{"id":9311,"date":"2012-05-22T18:33:36","date_gmt":"2012-05-22T17:33:36","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=9311"},"modified":"2021-12-29T22:48:17","modified_gmt":"2021-12-29T22:48:17","slug":"resolva-em-mathbbc-as-equacoes-2","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=9311","title":{"rendered":"Resolva, em $\\mathbb{C}$, as equa\u00e7\u00f5es"},"content":{"rendered":"<p><ul id='GTTabs_ul_9311' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_9311' class='GTTabs_curr'><a  id=\"9311_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_9311' ><a  id=\"9311_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_9311'>\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>${z^2} = 1 + i$<\/li>\n<li>${z^3} &#8211; iz = 0$<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_9311' onClick='GTTabs_show(1,9311)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_9311'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li>Como $w = 1 + i = \\sqrt 2 \\operatorname{cis} \\frac{\\pi }{4}$, temos:<br \/>\n$$\\begin{array}{*{20}{l}}<br \/>\n{{z^2} = 1 + i}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{z = \\sqrt {\\sqrt 2 } \\operatorname{cis} \\left( {\\frac{{\\tfrac{\\pi }{4}}}{2}} \\right)}&amp; \\vee &amp;{z = \\sqrt {\\sqrt 2 } \\operatorname{cis} \\left( {\\frac{{\\tfrac{\\pi }{4}}}{2} + \\frac{{2\\pi }}{2}} \\right)}<br \/>\n\\end{array}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{z = \\sqrt[4]{2}\\operatorname{cis} \\left( {\\frac{\\pi }{8}} \\right)}&amp; \\vee &amp;{z = \\sqrt[4]{2}\\operatorname{cis} \\left( {\\frac{{9\\pi }}{8}} \\right)}<br \/>\n\\end{array}}<br \/>\n\\end{array}$$<\/li>\n<li>Ora,<br \/>\n$$\\begin{array}{*{20}{l}}<br \/>\n{{z^3} &#8211; iz = 0}&amp; \\Leftrightarrow &amp;{z\\left( {{z^2} &#8211; i} \\right) = 0} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{z = 0}&amp; \\vee &amp;{{z^2} = i}<br \/>\n\\end{array}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{z = 0}&amp; \\vee &amp;{{z^2} = \\operatorname{cis} \\frac{\\pi }{2}}<br \/>\n\\end{array}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{z = 0}&amp; \\vee &amp;{z = \\sqrt 1 \\operatorname{cis} \\left( {\\frac{{\\tfrac{\\pi }{2}}}{2}} \\right)}&amp; \\vee &amp;{z = \\sqrt 1 \\operatorname{cis} \\left( {\\frac{{\\tfrac{\\pi }{2}}}{2} + \\frac{{2\\pi }}{2}} \\right)}<br \/>\n\\end{array}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{z = 0}&amp; \\vee &amp;{z = \\operatorname{cis} \\left( {\\frac{\\pi }{4}} \\right)}&amp; \\vee &amp;{z = \\operatorname{cis} \\left( {\\frac{{5\\pi }}{4}} \\right)}<br \/>\n\\end{array}}<br \/>\n\\end{array}$$<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_9311' onClick='GTTabs_show(0,9311)'>&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: ${z^2} = 1 + i$ ${z^3} &#8211; iz = 0$ Resolu\u00e7\u00e3o &gt;&gt; Resolu\u00e7\u00e3o &lt;&lt; Enunciado<\/p>\n","protected":false},"author":1,"featured_media":19178,"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-9311","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":1896,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2021\/12\/Mat69.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/9311","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=9311"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/9311\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19178"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=9311"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=9311"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=9311"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=9311"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}