{"id":84773,"date":"2022-12-28T11:00:58","date_gmt":"2022-12-28T14:00:58","guid":{"rendered":"https:\/\/vestibulares.estrategia.com\/portal\/?p=84773"},"modified":"2023-01-16T14:13:37","modified_gmt":"2023-01-16T17:13:37","slug":"angulos-notaveis","status":"publish","type":"post","link":"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/angulos-notaveis\/","title":{"rendered":"\u00c2ngulos not\u00e1veis: tabela, demonstra\u00e7\u00e3o, dicas e quest\u00f5es"},"content":{"rendered":"<p class=\"wp-block-paragraph\">Os &acirc;ngulos not&aacute;veis s&atilde;o valores de &acirc;ngulos e suas fun&ccedil;&otilde;es trigonom&eacute;tricas que aparecem com frequ&ecirc;ncia no cotidiano e nas provas de vestibulares. Conhecer esses n&uacute;meros facilita no desenvolvimento de c&aacute;lculos em diversas &aacute;reas da matem&aacute;tica, como geometria plana, espacial, anal&iacute;tica, <span id=\"urn:enhancement-a612fa34-689f-4b71-b0aa-18a4ecac1c03\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/trigonometria\">trigonometria<\/span>, f&iacute;sica &oacute;ptica e muito mais.&nbsp;<p class=\"wp-block-paragraph\">Neste resumo, voc&ecirc; encontra os principais &acirc;ngulos matem&aacute;ticos e seus valores de <span id=\"urn:enhancement-55913ece-f26f-4695-941a-8755daf94579\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/sen\">seno<\/span>, <span id=\"urn:enhancement-7e19c95f-04a5-4926-9278-eb39d4351d36\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cosseno\">cosseno<\/span> e <span id=\"urn:enhancement-d7c1c390-84da-489a-bce9-e531a0e8147e\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/tangente\">tangente<\/span>. Veja tamb&eacute;m como encontrar esses n&uacute;meros e onde eles est&atilde;o situados no <span id=\"urn:enhancement-9619f191-b68f-4024-861b-75b95639a091\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/ciclo-trigonometrico-2\">ciclo trigonom&eacute;trico<\/span>. Depois, vamos descobrir dicas e macetes importantes com a resolu&ccedil;&atilde;o comentada de exerc&iacute;cios de prova. 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\/angulos-notaveis\/#Quais-sao-os-angulos-notaveis\" >Quais s&atilde;o os &acirc;ngulos not&aacute;veis?<\/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\/angulos-notaveis\/#Tabela-de-angulos-notaveis\" >Tabela de &acirc;ngulos not&aacute;veis<\/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\/angulos-notaveis\/#Como-sao-encontradas-as-funcoes-dos-angulos-notaveis\" >Como s&atilde;o encontradas as fun&ccedil;&otilde;es dos &acirc;ngulos not&aacute;veis?<\/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\/angulos-notaveis\/#Outros-angulos-importantes\" >Outros &acirc;ngulos importantes<\/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\/angulos-notaveis\/#Questao-de-vestibular-sobre-angulos-notaveis\" >Quest&atilde;o de vestibular sobre &acirc;ngulos not&aacute;veis<\/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\/angulos-notaveis\/#Aprenda-mais-matematica-com-o-Estrategia\" >Aprenda mais matem&aacute;tica com o Estrat&eacute;gia&nbsp;<\/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\/angulos-notaveis\/#Veja-tambem\" >Veja tamb&eacute;m:<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\" id=\"h-quais-sao-os-angulos-notaveis\"><span class=\"ez-toc-section\" id=\"Quais-sao-os-angulos-notaveis\"><\/span>Quais s&atilde;o os &acirc;ngulos not&aacute;veis?<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\">Os principais &acirc;ngulos, ou &acirc;ngulos not&aacute;veis, s&atilde;o aqueles que aparecem com mais frequ&ecirc;ncia em diversos problemas matem&aacute;ticos. Em geral, &eacute; importante conhec&ecirc;-los e decorar algumas de suas caracter&iacute;sticas tamb&eacute;m.&nbsp;<\/p><p class=\"wp-block-paragraph\">Vamos tratar, principalmente, dos valores de 30&ordm;, 45&ordm; e 60&ordm;. Mas saiba, tamb&eacute;m ,que &eacute; importante conhecer &acirc;ngulos com 0&ordm;, 90&ordm;, 180&ordm; e 360&ordm;, que ser&atilde;o discutidos mais adiante.