\"ez-toc-section\" id=\"Subtracao-de-arcos\"><\/span>Subtra&ccedil;&atilde;o de arcos&nbsp;<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\">As f&oacute;rmulas utilizadas para a ncement-368a321d-09d6-4249-afea-76f3f9fc9f12&Prime; class=&rdquo;textannotation disambiguated wl-thing&rdquo; itemid=&rdquo;https:\/\/data.wordlift.io\/wl110249\/entity\/adicao&rdquo;&gt;soma e subtra&ccedil;&atilde;o de <span id=\"urn:enhanceme&lt;\/span&gt;nt-fdac72d3-c9bd-4996-a2b5-b8c561e94b45&Prime; class=&rdquo;textannotation disambiguated wl-thing&rdquo; itemid=&rdquo;https:\/\/data.wordlift.io\/wl110249\/entity\/arcos&rdquo;&gt;arcos&lt;\/span&gt; s&atilde;o as mesmas, com a diferencia&ccedil;&atilde;o dos sinais, como vimos no t&oacute;pico anterior. Veja-as a seguir.&lt;\/p&gt;\n\n\n\n&lt;h3 class=\" wp-block-heading>F&oacute;rmula para opera&ccedil;&otilde;es com arcos: subtra&ccedil;&atilde;o\n\n\n\n<h4 class=\"wp-block-heading\" id=\"h-seno-da-diferenca\">Seno da diferen&ccedil;a<\/h4>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>sen (a-b) = sen(a).cos(b) &ndash; sen (b).cos(a)<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Para aplicar a f&oacute;rmula, &eacute; preciso tomar cuidado de sempre manter os valores de a e b, at&eacute; que o valor do sen (a-b) seja encontrado. Ou seja, n&atilde;o realize a opera&ccedil;&atilde;o dentro do par&ecirc;nteses antes de solucionar a f&oacute;rmula. Como demonstrado a seguir:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">sen (a-b) = sen(a).cos(b) &ndash; sen (b).cos(a)<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">a = 90&ordm;<br>b = 30&ordm;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Vamos substituir a e b com cautela dentro da f&oacute;rmula:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">sen (90-30) = sen(90&ordm;).cos(30&ordm;) &ndash; sen (30&ordm;).cos(90&ordm;)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Embora 90&ordm; n&atilde;o esteja dentro dos &acirc;ngulos not&aacute;veis da tabela trigonom&eacute;trica, no <span id=\"urn:enhancement-9c2436c4-b889-4b86-b749-d403d24678c3\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/ciclo-trigonometrico-2\">ciclo trigonom&eacute;trico<\/span>, ele preenche completamente o eixo y. Ent&atilde;o, sen 90&ordm; = 1. Ao mesmo tempo, seu posicionamento determina que cos 90&ordm; = 0.<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">sen (90-30) = 1.cos(30&ordm;) &ndash; sen (30&ordm;).0<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Observe, com a substitui&ccedil;&atilde;o, que n&atilde;o ser&aacute; necess&aacute;rio preencher o valor de sen 30&ordm;, uma vez que ser&aacute; multiplicado por 0.&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">sen (90-30) = 1.cos(30&ordm;) &ndash; 0<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Ent&atilde;o, segue-se o c&aacute;lculo considerando o valor de cos 30&ordm; = &radic;3\/2<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">sen (90-30) = 1.&radic;3\/2&nbsp;<br>sen (90-30) = &radic;3\/2<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Agora que o c&aacute;lculo est&aacute; conclu&iacute;do, a subtra&ccedil;&atilde;o dentro do par&ecirc;nteses do seno pode ser conclu&iacute;da sem que haja confus&atilde;o entre os valores de a e b.&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">sen (90-30) = &radic;3\/2<br>sen (60&ordm;) = &radic;3\/2<br>sen (90-30) = sen (60&ordm;) = &radic;3\/2<\/p>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"h-cosseno-da-diferenca\">Cosseno da diferen&ccedil;a<\/h4>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>cos (a-b) = cos(a).cos(b) + sen(a).sen(b)<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">No geral, a aplica&ccedil;&atilde;o da f&oacute;rmula depende da substitui&ccedil;&atilde;o correta dos valores. Acompanhe a resolu&ccedil;&atilde;o de uma <strong>quest&atilde;o adaptada da Unitins 2016<\/strong>:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">&Eacute; correto afirmar que: I &ndash; o valor de sen (&pi; + x) + cos (&pi;\/2 &ndash; x) para todo x &#1013; R &eacute; igual a zero?