{"id":5112,"date":"2010-11-02T00:30:46","date_gmt":"2010-11-02T00:30:46","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=5112"},"modified":"2022-01-13T19:44:13","modified_gmt":"2022-01-13T19:44:13","slug":"rascunho-14","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=5112","title":{"rendered":"Duas condi\u00e7\u00f5es trigonom\u00e9tricas"},"content":{"rendered":"<p><ul id='GTTabs_ul_5112' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_5112' class='GTTabs_curr'><a  id=\"5112_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_5112' ><a  id=\"5112_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_5112'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Resolva, utilizando as capacidades da sua calculadora gr\u00e1fica, as seguintes condi\u00e7\u00f5es:<\/p>\n<ol>\n<li>$sen\\,x=0,4$<\/li>\n<li>$sen\\,x&gt;0,3$<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_5112' onClick='GTTabs_show(1,5112)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_5112'>\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\/11\/04-11-2010-Ecra001.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"5119\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=5119\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/04-11-2010-Ecra001.jpg\" data-orig-size=\"666,452\" 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;}\" data-image-title=\"04-11-2010-Ecra001\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/04-11-2010-Ecra001.jpg\" class=\"alignright wp-image-5119\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/04-11-2010-Ecra001.jpg\" alt=\"\" width=\"480\" height=\"326\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/04-11-2010-Ecra001.jpg 666w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/04-11-2010-Ecra001-300x203.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/04-11-2010-Ecra001-150x101.jpg 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/04-11-2010-Ecra001-400x271.jpg 400w\" sizes=\"auto, (max-width: 480px) 100vw, 480px\" \/><\/a>Consideremos duas fun\u00e7\u00f5es <em>f<sub>1<\/sub><\/em> e <em>f<sub>2<\/sub><\/em>, reais de vari\u00e1vel real, de dom\u00ednio R, definidas por\u00a0${{f}_{1}}(x)=sen\\,x$ e ${{f}_{2}}(x)=0,4$.\n<p>Efetuada a representa\u00e7\u00e3o gr\u00e1fica destas fun\u00e7\u00f5es no intervalo $\\left[ -2\\pi ,2\\pi\u00a0 \\right]$, verificamos que, no intervalo $\\left[ 0,2\\pi\u00a0 \\right]$, os gr\u00e1ficos intersectam-se nos pontos A e B, de coordenadas indicadas na figura (a abcissa \u00e9 um valor aproximado \u00e0s mil\u00e9simas).<\/p>\n<p>Sabendo que a fun\u00e7\u00e3o\u00a0<em>f<sub>1<\/sub><\/em> \u00e9 peri\u00f3dica, com per\u00edodo positivo m\u00ednimo $2\\pi $, conclui-se que a condi\u00e7\u00e3o dada tem uma infinidade de solu\u00e7\u00f5es, cujos valores, aproximados \u00e0s mil\u00e9simas, s\u00e3o dados por: $\\begin{matrix} \u00a0\u00a0 x\\simeq 0,412+2k\\pi\u00a0 &amp; \\vee\u00a0 &amp; x\\simeq 2,730+2k\\pi \\,,\\,\\,k\\in \\mathbb{Z}\u00a0 \\\\ \\end{matrix}$.<br \/>\n\u00ad<\/p>\n<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"5122\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=5122\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/04-11-2010-Ecra002.jpg\" data-orig-size=\"666,452\" 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;}\" data-image-title=\"04-11-2010-Ecra002\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/04-11-2010-Ecra002.jpg\" class=\"alignright wp-image-5122\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/04-11-2010-Ecra002-300x203.jpg\" alt=\"\" width=\"480\" height=\"326\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/04-11-2010-Ecra002-300x203.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/04-11-2010-Ecra002-150x101.jpg 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/04-11-2010-Ecra002-400x271.jpg 400w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/04-11-2010-Ecra002.jpg 666w\" sizes=\"auto, (max-width: 480px) 100vw, 480px\" \/>Apesar de ser poss\u00edvel resolver esta quest\u00e3o \u00e0 custa de duas fun\u00e7\u00f5es, vamos apenas utilizar uma.\n<p>Ora, $sen\\,x&gt;0,3\\Leftrightarrow sen\\,x-0,3&gt;0$.<br \/>\nConsideremos, ent\u00e3o, a fun\u00e7\u00e3o, real de vari\u00e1vel real, definida por ${{f}_{1}}(x)=sen\\,x-0,3$, com dom\u00ednio R.<\/p>\n<p>Efetuada a representa\u00e7\u00e3o gr\u00e1fica da fun\u00e7\u00e3o no intervalo $\\left[ -2\\pi ,2\\pi\u00a0 \\right]$, verificamos que, no intervalo $\\left[ 0,2\\pi\u00a0 \\right]$, o gr\u00e1fico intersecta o eixo <em>Ox<\/em> nos pontos A e B, com abcissas indicadas na figura (valores aproximados \u00e0s mil\u00e9simas).<\/p>\n<p>Sabendo que a fun\u00e7\u00e3o\u00a0<em>f<sub>1<\/sub><\/em> \u00e9 peri\u00f3dica, com per\u00edodo positivo m\u00ednimo $2\\pi $, conclui-se que a condi\u00e7\u00e3o dada tem uma infinidade de solu\u00e7\u00f5es, poss\u00edveis de agrupar numa infinidade de intervalos de n\u00fameros reais, que, usando\u00a0valores aproximados \u00e0s mil\u00e9simas, podem ser definidas por: $x\\in \\left] 0,305+2k\\pi ;2,840+2k\\pi\u00a0 \\right[,\\,\\,k\\in \\mathbb{Z}$.<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_5112' onClick='GTTabs_show(0,5112)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Resolva, utilizando as capacidades da sua calculadora gr\u00e1fica, as seguintes condi\u00e7\u00f5es: $sen\\,x=0,4$ $sen\\,x&gt;0,3$ Resolu\u00e7\u00e3o &gt;&gt; Resolu\u00e7\u00e3o &lt;&lt; Enunciado<\/p>\n","protected":false},"author":1,"featured_media":19485,"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-5112","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":2308,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/Calculadoras-Graficas_2.png.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/5112","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=5112"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/5112\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19485"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5112"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5112"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5112"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=5112"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}