&nbsp;<\/p><p class=\"wp-block-paragraph\">Lembre-se, quando tratamos sobre o grau de abertura de um v&eacute;rtice, estamos no campo da <span id=\"urn:enhancement-c3f51cea-516d-4c51-8d69-f5c149ff2952\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/trigonometria\">trigonometria<\/span>. Nesse sentido, saiba que <span id=\"urn:enhancement-a6fdc2b0-39df-42bf-92e7-52cb8fba92e8\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/sen\">seno<\/span>, <span id=\"urn:enhancement-9f6ce4dc-dd7a-4b52-89ce-fad2988bbd3a\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cosseno\">cosseno<\/span> e <span id=\"urn:enhancement-90874e7d-c890-4d3d-8037-03dcdf951eb6\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/tangente\">tangente<\/span> s&atilde;o raz&otilde;es trigonom&eacute;tricas que relacionam um &acirc;ngulo com as proje&ccedil;&otilde;es de sua reta nos eixos cartesianos. Essa padroniza&ccedil;&atilde;o &eacute; obtida por meio do <span id=\"urn:enhancement-ca9559fa-4cf9-4c6a-9763-e52c32bd7230\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/ciclo-trigonometrico-2\">ciclo trigonom&eacute;trico<\/span>.<\/p><div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/9xQSQrXdg3NN-YCyD5ubs1cnukW6dz2m9ParsJ1MxeRYQCjVeX07ZxxYlueHmVpN7jEzrQTc6iyYNGi8y3mMSI0QXHLEx9j8S9_FDExQaPjPNRgRk389gmLZ4z5JEo_LLE_JelGMhFOQ8u878gOiNbAwA1HdSHRgq3zcVhtkuyciYyUk8Z5A30w3GOP8\" alt=\"Ciclo trigonom&eacute;trico, importante para encontrar raz&otilde;es em &Acirc;ngulos not&aacute;veis\"><figcaption class=\"wp-element-caption\">Imagem: Reprodu&ccedil;&atilde;o\/Wikimedia<\/figcaption><\/figure>\n<\/div><h2 class=\"wp-block-heading\" id=\"h-tabela-de-angulos-notaveis\"><span class=\"ez-toc-section\" id=\"Tabela-de-angulos-notaveis\"><\/span>Tabela de &acirc;ngulos not&aacute;veis<span class=\"ez-toc-section-end\"><\/span><\/h2><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-3c736fd6-8b98-4c71-9b60-f56ead51301a\" 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-cc9972d3-b8fa-471a-9ff3-5acdc48c7873\" 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\">Veja que a tabela relaciona os &acirc;ngulos not&aacute;veis com suas respectivas<a href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/funcoes-trigonometricas\/\" target=\"_blank\"> fun&ccedil;&otilde;es trigonom&eacute;tricas<\/a>. Em cada linha, voc&ecirc; pode descobrir o <span id=\"urn:enhancement-9330f50a-fcf8-4614-9c5d-ddea8b9f644e\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/sen\">seno<\/span>, o <span id=\"urn:enhancement-4c8d3cfa-3e49-43cf-8af1-769199c263ae\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cosseno\">cosseno<\/span> e a <span id=\"urn:enhancement-ecd73da3-d0e5-43ae-8dc2-811f2994d9ab\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/tangente\">tangente<\/span> do grau de abertura em quest&atilde;o.&nbsp;<\/p><p class=\"wp-block-paragraph\">Para construir essa tabela, existe uma par&oacute;dia da m&uacute;sica natalina Jingle Bells, que ajuda na memoriza&ccedil;&atilde;o dessas fun&ccedil;&otilde;es trigonom&eacute;tricas, veja:<\/p><p class=\"has-text-align-center wp-block-paragraph\"><em>&ldquo;Um, dois, tr&ecirc;s,<\/em><br><em>Tr&ecirc;s, dois, um,&nbsp;<\/em><br><em>todos sobre dois.<\/em><\/p><p class=\"has-text-align-center wp-block-paragraph\"><em>A raiz vai no tr&ecirc;s&nbsp;<\/em><br><em>e tamb&eacute;m no dois.<\/em><\/p><p class=\"has-text-align-center wp-block-paragraph\"><em>A <span id=\"urn:enhancement-3e248ef0-c126-4732-a13f-33c2b373c198\" 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:<\/em><br><em>raiz de tr&ecirc;s, sobre tr&ecirc;s&nbsp;<\/em><br><em>Um,<\/em><br><em>&nbsp;raiz de tr&ecirc;s.&rdquo;<\/em><\/p><p class=\"wp-block-paragraph\">A can&ccedil;&atilde;o deve ser entoada enquanto a tabela se monta. Primeiro, s&atilde;o formadas as duas primeiras colunas, colocando um n&uacute;mero em cada casela. Depois, todas as c&eacute;dulas torna-se fra&ccedil;&otilde;es, com divis&atilde;o por dois, assim:<\/p><div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/bJ5eGHMUxXyMt64oG4ZjObxwML7hyJyfGxyYH_nbyJYeW_1uOap8EYvKKqTEW7Sh_JdqK1Wwdw0Y9EoDoCaf6Yh7RCEb_OTkcuPIV1mxlwPUMjWG7qpmdp0dzKn-F9b6pivTst7Qh1KbPtbJmj5I_mWHlQdd7AEt1brB6ny3nrlo2VWsBHWEllGV1X8s\" alt=\"Constru&ccedil;&atilde;o de tabela, &acirc;ngulos not&aacute;veis\"><\/figure>\n<\/div><p class=\"wp-block-paragraph\">Depois, em todos os n&uacute;meros dois ou tr&ecirc;s que estejam no numerador das fra&ccedil;&otilde;es, ser&atilde;o colocados sobre ra&iacute;zes quadradas<\/p><div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/7JfpwCJcmO7pKw7N6RhkJsZppL8xlIl2HK8qxA2kLxLJK8KzFkEFgd494ZLRCC6l_0OHE2tRSqMr7JcYHZQ5Uu33ysa30f0BzB6AJDdXlu7mtB11OsQnlMt7dJ3LzPdlVY4tYls33GCePkgXsm823LsbGvjjjd36qb7eMknam3g9_ZdZDvhCkOxnhQ8-XQ\" alt=\"Dicas de tabela, &acirc;ngulos not&aacute;veis\"><\/figure>\n<\/div><p class=\"wp-block-paragraph\">Depois, completa-se a &uacute;ltima coluna da tabela, exatamente como est&aacute; descrito nos versos da &uacute;ltima estrofe.