{"id":14127,"date":"2018-03-12T01:39:04","date_gmt":"2018-03-12T01:39:04","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=14127"},"modified":"2022-01-11T10:41:19","modified_gmt":"2022-01-11T10:41:19","slug":"observa-o-triangulo-abc-retangulo-em-a","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=14127","title":{"rendered":"Observa o tri\u00e2ngulo [ABC], ret\u00e2ngulo em A"},"content":{"rendered":"<p><ul id='GTTabs_ul_14127' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_14127' class='GTTabs_curr'><a  id=\"14127_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_14127' ><a  id=\"14127_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_14127'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag056-5.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"14128\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=14128\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag056-5.png\" data-orig-size=\"470,245\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"Tri\u00e2ngulo ret\u00e2ngulo\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag056-5.png\" class=\"alignright size-medium wp-image-14128\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag056-5-300x156.png\" alt=\"\" width=\"300\" height=\"156\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag056-5-300x156.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag056-5.png 470w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a>Observa o tri\u00e2ngulo [<em>ABC<\/em>], ret\u00e2ngulo em <em>A<\/em>.<\/p>\n<p>Sabendo que \\(\\cos \\alpha = \\frac{{12}}{{13}}\\), calcula:<\/p>\n<ol>\n<li>os valores exatos de\u00a0\\({{\\mathop{\\rm sen}\\nolimits} \\alpha }\\) e de \\({\\mathop{\\rm tg}\\nolimits} \\alpha \\).<\/li>\n<li>o comprimento da hipotnusa do tri\u00e2ngulo [<em>ABC<\/em>].<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_14127' onClick='GTTabs_show(1,14127)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_14127'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag056-5.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"14128\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=14128\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag056-5.png\" data-orig-size=\"470,245\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"Tri\u00e2ngulo ret\u00e2ngulo\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag056-5.png\" class=\"alignright wp-image-14128 size-medium\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag056-5-300x156.png\" alt=\"\" width=\"300\" height=\"156\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag056-5-300x156.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag056-5.png 470w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a>Aplicando a F\u00f3rmula Fundamental da Trigonometria, vem:<br \/>\n\\[\\begin{array}{*{20}{l}}{{{{\\mathop{\\rm sen}\\nolimits} }^2}\\alpha + {{\\left( {\\frac{{12}}{{13}}} \\right)}^2} = 1}&amp; \\Leftrightarrow &amp;{{{{\\mathop{\\rm sen}\\nolimits} }^2}\\alpha = 1 &#8211; \\frac{{144}}{{169}}}\\\\{}&amp; \\Leftrightarrow &amp;{{{{\\mathop{\\rm sen}\\nolimits} }^2}\\alpha = \\frac{{25}}{{169}}}\\end{array}\\]<br \/>\nComo\u00a0\\({\\mathop{\\rm sen}\\nolimits} \\alpha &gt; 0\\), ent\u00e3o\u00a0\\({\\mathop{\\rm sen}\\nolimits} \\alpha = \\sqrt {\\frac{{25}}{{169}}} = \\frac{5}{{13}}\\).<br \/>\nPor \u00faltimo, vem: \\({\\mathop{\\rm tg}\\nolimits} \\alpha = \\frac{{{\\mathop{\\rm sen}\\nolimits} \\alpha }}{{\\cos \\alpha }} = \\frac{{{\\textstyle{5 \\over {13}}}}}{{{\\textstyle{{12} \\over {13}}}}} = \\frac{5}{{13}} \\times \\frac{{13}}{{12}} = \\frac{5}{{12}}\\).<br \/>\n\u00ad<\/li>\n<li>Ora,<br \/>\n\\[\\begin{array}{*{20}{l}}{{\\mathop{\\rm sen}\\nolimits} \\alpha = \\frac{{\\overline {AC} }}{{\\overline {BC} }}}&amp; \\Leftrightarrow &amp;{\\frac{5}{{13}} = \\frac{{16}}{{\\overline {BC} }}}\\\\{}&amp; \\Leftrightarrow &amp;{\\overline {BC} = \\frac{{13 \\times 16}}{5}}\\\\{}&amp; \\Leftrightarrow &amp;{\\overline {BC} = 41,6}\\end{array}\\]<br \/>\nPortanto,\u00a0\\(\\overline {BC} = 41,6\\) cm.<\/li>\n<\/ol><\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_14127' onClick='GTTabs_show(0,14127)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Observa o tri\u00e2ngulo [ABC], ret\u00e2ngulo em A. Sabendo que \\(\\cos \\alpha = \\frac{{12}}{{13}}\\), calcula: os valores exatos de\u00a0\\({{\\mathop{\\rm sen}\\nolimits} \\alpha }\\) e de \\({\\mathop{\\rm tg}\\nolimits} \\alpha \\). o comprimento da hipotnusa&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":14130,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[213,97,489],"tags":[426,491,493,490,492],"series":[],"class_list":["post-14127","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-9--ano","category-aplicando","category-trigonometria-9--ano","tag-9-o-ano","tag-cosseno","tag-razao-trigonometrica","tag-seno","tag-tangente"],"views":3417,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag056-5a.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/14127","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=14127"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/14127\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/14130"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=14127"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=14127"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=14127"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=14127"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}