{"id":3981,"date":"2010-10-10T01:56:32","date_gmt":"2010-10-10T00:56:32","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=3981"},"modified":"2022-01-21T02:08:08","modified_gmt":"2022-01-21T02:08:08","slug":"rascunho-9","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=3981","title":{"rendered":"Outro hex\u00e1gono regular"},"content":{"rendered":"<p><ul id='GTTabs_ul_3981' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_3981' class='GTTabs_curr'><a  id=\"3981_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_3981' ><a  id=\"3981_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_3981'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/48.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"3990\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=3990\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/48.jpg\" data-orig-size=\"328,375\" 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=\"Hex\u00e1gono regular\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/48.jpg\" class=\"alignright wp-image-3990\" title=\"Hex\u00e1gono regular\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/48-262x300.jpg\" alt=\"\" width=\"240\" height=\"274\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/48-262x300.jpg 262w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/48-131x150.jpg 131w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/48.jpg 328w\" sizes=\"auto, (max-width: 240px) 100vw, 240px\" \/><\/a>Sobre o c\u00edrculo trigonom\u00e9trico de centro O da figura est\u00e1 representado um hex\u00e1gono regular. A amplitude positiva m\u00ednima do \u00e2ngulo generalizado AOB \u00e9 $\\frac{\\pi }{9}$ radianos.<\/p>\n<ol>\n<li>Qual \u00e9, em radianos, a express\u00e3o geral das amplitudes do \u00e2ngulo AOB?<\/li>\n<li>Prove que $\\frac{4\\pi }{9}$ radianos \u00e9 uma amplitude do \u00e2ngulo orientado AOC.<\/li>\n<li>Indique, em radianos, a amplitude dos seguintes \u00e2ngulos generalizados: AOE, AOF e AOG.<\/li>\n<li>Determine, com aproxima\u00e7\u00e3o \u00e0s d\u00e9cimas, as coordenadas dos pontos E e G.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_3981' onClick='GTTabs_show(1,3981)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_3981'>\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\/48.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"3990\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=3990\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/48.jpg\" data-orig-size=\"328,375\" 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=\"Hex\u00e1gono regular\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/48.jpg\" class=\"alignright wp-image-3990\" title=\"Hex\u00e1gono regular\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/48-262x300.jpg\" alt=\"\" width=\"240\" height=\"274\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/48-262x300.jpg 262w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/48-131x150.jpg 131w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/48.jpg 328w\" sizes=\"auto, (max-width: 240px) 100vw, 240px\" \/><\/a>Como a amplitude positiva m\u00ednima do \u00e2ngulo generalizado AOB \u00e9 $\\frac{\\pi }{9}$ radianos, ent\u00e3o a express\u00e3o geral das amplitudes desse \u00e2ngulo, em radianos, \u00e9: \\[\\frac{\\pi }{9}+2k\\pi ,\\,\\,k\\in \\mathbb{Z}\\]<br \/>\n\u00ad<\/li>\n<li>Como o hex\u00e1gono \u00e9 regular, ent\u00e3o $B\\hat{O}C=60{}^\\text{o}=\\frac{\\pi }{3}rad$.<br \/>\nAssim, uma das amplitudes, em radianos, do \u00e2ngulo orientado AOC \u00e9: \\[A\\hat{O}C=A\\hat{O}B+B\\hat{O}C=\\frac{\\pi }{9}+\\frac{\\pi }{3}=\\frac{4\\pi }{9}\\]<br \/>\n\u00ad<\/li>\n<li>Como $A\\hat{O}E=A\\hat{O}B+B\\hat{O}E=\\frac{\\pi }{9}+3\\times \\frac{\\pi }{3}=\\frac{10\\pi }{9}$ radianos, ent\u00e3o a amplitude, em radianos, do \u00e2ngulo generalizado AOE \u00e9: \\[\\frac{10\\pi }{9}+2k\\pi ,\\,\\,k\\in \\mathbb{Z}\\]<br \/>\nComo $A\\hat{O}F=A\\hat{O}B+B\\hat{O}F=\\frac{\\pi }{9}+4\\times \\frac{\\pi }{3}=\\frac{13\\pi }{9}$ radianos, ent\u00e3o a amplitude, em radianos, do \u00e2ngulo generalizado AOF \u00e9: \\[\\frac{13\\pi }{9}+2k\\pi ,\\,\\,k\\in \\mathbb{Z}\\]<br \/>\nComo $A\\hat{O}G=A\\hat{O}B+B\\hat{O}G=\\frac{\\pi }{9}+5\\times \\frac{\\pi }{3}=\\frac{16\\pi }{9}$ radianos, ent\u00e3o a amplitude, em radianos, do \u00e2ngulo generalizado AOG \u00e9: \\[\\frac{16\\pi }{9}+2k\\pi ,\\,\\,k\\in \\mathbb{Z}\\]<br \/>\n\u00ad<\/li>\n<li>$B\\,(\\cos \\frac{10\\pi }{9},sen\\,\\frac{10\\pi }{9})$ e $G\\,(\\cos \\frac{16\\pi }{9},sen\\,\\frac{16\\pi }{9})$.<br \/>\nLogo, as coordenadas destes pontos, com aproxima\u00e7\u00e3o \u00e0s d\u00e9cimas, s\u00e3o $(-0,9;-0,3)$ e $(0,8;-0,6)$, respetivamente.<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_3981' onClick='GTTabs_show(0,3981)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Sobre o c\u00edrculo trigonom\u00e9trico de centro O da figura est\u00e1 representado um hex\u00e1gono regular. A amplitude positiva m\u00ednima do \u00e2ngulo generalizado AOB \u00e9 $\\frac{\\pi }{9}$ radianos. Qual \u00e9, em radianos, a&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20802,"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-3981","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":3178,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/11V1Pag095-48_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/3981","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=3981"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/3981\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20802"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3981"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=3981"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=3981"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=3981"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}