<\/p><div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/dpPMWz1EORFJTnnoZl2ZWAEFot051KTsZOFD8WEEap_uC7pqE1iqQYI-AFX4m5ZqrO2LjV82kxMer5kZ43v0GZsYNRkp6uUGmadFDn0Pf7vqM8vAsnKhtXOorLZ6TWG86r7vpPalIzc4Ha_AOyQdwfakWH87u_S83Ft3UJbh5puouZhPXma014acNKNpKw\" alt=\"tabela: &acirc;ngulos not&aacute;veis\"><\/figure>\n<\/div><h2 class=\"wp-block-heading\" id=\"h-como-sao-encontradas-as-funcoes-dos-angulos-notaveis\"><span class=\"ez-toc-section\" id=\"Como-sao-encontradas-as-funcoes-dos-angulos-notaveis\"><\/span>Como s&atilde;o encontradas as fun&ccedil;&otilde;es dos &acirc;ngulos not&aacute;veis?<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\">As raz&otilde;es trigonom&eacute;tricas <span id=\"urn:enhancement-0fe6f7b6-4855-4c7a-9852-3c9b9b7cb965\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/sen\">seno<\/span>, <span id=\"urn:enhancement-04d2bbca-1a68-4af5-93f4-6b2c22966eb6\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cosseno\">cosseno<\/span> e <span id=\"urn:enhancement-84267769-1f7f-407f-a65d-62c84df0a8f9\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/tangente\">tangente<\/span> podem ser encontradas por meio do <span id=\"urn:enhancement-7d26b6bd-5219-4a92-9c6b-3fbbf48741c5\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/ciclo-trigonometrico-2\">ciclo trigonom&eacute;trico<\/span>. Mas o princ&iacute;pio que norteia esse conhecimento parte do estudo da <span id=\"urn:enhancement-e89db357-1c74-4712-bb35-872dc6408f3a\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/trigonometria\">trigonometria<\/span> no <span id=\"urn:local-annotation-377829\" class=\"textannotation disambiguated\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/triangulos-retangulos\">tri&acirc;ngulo ret&acirc;ngulo<\/span>.&nbsp;<\/p><div class=\"wp-block-image\">\n<figure class=\"aligncenter is-resized\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/tW9J2Nmvy1v7p_JIaD29aGYEwBSEAb4VVDc0HEKZ64TwfvZgyD_v1xHiXf2UKq16BHwyyKsTy8HIBEvyPo0e3cjJWZZodYYJiWJQqsFGiDYmGyZHyG9Z-4xJBTTV0AElr8csI2gqcpL8yh2zl84UzZW88TtTWUmuG4PHt3F9PtXuLRsQ6ZfsxIXLDduRLA\" alt=\"\" width=\"-113\" height=\"-113\"><figcaption class=\"wp-element-caption\">Imagem: Reprodu&ccedil;&atilde;o\/Wikimedia<\/figcaption><\/figure>\n<\/div><p class=\"wp-block-paragraph\">Em um tri&acirc;ngulo ret&acirc;ngulo, o lado que fica imediatamente oposto ao &acirc;ngulo reto &eacute; chamado de <span id=\"urn:enhancement-245afade-e175-4415-b8f7-6e4339d6bef6\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/hipotenusa\">hipotenusa<\/span>. Depois, &eacute; preciso definir um &acirc;ngulo de destaque para encontrar as raz&otilde;es trigonom&eacute;tricas.&nbsp;<\/p><p class=\"wp-block-paragraph\">No exemplo acima, considere que &Acirc; &eacute; o &acirc;ngulo em quest&atilde;o. Ent&atilde;o, o lado do <a href=\"http:\/\/Tri%C3%A2ngulos:%20defini%C3%A7%C3%A3o,%20per%C3%ADmetro,%20%C3%A1rea,%20classifica%C3%A7%C3%B5es\" target=\"_blank\" rel=\"noopener noreferrer\">tri&acirc;ngulo<\/a> que est&aacute; oposto a ele &eacute; chamado de <span id=\"urn:enhancement-ee4aead6-4eb8-4e2b-af3a-4aadfd29776a\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cateto\">cateto<\/span> oposto, enquanto que o lado que forma o &acirc;ngulo &Acirc; juntamente com a <span id=\"urn:enhancement-c699676a-0117-4d5c-a392-5ecceac038d2\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/hipotenusa\">hipotenusa<\/span> &eacute; o <span id=\"urn:enhancement-bd718a25-e377-4757-9b6e-8be1fe82d40d\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cateto\">cateto<\/span> adjacente. A partir desses conceitos, admite-se que:<\/p><p class=\"wp-block-paragraph\"><span id=\"urn:enhancement-469126a4-93d3-4e7a-9f7e-0d44ff79b462\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/sen\">seno<\/span> &Acirc; = <span id=\"urn:enhancement-0b860210-32f3-4ae6-ab78-7fa0159483be\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cateto\">cateto<\/span> oposto \/ <span id=\"urn:enhancement-903dfdcc-c3dd-4578-91ce-d390f11f1e15\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/hipotenusa\">hipotenusa<\/span><br><span id=\"urn:enhancement-af50ba50-2c7a-4f87-9ec0-29f4365a731b\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cosseno\">cosseno<\/span> &Acirc; = <span id=\"urn:enhancement-8ea575d2-847f-476e-a811-2b29564eeb22\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cateto\">cateto<\/span> adjacente \/ <span id=\"urn:enhancement-4a2fda70-f35d-415e-9f8e-f7b684cedff6\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/hipotenusa\">hipotenusa<\/span><br><span id=\"urn:enhancement-ed9cac84-9613-42e8-b2c3-1d807a9d3a61\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/tangente\">tangente<\/span> &Acirc; = <span id=\"urn:enhancement-39bba120-0875-44c4-bef5-b229e4931601\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cateto\">cateto<\/span> oposto \/ <span id=\"urn:enhancement-2151e805-2920-4173-bbce-c11620ff9508\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cateto\">cateto<\/span> adjacente<\/p><p class=\"wp-block-paragraph\">Com todas essas f&oacute;rmulas definidas e tamb&eacute;m a padroniza&ccedil;&atilde;o do <span id=\"urn:enhancement-c5bcf2bb-9f80-4d3f-bc32-5fd76eb52c7b\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/ciclo-trigonometrico-2\">ciclo trigonom&eacute;trico<\/span> de raio 1, fica mais f&aacute;cil definir as raz&otilde;es para os &acirc;ngulos not&aacute;veis. Acompanhe:<\/p><div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/sTyFNRxGAjY0XwPl4HPFWyQoEQLfxnHX0DvLZTt5DXB_gnSzyQlP7xeINWzc2baF8uTl---Dh2anolUTp2AMY3hzSHiMI4Ujtm3eIQ5Fm9_9lng8N9fYogNdA0slFOvo-5m6R0XhNNMPbPBqIwLWwETFiTwq_PLYSerxvKDXYYXTUXQHEp4pBs4gColIxA\" alt=\"demonstra&ccedil;&atilde;o &acirc;ngulos not&aacute;veis\"><figcaption class=\"wp-element-caption\">Imagem: Adapta&ccedil;&atilde;o\/Wikimedia<\/figcaption><\/figure>\n<\/div><p class=\"wp-block-paragraph\">Primeiro, encontramos o &acirc;ngulo de 30&ordm; no <a href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/circulo-trigonometrico\/\" target=\"_blank\">ciclo trigonom&eacute;trico<\/a> e, a partir disso, ampliamos o tri&acirc;ngulo ret&acirc;ngulo formado entre o raio do c&iacute;rculo e as proje&ccedil;&otilde;es dessa reta sobre o eixo x e o eixo y.<\/p><p class=\"wp-block-paragraph\">Perceba que raio = 1 e, nesse caso, o raio representa a <span id=\"urn:enhancement-13865a58-8c9a-46ef-91a5-2d7635cd1bbc\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/hipotenusa\">hipotenusa<\/span> do tri&acirc;ngulo ret&acirc;ngulo. O valor de 60&ordm; foi encontrado porque, obrigatoriamente, a soma dos &acirc;ngulos internos de um tri&acirc;ngulo deve ser igual a 180&ordm;.<\/p><p class=\"wp-block-paragraph\">Agora, vamos estabelecer as fun&ccedil;&otilde;es trigonom&eacute;tricas para 30&ordm;. Olhe para o tri&acirc;ngulo ret&acirc;ngulo A e perceba que, o <span id=\"urn:enhancement-ff9ff743-b518-43b3-9430-022fd6bdf7fe\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cateto\">cateto<\/span> imediatamente oposto a ele mede &frac12; (co = 1\/2), o <span id=\"urn:enhancement-071a5e57-ce0a-4f8a-99cf-3d95a37d8f9d\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cateto\">cateto<\/span> que est&aacute; pr&oacute;ximo ao &acirc;ngulo tem &radic;3\/2 de comprimento (ca = &radic;3\/2). Com isso:<\/p><p class=\"wp-block-paragraph\"><span id=\"urn:enhancement-0385159e-57ba-4e5b-bd52-00a4fb46f1fa\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/sen\">seno<\/span> 30&ordm; = <span id=\"urn:enhancement-9da0451d-4070-4e06-852e-3bf70dd61694\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cateto\">cateto<\/span> oposto \/ <span id=\"urn:enhancement-a3a8ea6d-0cb4-47e0-8be1-9eb5a3adb5c6\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/hipotenusa\">hipotenusa<\/span> = (&frac12;)\/1 &rArr; <span id=\"urn:enhancement-3d4dcf3d-d22e-46ba-94cd-058feefde4b8\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/sen\">sen<\/span> 30&ordm; = 1\/2<br><span id=\"urn:enhancement-5fa26ce4-9052-437f-8537-d30b2df3bdec\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cosseno\">cosseno<\/span> 30&ordm; = <span id=\"urn:enhancement-a02ed36f-9ba8-4718-a7d5-f9b7f79237e3\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cateto\">cateto<\/span> adjacente \/ <span id=\"urn:enhancement-2a700313-3ae7-46ca-b939-47798050478d\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/hipotenusa\">hipotenusa<\/span> = (&radic;3\/2)\/1&rArr; <span id=\"urn:enhancement-4da714e8-d209-4ef2-8a65-fb17ee28989f\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cosseno\">cos<\/span> 30&ordm; = &radic;3\/2<br><span id=\"urn:enhancement-9c31de12-d189-473d-97fa-57c45594346e\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/tangente\">tangente<\/span> 30&ordm; = <span id=\"urn:enhancement-f764c58e-64bb-4893-a536-6cca4008597c\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cateto\">cateto<\/span> oposto \/ <span id=\"urn:enhancement-05947953-fe06-41a9-9db2-0e60de939381\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cateto\">cateto<\/span> adjacente = (1\/2) \/ (&radic;3\/2)&nbsp; &rArr; tg 30&ordm; = &radic;3\/3<\/p><p class=\"wp-block-paragraph\">Depois, com procedimentos semelhantes aos que foram utilizados anteriormente, vamos descobrir as raz&otilde;es trigonom&eacute;tricas para o &acirc;ngulo de 60&ordm;. Lembre-se de que, dessa vez, os catetos opostos e adjacentes t&ecirc;m valores invertidos: ca = &frac12; e co = &radic;3\/2.<\/p><p class=\"wp-block-paragraph\"><span id=\"urn:enhancement-7d5f2ad8-d6d9-425e-8c38-db5b0e80a80b\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/sen\">seno<\/span> 60&ordm; = <span id=\"urn:enhancement-455f1612-40a5-474d-822d-6de20db996ca\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cateto\">cateto<\/span> oposto \/ <span id=\"urn:enhancement-91662ff7-1f39-401b-a3fa-299db80f7432\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/hipotenusa\">hipotenusa<\/span> = (&radic;3\/2)\/1 &rArr; <span id=\"urn:enhancement-3fe57bae-2377-4309-8849-42914e7f4490\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/sen\">sen<\/span> 60&ordm; = &radic;3\/2<br><span id=\"urn:enhancement-26458dc8-1fb9-48c7-a5bd-d66ada4f0d84\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cosseno\">cosseno<\/span> 60&ordm; = <span id=\"urn:enhancement-ba0dbb34-8f51-4bfb-be8b-ac68f7dafd6e\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cateto\">cateto<\/span> adjacente \/ <span id=\"urn:enhancement-8018442d-ea57-4666-b7db-306f6c337dc9\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/hipotenusa\">hipotenusa<\/span> = (&frac12;)\/1&nbsp; &rArr; <span id=\"urn:enhancement-cddb9b38-46e5-4f09-b322-9789e59fe604\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cosseno\">cos<\/span> 60&ordm; =&nbsp; &frac12;<br><span id=\"urn:enhancement-bda548ff-44ea-4646-8188-67a0009eead5\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/tangente\">tangente<\/span> 60&ordm; = <span id=\"urn:enhancement-4d07fcab-552c-4f2d-b702-9722ea4c27da\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cateto\">cateto<\/span> oposto \/ <span id=\"urn:enhancement-575382e2-4d7f-4eeb-96cb-ded0e47d3cac\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cateto\">cateto<\/span> adjacente = (&radic;3\/2) \/ (1\/2) &rArr; tg 60&ordm; = &radic;3<\/p><p class=\"wp-block-paragraph\">&Eacute; importante notar que os valores de <span id=\"urn:enhancement-112b7418-b581-40d2-afc9-5cfb396ef361\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/sen\">seno<\/span> e <span id=\"urn:enhancement-643ba5d7-e724-4cb4-9272-0ac19849a1d2\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cosseno\">cosseno<\/span> de &acirc;ngulos complementares (que juntos somam 90&ordm;) ser&atilde;o invertidos, ou seja, o <span id=\"urn:enhancement-3dc6ca9e-6976-46f7-bdc7-8769a344885e\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/sen\">seno<\/span> de um &eacute; o <span id=\"urn:enhancement-52dc6ad0-bb90-4280-b8ce-525949969f73\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cosseno\">cosseno<\/span> do outro. &Eacute; exatamente isso que observamos entre 30&ordm; e 60&ordm;.<\/p><p class=\"wp-block-paragraph\">Por fim, vamos encontrar as raz&otilde;es trigonom&eacute;tricas de 45&ordm;. Em um quadrado, uma diagonal que parte de um v&eacute;rtice a outro divide o &acirc;ngulo de 90&ordm; graus em dois &acirc;ngulos de 45&ordm;. Diante disso, s&atilde;o formados dois tri&acirc;ngulos ret&acirc;ngulos. Vamos estudar um desses, para um quadrado de lado igual a 1.