{"id":9358,"date":"2012-05-23T15:54:35","date_gmt":"2012-05-23T14:54:35","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=9358"},"modified":"2021-12-29T23:48:31","modified_gmt":"2021-12-29T23:48:31","slug":"resolva-em-mathbbc-as-equacoes-4","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=9358","title":{"rendered":"Resolva, em $\\mathbb{C}$, as equa\u00e7\u00f5es"},"content":{"rendered":"<p><ul id='GTTabs_ul_9358' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_9358' class='GTTabs_curr'><a  id=\"9358_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_9358' ><a  id=\"9358_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_9358'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Resolva, em $\\mathbb{C}$, as equa\u00e7\u00f5es:<\/p>\n<ol>\n<li>${z^4}.\\overline z\u00a0 = 32i$<\/li>\n<li>${z^3} + \\left( {\\sqrt 3\u00a0 + i} \\right)z = 0$<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_9358' onClick='GTTabs_show(1,9358)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_9358'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li>$${z^4}.\\overline z\u00a0 = 32i$$<br \/>\nConsiderando $z = \\rho \\operatorname{cis} \\theta $, temos:<br \/>\n$$\\begin{array}{*{20}{l}}<br \/>\n{{{\\left( {\\rho \\operatorname{cis} \\theta } \\right)}^4} \\times \\overline {\\rho \\operatorname{cis} \\theta }\u00a0 = 32i}&amp; \\Leftrightarrow &amp;{{\\rho ^4}\\operatorname{cis} \\left( {4\\theta } \\right) \\times \\rho \\operatorname{cis} \\left( { &#8211; \\theta } \\right) = 32\\operatorname{cis} \\left( {\\frac{\\pi }{2}} \\right)} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{{\\rho ^5}\\operatorname{cis} \\left( {3\\theta } \\right) = 32\\operatorname{cis} \\left( {\\frac{\\pi }{2}} \\right)} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\left\\{ {\\begin{array}{*{20}{l}}<br \/>\n{{\\rho ^5} = 32} \\\\<br \/>\n{3\\theta\u00a0 = \\frac{\\pi }{2} + 2k\\pi ,k \\in \\mathbb{Z}}<br \/>\n\\end{array}} \\right.} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\left\\{ {\\begin{array}{*{20}{l}}<br \/>\n{\\rho\u00a0 = \\sqrt[5]{{32}}} \\\\<br \/>\n{\\theta\u00a0 = \\frac{\\pi }{6} + \\frac{{2k\\pi }}{3},k \\in \\mathbb{Z}}<br \/>\n\\end{array}} \\right.} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\left\\{ {\\begin{array}{*{20}{l}}<br \/>\n{\\rho\u00a0 = 2} \\\\<br \/>\n{\\theta\u00a0 = \\frac{\\pi }{6} + \\frac{{2k\\pi }}{3},k \\in \\mathbb{Z}}<br \/>\n\\end{array}} \\right.}<br \/>\n\\end{array}$$<br \/>\nLogo, $$\\begin{array}{*{20}{l}}<br \/>\n{{z^4}.\\overline z\u00a0 = 32i}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{z = 2\\operatorname{cis} \\frac{\\pi }{6}}&amp; \\vee &amp;{z = 2\\operatorname{cis} \\frac{{5\\pi }}{6}}&amp; \\vee &amp;{z = 2\\operatorname{cis} \\frac{{3\\pi }}{2}}<br \/>\n\\end{array}}<br \/>\n\\end{array}$$<br \/>\n\u00ad<\/li>\n<li>Ora,<br \/>\n$$\\begin{array}{*{20}{l}}<br \/>\n{{z^3} + \\left( {\\sqrt 3\u00a0 + i} \\right)z = 0}&amp; \\Leftrightarrow &amp;{\\left( {{z^2} + \\sqrt 3\u00a0 + i} \\right)z = 0} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{z = 0}&amp; \\vee &amp;{{z^2} + \\sqrt 3\u00a0 + i = 0}<br \/>\n\\end{array}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{z = 0}&amp; \\vee &amp;{{z^2} =\u00a0 &#8211; \\sqrt 3\u00a0 &#8211; i}<br \/>\n\\end{array}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{z = 0}&amp; \\vee &amp;{{z^2} = 2\\operatorname{cis} \\frac{{7\\pi }}{6}}<br \/>\n\\end{array}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{z = 0}&amp; \\vee &amp;{z = \\sqrt 2 \\operatorname{cis} \\left( {\\frac{{\\tfrac{{7\\pi }}{6}}}{2}} \\right)}&amp; \\vee &amp;{z = \\sqrt 2 \\operatorname{cis} \\left( {\\frac{{\\tfrac{{7\\pi }}{6}}}{2} + \\frac{{2\\pi }}{2}} \\right)}<br \/>\n\\end{array}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{z = 0}&amp; \\vee &amp;{z = \\sqrt 2 \\operatorname{cis} \\left( {\\frac{{7\\pi }}{{12}}} \\right)}&amp; \\vee &amp;{z = \\sqrt 2 \\operatorname{cis} \\left( {\\frac{{19\\pi }}{{12}}} \\right)}<br \/>\n\\end{array}}<br \/>\n\\end{array}$$<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_9358' onClick='GTTabs_show(0,9358)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Resolva, em $\\mathbb{C}$, as equa\u00e7\u00f5es: ${z^4}.\\overline z\u00a0 = 32i$ ${z^3} + \\left( {\\sqrt 3\u00a0 + i} \\right)z = 0$ Resolu\u00e7\u00e3o &gt;&gt; Resolu\u00e7\u00e3o &lt;&lt; Enunciado<\/p>\n","protected":false},"author":1,"featured_media":19189,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[226,97,310],"tags":[427,18],"series":[],"class_list":["post-9358","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-12--ano","category-aplicando","category-numeros-complexos-12--ano","tag-12-o-ano","tag-numeros-complexos"],"views":1685,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2021\/12\/Mat75.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/9358","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=9358"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/9358\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=9358"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=9358"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=9358"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=9358"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}