{"id":5134,"date":"2010-11-04T22:49:22","date_gmt":"2010-11-04T22:49:22","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=5134"},"modified":"2021-12-25T19:53:59","modified_gmt":"2021-12-25T19:53:59","slug":"duas-funcoes-trigonometricas","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=5134","title":{"rendered":"Duas fun\u00e7\u00f5es trigonom\u00e9tricas"},"content":{"rendered":"<p><ul id='GTTabs_ul_5134' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_5134' class='GTTabs_curr'><a  id=\"5134_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_5134' ><a  id=\"5134_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_5134'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Na figura est\u00e3o as representa\u00e7\u00f5es gr\u00e1ficas de duas fun\u00e7\u00f5es, <em>f<\/em> e <em>g<\/em>, no intervalo $\\left[ -2\\pi ,2\\pi\u00a0 \\right]$.<\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/sinx-cos3x_a.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"12459\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=12459\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/sinx-cos3x_a.png\" data-orig-size=\"955,223\" 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=\"Duas fun\u00e7\u00f5es trigonom\u00e9tricas\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/sinx-cos3x_a.png\" class=\"aligncenter wp-image-12459 size-full\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/sinx-cos3x_a.png\" alt=\"Duas fun\u00e7\u00f5es trigonom\u00e9tricas\" width=\"955\" height=\"223\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/sinx-cos3x_a.png 955w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/sinx-cos3x_a-300x70.png 300w\" sizes=\"auto, (max-width: 955px) 100vw, 955px\" \/><\/a> Sabe-se que:<\/p>\n<ul>\n<li><em>f<\/em> \u00e9 definida por $f(x)=sen\\,x$;<\/li>\n<li><em>g<\/em> \u00e9 definida por $g(x)=\\cos (3x)$;<\/li>\n<li>A \u00e9 um ponto de intersec\u00e7\u00e3o dos gr\u00e1ficos de <em>f<\/em> e de <em>g<\/em>.<\/li>\n<\/ul>\n<p>Sem recorrer \u00e0 calculadora, a n\u00e3o ser para efectuar eventuais c\u00e1lculos num\u00e9ricos, determine a abcissa (valor exacto) de A.<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_5134' onClick='GTTabs_show(1,5134)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_5134'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/sinx-cos3x_a.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"12459\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=12459\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/sinx-cos3x_a.png\" data-orig-size=\"955,223\" 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=\"Duas fun\u00e7\u00f5es trigonom\u00e9tricas\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/sinx-cos3x_a.png\" class=\"aligncenter wp-image-12459 size-full\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/sinx-cos3x_a.png\" alt=\"Duas fun\u00e7\u00f5es trigonom\u00e9tricas\" width=\"955\" height=\"223\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/sinx-cos3x_a.png 955w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/sinx-cos3x_a-300x70.png 300w\" sizes=\"auto, (max-width: 955px) 100vw, 955px\" \/><\/a><\/p>\n<\/p>\n<p>A abcissa do ponto A satisfaz a seguinte condi\u00e7\u00e3o: $f(x)=g(x)\\wedge x\\in \\left] \\frac{\\pi }{2},\\pi\u00a0 \\right[$. Ora, \\[\\begin{array}{*{35}{l}} \u00a0\u00a0 sen\\,x=\\cos (3x) &amp; \\Leftrightarrow\u00a0 &amp; sen\\,x=sen\\,(\\frac{\\pi }{2}-3x)\u00a0 \\\\ \u00a0\u00a0 {} &amp; \\Leftrightarrow\u00a0 &amp; \\begin{matrix} \u00a0\u00a0 x=(\\frac{\\pi }{2}-3x)+2k\\pi\u00a0 &amp; \\vee\u00a0 &amp; x=\\pi -(\\frac{\\pi }{2}-3x)+2k\\pi \\,,\\,\\,k\\in \\mathbb{Z}\u00a0 \\\\ \\end{matrix}\u00a0 \\\\ \u00a0\u00a0 {} &amp; \\Leftrightarrow\u00a0 &amp; \\begin{matrix} \u00a0\u00a0 4x=\\frac{\\pi }{2}+2k\\pi\u00a0 &amp; \\vee\u00a0 &amp; -2x=\\frac{\\pi }{2}+2k\\pi \\,,\\,\\,k\\in \\mathbb{Z}\u00a0 \\\\ \\end{matrix}\u00a0 \\\\ \u00a0\u00a0 {} &amp; \\Leftrightarrow\u00a0 &amp; \\begin{matrix} \u00a0\u00a0 x=\\frac{\\pi }{8}+\\frac{k\\pi }{2} &amp; \\vee\u00a0 &amp; x=-\\frac{\\pi }{4}+k\\pi \\,,\\,\\,k\\in \\mathbb{Z}\u00a0 \\\\ \\end{matrix}\u00a0 \\\\ \\end{array}\\] Atribuindo valores convenientes a k, obt\u00e9m-se: \\[\\begin{matrix} \u00a0\u00a0 k=0: &amp; x=\\frac{\\pi }{8} &amp; \\vee\u00a0 &amp; x=-\\frac{\\pi }{4}\u00a0 \\\\ \u00a0\u00a0 k=1: &amp; x=\\frac{5\\pi }{8} &amp; \\vee\u00a0 &amp; x=\\frac{3\\pi }{4}\u00a0 \\\\ \u00a0\u00a0 k=2: &amp; x=\\frac{9\\pi }{8} &amp; \\vee\u00a0 &amp; x=\\frac{7\\pi }{4}\u00a0 \\\\ \\end{matrix}\\] H\u00e1 duas solu\u00e7\u00f5es pertencentes ao intervalo $\\left] \\frac{\\pi }{2},\\pi\u00a0 \\right[$ (para $k=1$), ali\u00e1s, situa\u00e7\u00e3o observ\u00e1vel na representa\u00e7\u00e3o gr\u00e1fica dada. Ora, o valor procurado \u00e9 o maior dos dois, portanto, a abcissa do ponto A \u00e9 $\\frac{3\\pi }{4}$.<\/p>\n<\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/sinx-cos3x.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"12461\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=12461\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/sinx-cos3x.png\" data-orig-size=\"964,225\" 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=\"sinx-cos3x\" data-image-description=\"\" data-image-caption=\"&lt;p&gt;Duas fun\u00e7\u00f5es trigonom\u00e9tricas &amp;#8211; Solu\u00e7\u00e3o&lt;\/p&gt;\n\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/sinx-cos3x.png\" class=\"aligncenter wp-image-12461 size-full\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/sinx-cos3x.png\" alt=\"Duas fun\u00e7\u00f5es trigo nom\u00e9tricas - Solu\u00e7\u00e3o\" width=\"964\" height=\"225\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/sinx-cos3x.png 964w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/sinx-cos3x-300x70.png 300w\" sizes=\"auto, (max-width: 964px) 100vw, 964px\" \/><\/a><\/p>\n<p>Note:<\/p>\n<ul>\n<li>$\\frac{3\\pi }{4}\\simeq 2,35619$<\/li>\n<li>$f(\\frac{3\\pi }{4})=sen\\,(\\frac{3\\pi }{4})=\\frac{\\sqrt{2}}{2}\\simeq 0,70711$<\/li>\n<li>$g(\\frac{3\\pi }{4})=\\cos (3\\times \\frac{3\\pi }{4})=\\cos (2\\pi +\\frac{\\pi }{4})=\\cos (\\frac{\\pi }{4})=\\frac{\\sqrt{2}}{2}\\simeq 0,70711$<\/li>\n<\/ul>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_5134' onClick='GTTabs_show(0,5134)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Na figura est\u00e3o as representa\u00e7\u00f5es gr\u00e1ficas de duas fun\u00e7\u00f5es, f e g, no intervalo $\\left[ -2\\pi ,2\\pi\u00a0 \\right]$. Sabe-se que: f \u00e9 definida por $f(x)=sen\\,x$; g \u00e9 definida por $g(x)=\\cos (3x)$;&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":19463,"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-5134","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":3976,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2021\/12\/Mat127.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/5134","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=5134"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/5134\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19463"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5134"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5134"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5134"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=5134"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}