<\/p><div class=\"wp-block-image\">\n<figure class=\"aligncenter is-resized\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/X7S0HfwvnNxBwwAaidcIkiFE5AKFJ9EfrrUtNsgYEPtsCCT2v5dKx737FXqOHd55yxCMbw6xxxtbZn5NlRh_hcgIdZfNhlushj3kVyZj2En3VZCLjL2y4tLz1zyf-udY5k7V5XN7OQue5H-eJLkHikEV_4qi2FtSWTbpe-g72ZPMO1iJYzFedL8cDPf0MA\" alt=\"tri&acirc;ngulo\" width=\"-169\" height=\"-174\"><figcaption class=\"wp-element-caption\">Imagem: Reprodu&ccedil;&atilde;o\/Wikimedia<\/figcaption><\/figure>\n<\/div><p class=\"wp-block-paragraph\">Perceba que, em tri&acirc;ngulos como esse, os catetos possuem valores iguais. Ent&atilde;o, seguimos o estudo substituindo ca = co = 1. Ao mesmo tempo, est&aacute; definido que a diagonal de um quadrado &eacute; igual a L.&radic;2, por isso, nesse caso, a <span id=\"urn:enhancement-0671e193-45b2-4b22-832b-e36188ccf8f1\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/hipotenusa\">hipotenusa<\/span> &eacute; igual a 1.&radic;2 &rArr; <span id=\"urn:enhancement-cb246476-4daa-4466-8a0f-5653d42dc6fb\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/hipotenusa\">hipotenusa<\/span> = &radic;2,<\/p><p class=\"wp-block-paragraph\"><span id=\"urn:enhancement-b0db3f8b-5dda-4e7f-bcab-a8a1e7e43460\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/sen\">seno<\/span> 45&ordm; = <span id=\"urn:enhancement-eec38e10-4310-4cd7-b6c4-95d0f96db5e5\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cateto\">cateto<\/span> oposto \/ <span id=\"urn:enhancement-470121b1-f49e-4380-a139-e3eb52679161\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/hipotenusa\">hipotenusa<\/span> = 1\/&radic;2 &rArr; <span id=\"urn:enhancement-8082e405-292b-4bf6-8acd-0b5c0d1dbde0\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/sen\">sen<\/span> 45&ordm; = &radic;2\/2<\/p><p class=\"wp-block-paragraph\"><span id=\"urn:enhancement-7f53b9ac-f02e-47a0-833c-c3a0ed687e8d\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cosseno\">cosseno<\/span> 45&ordm; = <span id=\"urn:enhancement-5eeaf3d8-a6d4-4b23-b1b1-5313d69416b4\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cateto\">cateto<\/span> adjacente \/ <span id=\"urn:enhancement-82e6dab4-939c-478e-9aa1-a1a798bdee85\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/hipotenusa\">hipotenusa<\/span> =&nbsp; 1\/&radic;2 &rArr; <span id=\"urn:enhancement-b9a49926-70fc-436d-b7f6-ac98077a84b4\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cosseno\">cos<\/span> 45&ordm; = &radic;2\/2<\/p><p class=\"wp-block-paragraph\"><span id=\"urn:enhancement-3a859d05-d7b2-468c-8fab-05c915bb9ec2\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/tangente\">tangente<\/span> 45&ordm; = <span id=\"urn:enhancement-1a1d2e23-c6ef-4768-b153-2f3caf2a431d\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cateto\">cateto<\/span> oposto \/ <span id=\"urn:enhancement-9f14ccdb-d948-4fbd-8673-aecad4689384\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cateto\">cateto<\/span> adjacente = &radic;2\/&radic;2 &rArr; tg 45&ordm; = 1<\/p><p class=\"wp-block-paragraph\">+ Veja tamb&eacute;m: <a href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/arcos-e-angulos\/\" target=\"_blank\">Arcos e &acirc;ngulos: o que s&atilde;o, circunfer&ecirc;ncia e como medir<\/a><\/p><h2 class=\"wp-block-heading\" id=\"h-outros-angulos-importantes\"><span class=\"ez-toc-section\" id=\"Outros-angulos-importantes\"><\/span>Outros &acirc;ngulos importantes<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\">Por fim, conhe&ccedil;a tamb&eacute;m outros &acirc;ngulos importantes e seus valores de <span id=\"urn:enhancement-6e6bd0cb-3fbc-4504-8555-7e0129e73be5\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/sen\">seno<\/span> e <span id=\"urn:enhancement-046c50f8-f996-442f-b75e-fecd56c9cba0\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cosseno\">cosseno<\/span>:<\/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-701e1408-d8ca-4f23-ae72-f82529f0ce60\" 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-d8083f35-f95a-4acb-9a3c-f76a75343c77\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cosseno\">cos<\/span><\/strong><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><strong>0&ordm;<\/strong><\/td><td class=\"has-text-align-center\" data-align=\"center\">0<\/td><td