<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Primeiramente<\/strong>, aplica-se o conhecimento sobre seno da <span id=\"urn:enhancement-ee216aba-d1a9-4b88-b422-d4c67ef26a07\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/adicao\">soma<\/span>:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">&#119904;&#119890;&#119899;(&#120587;+&#119909;)=&#119904;&#119890;&#119899;(&#120587;).cos(&#119909;)+&#119904;&#119890;&#119899;(&#119909;).cos (&#120587;)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Lembre-se que os valores de l-thing&rdquo; itemid=&rdquo;https:\/\/data.wordlift.io\/wl110249\/entity\/angulo&rdquo;&gt;&acirc;ngulo podem ser convertidos em radianos. De <span id=\"urn:enhancement-54d3c48e-aa91-4c78-9755-827b44f166&lt;\/span&gt;19&Prime; class=&rdquo;textannotation disambiguated wl-thing&rdquo; itemid=&rdquo;https:\/\/data.wordlift.io\/wl110249\/entity\/geometria&rdquo;&gt;forma&lt;\/span&gt; que &#120587; radianos = 180&ordm;&lt;\/p&gt;\n\n\n\n&lt;p class=\" has-text-align-center wp-block-paragraph>sen &#120587; = sen 180&ordm; = 0<br>cos &#120587; = cos&nbsp; 180&ordm; = -1<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Basta substituir esses valores o c&aacute;lculo anterior:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">&#119904;&#119890;&#119899;(&#120587;+&#119909;)=0.cos(&#119909;) + &#119904;&#119890;&#119899;(&#119909;).(-1)<br>&#119904;&#119890;&#119899;(&#120587;+&#119909;) = &ndash; sen(&#119909;)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">O <strong>segundo passo <\/strong>&eacute; analisar o cosseno da diferen&ccedil;a entre &#120587;\/2 e x.<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">cos (&#120587;\/2-&#119909;) = cos(&#120587;\/2).cos(x) + &#119904;&#119890;&#119899;(&#120587;\/2).sen(x)<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">&nbsp;&#120587;\/2 = 90&ordm;<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">sen &#120587;\/2 = 1<br>cos &#120587;\/2 = 0<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Com as substitui&ccedil;&otilde;es corretas:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">cos (&#120587;\/2-&#119909;) = cos(&#120587;\/2).cos(x) + &#119904;&#119890;&#119899;(&#120587;\/2).sen(x)<br>cos (&#120587;\/2-&#119909;) = 0.cos(x) + 1.sen(x)<br>cos (&#120587;\/2-&#119909;) = 1 sen(x)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Ao estabelecer o c&aacute;lculo fornecido pela afirma&ccedil;&atilde;o teremos que:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">sen (&pi; + x) + cos (&pi;\/2 &ndash; x) =<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">&ndash; sen(&#119909;) + sen (x) = 0, ent&atilde;o a afirmativa est&aacute; correta.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"h-tangente-da-diferenca\">Tangente da diferen&ccedil;a<\/h4>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/y2K8A4TIo1oXOuhsao2-sKeoKxhhLzg5ibVuuHU-vbCdP1DGyHN5wpmJ4_BE9h8pBp-_dinKnq74xjdCy7ErQHNnbNA8XsNpxzjb1vGYO4XuCIwUydJDquYCp2mJlqX8Lqpin5JbYZBWDbp1fqO58A\" alt=\"tangente da diferen&ccedil;a\"><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">A quest&atilde;o abaixo trata exatamente sobre a tangente da diferen&ccedil;a e, inclusive, a banca da UERJ forneceu a f&oacute;rmula completa para a resolu&ccedil;&atilde;o do exerc&iacute;cio.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">No esquema abaixo, est&atilde;o representados um quadrado ABCD e um <span id=\"urn:enhancement-cabdb994-9c15-46fd-808e-afaf2a0dff10\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/circulo\">c&iacute;rculo<\/span> de centro P e raio r, tangente &agrave;s retas AB e BC. O lado do quadrado mede 3r.