{"id":3826,"date":"2010-10-09T00:14:18","date_gmt":"2010-10-08T23:14:18","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=3826"},"modified":"2022-01-13T14:40:40","modified_gmt":"2022-01-13T14:40:40","slug":"sabendo-que","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=3826","title":{"rendered":"Sabendo que&#8230;"},"content":{"rendered":"<p><ul id='GTTabs_ul_3826' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_3826' class='GTTabs_curr'><a  id=\"3826_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_3826' ><a  id=\"3826_1\" onMouseOver=\"GTTabsShowLinks('Resolu\u00e7\u00e3o'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Resolu\u00e7\u00e3o<\/a><\/li>\n<\/ul>\n\n<div class='GTTabs_divs GTTabs_curr_div' id='GTTabs_0_3826'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<ol>\n<li>Sabendo que $tg\\,\\alpha =\\frac{3}{4}$ e $\\pi &lt;\\alpha &lt;\\frac{3\\pi }{2}$, determine $sen\\,\\alpha $ e $\\cos \\alpha $.<\/li>\n<li>Determine $sen\\,\\alpha $ e $\\cos \\alpha $, sabendo que\u00a0$tg\\,\\alpha =-2$ e $-\\frac{\\pi }{2}&lt;\\alpha &lt;0$.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_3826' onClick='GTTabs_show(1,3826)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_3826'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li>Dado que $1+t{{g}^{2}}\\alpha =\\frac{1}{{{\\cos }^{2}}\\alpha }$, para $\\cos \\alpha \\ne 0$, obt\u00e9m-se para $tg\\,\\alpha =\\frac{3}{4}$:<br \/>\n\\[\\begin{array}{*{35}{l}}<br \/>\n1+{{\\left( \\frac{3}{4} \\right)}^{2}}=\\frac{1}{{{\\cos }^{2}}\\alpha } &amp; \\Leftrightarrow\u00a0 &amp; \\frac{25}{16}=\\frac{1}{{{\\cos }^{2}}\\alpha }\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; {{\\cos }^{2}}\\alpha =\\frac{16}{25}\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\nComo $\\pi &lt;\\alpha &lt;\\frac{3\\pi }{2}$, ent\u00e3o (o cosseno \u00e9 negativo no 3.\u00ba Q): \\[\\cos \\alpha =-\\sqrt{\\frac{16}{25}}=-\\frac{4}{5}\\]<br \/>\nFinalmente, substituindo os valores conhecidos na rela\u00e7\u00e3o $tg\\,\\alpha =\\frac{sen\\,\\alpha }{\\cos \\alpha }$, vem:<br \/>\n\\[\\begin{array}{*{35}{l}}<br \/>\n\\frac{3}{4}=\\frac{sen\\,\\alpha }{-\\frac{4}{5}} &amp; \\Leftrightarrow\u00a0 &amp; sen\\,\\alpha =\\frac{3\\times (-\\frac{4}{5})}{4}\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; sen\\,\\alpha =-\\frac{3}{5}\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\n\u00ad<\/li>\n<li>Dado que $1+t{{g}^{2}}\\alpha =\\frac{1}{{{\\cos }^{2}}\\alpha }$, para $\\cos \\alpha \\ne 0$, obt\u00e9m-se para $tg\\,\\alpha =-2$:<br \/>\n\\[1+{{(-2)}^{2}}=\\frac{1}{{{\\cos }^{2}}\\alpha }\\Leftrightarrow {{\\cos }^{2}}\\alpha =\\frac{1}{5}\\]<br \/>\nComo $-\\frac{\\pi }{2}&lt;\\alpha &lt;0$, ent\u00e3o (o cosseno \u00e9 positivo no 4.\u00ba Q): \\[\\cos \\alpha =+\\sqrt{\\frac{1}{5}}=\\frac{\\sqrt{5}}{5}\\]<br \/>\nFinalmente, substituindo os valores conhecidos na rela\u00e7\u00e3o $tg\\,\\alpha =\\frac{sen\\,\\alpha }{\\cos \\alpha }$, vem:<br \/>\n\\[\\begin{array}{*{35}{l}}<br \/>\n-2=\\frac{sen\\,\\alpha }{\\frac{\\sqrt{5}}{5}} &amp; \\Leftrightarrow\u00a0 &amp; sen\\,\\alpha =-\\frac{2\\sqrt{5}}{5}\u00a0 \\\\<br \/>\n\\end{array}\\]<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_3826' onClick='GTTabs_show(0,3826)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Sabendo que $tg\\,\\alpha =\\frac{3}{4}$ e $\\pi &lt;\\alpha &lt;\\frac{3\\pi }{2}$, determine $sen\\,\\alpha $ e $\\cos \\alpha $. Determine $sen\\,\\alpha $ e $\\cos \\alpha $, sabendo que\u00a0$tg\\,\\alpha =-2$ e $-\\frac{\\pi }{2}&lt;\\alpha &lt;0$. Resolu\u00e7\u00e3o&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":19468,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[98,97,99],"tags":[422,423],"series":[],"class_list":["post-3826","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-11--ano","category-aplicando","category-trigonometria","tag-11-o-ano","tag-trigonometria"],"views":1720,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2021\/12\/Mat132.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/3826","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=3826"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/3826\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19468"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3826"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=3826"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=3826"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=3826"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}