class=\"has-text-align-center\" data-align=\"center\">1<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><strong>90&ordm;<\/strong><\/td><td class=\"has-text-align-center\" data-align=\"center\">1<\/td><td class=\"has-text-align-center\" data-align=\"center\">0<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><strong>180&ordm;<\/strong><\/td><td class=\"has-text-align-center\" data-align=\"center\">0<\/td><td class=\"has-text-align-center\" data-align=\"center\">-1<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><strong>270&ordm;<\/strong><\/td><td class=\"has-text-align-center\" data-align=\"center\">-1<\/td><td class=\"has-text-align-center\" data-align=\"center\">0<\/td><\/tr><\/tbody><\/table><\/figure><h2 class=\"wp-block-heading\" id=\"h-questao-de-vestibular-sobre-angulos-notaveis\"><span class=\"ez-toc-section\" id=\"Questao-de-vestibular-sobre-angulos-notaveis\"><\/span>Quest&atilde;o de vestibular sobre &acirc;ngulos not&aacute;veis<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\">(Enem\/2010)<\/p><p class=\"wp-block-paragraph\">Um bal&atilde;o atmosf&eacute;rico, lan&ccedil;ado em Bauru (343 quil&ocirc;metros a Noroeste de S&atilde;o Paulo), na noite do &uacute;ltimo domingo, caiu nesta segunda-feira em Cuiab&aacute; Paulista, na regi&atilde;o de Presidente Prudente, assustando agricultores da regi&atilde;o. O artefato faz parte do programa Projeto Hibiscus, desenvolvido por Brasil, Fran&ccedil;a, Argentina, Inglaterra e It&aacute;lia, para a medi&ccedil;&atilde;o do comportamento da camada de oz&ocirc;nio, e sua descida se deu ap&oacute;s o cumprimento do tempo previsto de medi&ccedil;&atilde;o.<\/p><div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/57o4S8Yz4Ssk3glWf9vTus7BGOqBt6ywaT1vzJDKq485l0scuBLqyiYTdRmcHFNCoh6kPmmF3SJ1_UcU1VRriQ7_poeUVq1KLQhy1OpN9fEFAl9wZQALuIUuEoOFj4gikB4exYbAafWkncKlYGnkTJut2YZNNttbsmy7Btx9xqDUO9OEEPeqUeI3vX73TA\" alt=\"\"><\/figure>\n<\/div><p class=\"wp-block-paragraph\">Na data do acontecido, duas pessoas avistaram o bal&atilde;o. Uma estava a 1,8 km da posi&ccedil;&atilde;o vertical do bal&atilde;o e o avistou sob um &acirc;ngulo de 60&ordm;; a outra estava a 5,5 km da posi&ccedil;&atilde;o vertical do bal&atilde;o, alinhada com a primeira, e no mesmo sentido, conforme se v&ecirc; na figura, e o avistou sob um &acirc;ngulo de 30&ordm;.<\/p><p class=\"wp-block-paragraph\">Qual a altura aproximada em que se encontrava o bal&atilde;o?<\/p><p class=\"wp-block-paragraph\">a) 1,8km<br>b) 1,9km<br>c) 3,1km<br>d) 3,7km<br>e) 5,5km<\/p><p class=\"wp-block-paragraph\">Considera-se que a altura do bal&atilde;o &eacute; o <span id=\"urn:enhancement-a15f4aee-1003-4566-bf5e-a4832da5e37c\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cateto\">cateto<\/span> oposto para o &acirc;ngulo de 60&ordm;, com <span id=\"urn:enhancement-62dc6c22-25bf-4573-a5d9-7d126b616f3d\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cateto\">cateto<\/span> adjacente de 1,8km. Ao mesmo tempo, &eacute; tamb&eacute;m <span id=\"urn:enhancement-ab65c0c8-98d4-4e71-b40a-550cbf3c575b\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cateto\">cateto<\/span> oposto de 30&ordm;, quando o <span id=\"urn:enhancement-c54e7f3a-fbd9-4b4d-b42b-b134ec06ee2b\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cateto\">cateto<\/span> adjacente mede 1,8 + 3,7 km.&nbsp;<\/p><p class=\"wp-block-paragraph\">A partir disso, vamos utilizar as raz&otilde;es trigonom&eacute;tricas dos &acirc;ngulos not&aacute;veis e descobrir a altura de queda.&nbsp;<\/p><p class=\"wp-block-paragraph\">&nbsp;tg 60&ordm; = &radic;3<\/p><p class=\"wp-block-paragraph\"><span id=\"urn:enhancement-61b44043-cb92-4726-8d99-1ccc07fbea2d\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/tangente\">tangente<\/span> 60&ordm; = <span id=\"urn:enhancement-5b3e7855-8b87-44f0-b2f6-26a12d7a8a85\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cateto\">cateto<\/span> oposto \/ <span id=\"urn:enhancement-8792618c-95d7-47f5-9e56-32367d0fc74d\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cateto\">cateto<\/span> adjacente<\/p><p class=\"wp-block-paragraph\">&radic;3 = altura \/ 1,8&nbsp;<br>&radic;3.1,8 = altura<br>altura = 3,11 km<\/p><p class=\"wp-block-paragraph\">tg 30&ordm; = &radic;3\/3<\/p><p class=\"wp-block-paragraph\"><span