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/OJpcTZfvfayhDr6Lo4vqi2T9L2W5hzZ0I-ccJgI4emcmcBlivJkN5qvdV_iyUv-TeY1IUETxzdIXmf3OkNH3CJ-IfyWZ8R9AfWqs0XEiqkyww0RppfRXcqVN6eCP2P8abi_HmgK7XSL94y6PjDZ9rhI\" alt=\"quest&atilde;o sobre opera&ccedil;&otilde;es com arcos\"><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">A medida &theta; do <span id=\"urn:enhancement-5d0919f5-2a03-40ba-adc9-241af3e4b1bf\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/angulo\">&acirc;ngulo<\/span> C&Acirc;P pode ser determinada a partir da seguinte identidade trigonom&eacute;trica:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/E1Iiyu4IlSJeaufDNmgN8ZeAInC-GprzvyVZu4K2zWe0AEMxGw4HbOw2rHVpDxAC06e82xg8KQG2eGF90HS_FX49qmm1vzSHguFAIvW0BMOVFKYJaGPg99JRpkUxN-G6ewbVtmFbI4UC4Rp3OxHvm8k\" alt=\"\"><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">O valor da tangente de &theta; &eacute; igual a:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">a) 0,65<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">b) 0,60<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">c) 0,55<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">d) 0,50<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Esse exerc&iacute;cio nhancement-5d0919f5-2a03-40ba-adc9-241af3e4b1bf&rdquo; class=&rdquo;textannotation&rdquo;&gt;o&rdquo;&gt;soma<\/p><\/span> conhecimentos de <span id=\"urn:enhancement-062e817e-5cc7-444f-9704-84f781466698\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl1102&lt;\/span&gt;49\/entity\/trigonometria&rdquo;&gt;trigonometria&lt;\/span&gt;, &lt;span id=\" urn:enhancement-8d179ca4-f087-497e-bffe-0f831af4a1c7>geometria plana<\/span> e &acirc;ngulos. Como o segmento AC &eacute; uma diagonal de um quadrado, o <span id=\"urn:enhancement-7047c513-e336-43d7-ae4a-13360a2b90b5\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/angulo\">&acirc;ngulo<\/span> formado entre ele e a base AB mede 45&deg;<\/p><div class=\"wp-block-image\">\n<figure class=\"aligncenter is-resized\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/dqRpkANkTV0IiHzJOIPt4t2jyzsVHtcfhN_czWkZKWjmjRiqehSfol406HBTbTOVzOHha9qGMrXDsy9rteOfO-gOT-Nd-gTWhfzivjXXDwfXCLHzX4TqoXpQqBxIMfIFgrizmQf5-r1mPpDhLUnZjzY\" alt=\"\" style=\"width:206px;height:141px\"><\/figure>\n<\/div><p class=\"wp-block-paragraph\">Logo, temos que &#120637; + &#119961; = &#120786;&#120787;&deg;, ou &#120637; = &#120786;&#120787;&deg;&minus; &#119961;.<\/p><p class=\"wp-block-paragraph\">A tangente do <span id=\"urn:enhancement-0edd9e71-74aa-4eef-9b53-ad7ac5339d36\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/angulo\">&acirc;ngulo<\/span> &#119961; pode ser calculada pela rela&ccedil;&atilde;o trigonom&eacute;trica entre os catetos do tri&acirc;ngulo ABP, sendo:<\/p><div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/oy4O_m8He9Tu7IaarnoxTQUYnu-PxWfESNJzm2tYHDPAE-Gs03aUtptrDwHKIv-7jdbWAkP9HeG60O1sf9qc9EMUv5Cckfuatjrh3AsH2WdBwgM3YwWKh4mKul7Tkx-laYc7QOnjEpVK_I8VDUvFlXI\" alt=\"exerc&iacute;cio sobre opera&ccedil;&otilde;es com arcos\"><\/figure>\n<\/div><p class=\"wp-block-paragraph\">Sabemos que &#119957;&#119944; &#120637; = &#119957;&#119944; ( &#120786;&#120787;&deg; &minus; &#119961; )<\/p><div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/iFXu83rzzDTUoLxmB3zVcA06Vp7TLFBeCLPHliFQHXlNxTlT53DzxT42GI8uwgtFXdm99TLxuzCBtb5Gq3EFzLnYQZDyXpnTtIBJU8ohhtYHl-H_h90AXfBRfBt4uNaJZ40xi5zQ1Cpb_EGDSkhgNF8\" alt=\"\"><\/figure>\n<\/div><h2 class=\"wp-block-heading\" id=\"h-formulas-para-operacoes-com-arcos-duplos\"><span class=\"ez-toc-section\" id=\"Formulas-para-operacoes-com-arcos-duplos\"><\/span>F&oacute;rmulas para opera&ccedil;&otilde;es com arcos duplos<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\">Na <span id=\"urn:enhancement-1cd4f647-0e99-4507-acc2-c3b943dd69ef\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/trigonometria\">trigonometria<\/span>, um arco duplo &eacute; definido pela <span id=\"urn:enhancement-a91a355c-7d04-4286-a6a1-b8a8c1712e0d\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/multiplicacao\">multiplica&ccedil;&atilde;o<\/span> de um <span id=\"urn:enhancement-aa069aed-d399-4423-a54c-a5e64b094355\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/angulo\">&acirc;ngulo<\/span> em seu dobro, por exemplo, para 30&ordm;, o arco duplo vale 60&ordm;.