{"id":4402,"date":"2010-10-20T22:41:43","date_gmt":"2010-10-20T21:41:43","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=4402"},"modified":"2022-01-21T02:34:06","modified_gmt":"2022-01-21T02:34:06","slug":"um-circulo-trigonometrico","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=4402","title":{"rendered":"Um c\u00edrculo trigonom\u00e9trico"},"content":{"rendered":"<p><ul id='GTTabs_ul_4402' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_4402' class='GTTabs_curr'><a  id=\"4402_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_4402' ><a  id=\"4402_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_4402'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pag97-61.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"4405\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=4405\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pag97-61.jpg\" data-orig-size=\"323,317\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;HP pstc4380&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;}\" data-image-title=\"C\u00edrculo trigonom\u00e9trico\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pag97-61.jpg\" class=\"alignright wp-image-4405\" title=\"C\u00edrculo trigonom\u00e9trico\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pag97-61-300x294.jpg\" alt=\"\" width=\"240\" height=\"236\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pag97-61-300x294.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pag97-61-150x147.jpg 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pag97-61.jpg 323w\" sizes=\"auto, (max-width: 240px) 100vw, 240px\" \/><\/a>Na figura est\u00e1 representado um c\u00edrculo trigonom\u00e9trico.<\/p>\n<p>O segmento [OA] \u00e9 perpendicular a [OB]. O \u00e2ngulo COB tem amplitude $\\alpha $ radianos.<\/p>\n<ol>\n<li>Calcule as coordenadas do ponto A.<\/li>\n<li>Determine o valor exato da express\u00e3o: $tg\\,(\\pi -\\alpha )+\\cos (\\pi +\\alpha )$.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_4402' onClick='GTTabs_show(1,4402)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_4402'>\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\/2010\/10\/pag97-61.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"4405\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=4405\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pag97-61.jpg\" data-orig-size=\"323,317\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;HP pstc4380&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;}\" data-image-title=\"C\u00edrculo trigonom\u00e9trico\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pag97-61.jpg\" class=\"alignright wp-image-4405\" title=\"C\u00edrculo trigonom\u00e9trico\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pag97-61-300x294.jpg\" alt=\"\" width=\"240\" height=\"236\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pag97-61-300x294.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pag97-61-150x147.jpg 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pag97-61.jpg 323w\" sizes=\"auto, (max-width: 240px) 100vw, 240px\" \/><\/a>Como A \u00e9 um ponto da circunfer\u00eancia que define o c\u00edrculo trigonom\u00e9trico, ent\u00e3o as suas coordenadas s\u00e3o $A\\,\\left( \\cos (\\frac{\\pi }{2}-\\alpha ),sen\\,(\\frac{\\pi }{2}-\\alpha ) \\right)$, com $\\begin{matrix}<br \/>\nsen\\,(\\frac{\\pi }{2}-\\alpha )=\\frac{1}{2} &amp; \\wedge\u00a0 &amp; (\\frac{\\pi }{2}-\\alpha )\\in 1.{}^\\text{o}Q\u00a0 \\\\<br \/>\n\\end{matrix}$.<br \/>\nDeste modo, \\[\\cos (\\frac{\\pi }{2}-\\alpha )=\\frac{\\sqrt{3}}{2}\\] e, portanto, \\[A\\,(\\frac{\\sqrt{3}}{2},\\frac{1}{2})\\]<br \/>\n\u00ad<\/li>\n<li>Como\u00a0\\[sen\\,(\\frac{\\pi }{2}-\\alpha )=\\frac{1}{2}\\Leftrightarrow \\cos \\alpha =\\frac{1}{2}\\] e \\[\\cos (\\frac{\\pi }{2}-\\alpha )=\\frac{\\sqrt{3}}{2}\\Leftrightarrow sen\\,\\alpha =\\frac{\\sqrt{3}}{2}\\] temos: \\[\\begin{array}{*{35}{l}}<br \/>\ntg\\,(\\pi -\\alpha )+\\cos (\\pi +\\alpha ) &amp; = &amp; tg\\,(-\\alpha )-\\cos \\alpha\u00a0\u00a0 \\\\<br \/>\n{} &amp; = &amp; -tg\\,\\alpha -\\cos \\alpha\u00a0\u00a0 \\\\<br \/>\n{} &amp; = &amp; -\\frac{\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}}-\\frac{1}{2}\u00a0 \\\\<br \/>\n{} &amp; = &amp; -\\sqrt{3}-\\frac{1}{2}\u00a0 \\\\<br \/>\n\\end{array}\\]<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_4402' onClick='GTTabs_show(0,4402)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Na figura est\u00e1 representado um c\u00edrculo trigonom\u00e9trico. O segmento [OA] \u00e9 perpendicular a [OB]. O \u00e2ngulo COB tem amplitude $\\alpha $ radianos. Calcule as coordenadas do ponto A. Determine o valor&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20807,"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-4402","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":2633,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/11V1Pag097-61_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/4402","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=4402"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/4402\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20807"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4402"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4402"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4402"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=4402"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}