id=\"urn:enhancement-6603a46c-3718-4040-966c-9d11a4520c76\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/tangente\">tangente<\/span> 30&ordm; = <span id=\"urn:enhancement-38cb44fb-141a-4307-9b1c-88a7cc4373f3\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cateto\">cateto<\/span> oposto \/ <span id=\"urn:enhancement-bc55910f-ede4-45cc-8439-1ea38b0a86ff\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/cateto\">cateto<\/span> adjacente<\/p><p class=\"wp-block-paragraph\">&radic;3\/3 = altura \/ 1,8 + 3,7<br>&radic;3\/3 = altura \/ 5,5<br>5,5 . (&radic;3\/3)&nbsp; = altura<br>altura &#8771; 3,1 km<\/p><p class=\"wp-block-paragraph\">Note que o conhecimento sobre o valor da <span id=\"urn:enhancement-4d795029-ec36-40fd-a428-54c6383d7c7c\" class=\"textannotation disambiguated wl-thing wl-no-link\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/tangente\">tangente<\/span> de pelo menos um dos &acirc;ngulos not&aacute;veis seria suficiente para responder a quest&atilde;o com a alternativa correta, a letra <strong>C<\/strong>.&nbsp;<\/p><h2 class=\"wp-block-heading\" id=\"h-aprenda-mais-matematica-com-o-estrategia\"><span class=\"ez-toc-section\" id=\"Aprenda-mais-matematica-com-o-Estrategia\"><\/span>Aprenda mais matem&aacute;tica com o Estrat&eacute;gia&nbsp;<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\">Com as trilhas estrat&eacute;gias dos cursos preparat&oacute;rios da Coruja, voc&ecirc; pode percorrer as disciplinas necess&aacute;rias para o seu vestibular. Elas s&atilde;o um guia do que e quando estudar os temas, preparadas por alunos aprovados nas institui&ccedil;&otilde;es mais concorridas do pa&iacute;s. Clique no banner abaixo e saiba mais sobre nossos cursos:<\/p><p class=\"wp-block-paragraph\"><a id=\"cta\" class=\"cta-imagem\" href=\"https:\/\/vestibulares.estrategia.com\/cursos\/\" target=\"blank\">\n                <img decoding=\"async\" width=\"100%\" src=\"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2025\/07\/ESTRATEGIA.jpg\" alt=\"\" title=\"ESTRAT\u00c9GIA\">\n        <\/a>\n\n\n\n<\/p><h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Veja-tambem\"><\/span>Veja tamb&eacute;m:<span class=\"ez-toc-section-end\"><\/span><\/h2><ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/vestibulares.estrategia.com\/portal\/matematica\/trigonometria\/\">Trigonometria: defini&ccedil;&atilde;o e principais conceitos!<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/circulo-trigonometrico\/\">C&iacute;rculo trigonom&eacute;trico: o que &eacute;, exemplos, usos e aplica&ccedil;&otilde;es<\/a>&nbsp;<\/li>\n\n\n\n<li><a href=\"https:\/\/vestibulares.estrategia.com\/portal\/matematica\/aproximacao-senx-tgx-x-e-suas-aplicacoes-no-vestibular\/\">Aproxima&ccedil;&atilde;o sen(x) = tg(x) = xe suas aplica&ccedil;&otilde;es no vestibular<\/a>&nbsp;<\/li>\n\n\n\n<li><a href=\"https:\/\/vestibulares.estrategia.com\/portal\/matematica\/seno-cosseno-tangente\/\">Seno, cosseno e tangente: como calcular, tabela e formulas<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/vestibulares.estrategia.com\/portal\/matematica\/operacoes-com-arcos\/\">Opera&ccedil;&otilde;es com arcos: o que s&atilde;o, ciclo trigonom&eacute;trico e como fazer<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/vestibulares.estrategia.com\/portal\/matematica\/funcoes-trigonometricas\/\">Fun&ccedil;&otilde;es trigonom&eacute;tricas: f&oacute;rmulas, gr&aacute;ficos e aplica&ccedil;&otilde;es<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/trigonometria-no-triangulo-retangulo\/\">Trigonometria no tri&acirc;ngulo ret&acirc;ngulo: f&oacute;rmulas e quest&atilde;o<\/a>&nbsp;<\/li>\n<\/ul><\/p>\n","protected":false},"excerpt":{"rendered":"Os &acirc;ngulos not&aacute;veis s&atilde;o valores de &acirc;ngulos e suas fun&ccedil;&otilde;es trigonom&eacute;tricas que aparecem com frequ&ecirc;ncia no cotidiano e&hellip;\n","protected":false},"author":23,"featured_media":86323,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[29],"tags":[],"class_list":["post-84773","post","type-post","status-publish","format-standard","has-post-thumbnail","category-matematica"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v25.9 (Yoast SEO v28.2) - https:\/\/yoast.com\/product\/yoast-seo-premium-wordpress\/ -->\n<title>\u00c2ngulos not\u00e1veis: 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