<\/p><p class=\"wp-block-paragraph\">Ao somar ou subtrair dois ated wl-thing&rdquo; itemid=&rdquo;https:\/\/data.wordlift.io\/wl110249\/entity\/arcos&rdquo;&gt;arcos de &acirc;ngulos iguais, as f&oacute;rmulas podem ser sintetizadas, da seguinte <span id=\"urn:enhancement-ba0ab678-cbb7-49f5-&lt;\/span&gt;9511-229b47351e34&Prime; class=&rdquo;textannotation disambiguated wl-thing&rdquo; itemid=&rdquo;https:\/\/data.wordlift.io\/wl110249\/entity\/geometria&rdquo;&gt;forma&lt;\/span&gt;:&lt;\/p&gt;\n\n\n\n&lt;ul class=\" wp-block-list>\n<li>Seno do arco duplo<\/li>\n\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">sen(a+a) = sen(a).cos(a) + sen (a).cos(a)<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>sen(a+a) = 2.sen(a).cos(a)<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Cosseno do arco duplo&nbsp;<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">cos (a+a) = cos(a).cos(a) &ndash; sen(a).sen(a)<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>cos (a+a) = cos(a)<\/strong><strong><sup>2<\/sup><\/strong><strong> &ndash; sen(a)<\/strong><strong><sup>2<\/sup><\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Tangente do arco duplo<\/li>\n<\/ul>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/xEBWzSoyuAHlXjvswCoGIvnZPlf8OOQgbqfm-XVddg3x6D74EFFlyj71yXlHmiBUmr5KmVePxh8lD81kSWrJUjSOIMSNuxFqoBbFMFg5UOpkRvwKP7_ADjdVII0cp1MO12CJ1LMY5A0b5dcHt5QHWQ\" alt=\"\"><\/figure>\n<\/div>\n\n\n<h2 class=\"wp-block-heading\" id=\"h-exemplo-de-operacoes-com-arcos-duplos\"><span class=\"ez-toc-section\" id=\"Exemplo-de-operacoes-com-arcos-duplos\"><\/span>Exemplo de opera&ccedil;&otilde;es com arcos duplos<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">As opera&ccedil;&otilde;es com temid=&rdquo;https:\/\/data.wordlift.io\/wl110249\/entity\/arcos&rdquo;&gt;arcos duplos s&atilde;o exploradas principalmente quando o enunciado requer a raz&atilde;o trigonom&eacute;trica de um <span id=\"urn:enhancem&lt;\/span&gt;ent-c6331063-ac08-4876-925a-a984865f3336&Prime; class=&rdquo;textannotation disambiguated wl-thing&rdquo; itemid=&rdquo;https:\/\/data.wordlift.io\/wl110249\/entity\/angulo&rdquo;&gt;&acirc;ngulo&lt;\/span&gt; pouco comum, como sen 120&ordm;. Aqui, &eacute; importante observar o &lt;span id=\" urn:enhancement-35088295-cb4b-4cae-979c-f9a9cd062026 class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/angulo\">&acirc;ngulo<\/span> &eacute; tentar transform&aacute;-lo em uma <span id=\"urn:enhancement-2e5b47cb-d62a-4793-8178-511aaa921410\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/adicao\">soma<\/span> de dois membros comuns.&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">sen 120&ordm; = sen (60 + 60) = sen (2.60)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Para encontrar o valor, ent&atilde;o, basta aplicar a f&oacute;rmula com aten&ccedil;&atilde;o.<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">sen(a+a) = 2.sen(a).cos(a)<br>sen(60 + 60) = 2.sen(60).cos(60)<br><br>sen 60&ordm; = &radic;3\/2<br>cos 60&ordm; = 1\/2<br><\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">sen(60 + 60) = 2. (&radic;3\/2) .cos(60)<br>sen(60 + 60) = &radic;3 .cos(60)<br><\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">sen(60 + 60) = &radic;3 . 1\/2<br>sen(60 + 60) = &radic;3\/2<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">sen 120&ordm; = &radic;3\/2<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-estude-com-o-estrategia-vestibulares\"><span class=\"ez-toc-section\" id=\"Estude-com-o-Estrategia-Vestibulares\"><\/span>Estude com o Estrat&eacute;gia Vestibulares<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">No curso preparat&oacute;rio da Coruja, voc&ecirc; poder&aacute; aprender com profundidade as mat&eacute;rias exatas, um <span id=\"urn:enhancement-05fe2471-f3cb-4bdd-92a7-48c7cf928ebd\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/pontos\">ponto<\/span> de grande interesse para os vestibulandos. Al&eacute;m disso, o Estrat&eacute;gia fornece aulas, simulados e trilhas estrat&eacute;gicas que te ajudam na programa&ccedil;&atilde;o e avalia&ccedil;&atilde;o do conhecimento em todas as disciplinas mais importantes para os vestibulares nacionais. 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