{"id":9193,"date":"2012-05-21T00:43:34","date_gmt":"2012-05-20T23:43:34","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=9193"},"modified":"2021-12-29T13:21:09","modified_gmt":"2021-12-29T13:21:09","slug":"calcule-3","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=9193","title":{"rendered":"Calcule"},"content":{"rendered":"<p><ul id='GTTabs_ul_9193' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_9193' class='GTTabs_curr'><a  id=\"9193_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_9193' ><a  id=\"9193_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_9193'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Calcule:<\/p>\n<ol>\n<li>$${\\left( { &#8211; 1 &#8211; \\sqrt 3 i} \\right)^6}$$<\/li>\n<li>$${\\left( {\\frac{{2 + 2i}}{{2 &#8211; 2i}}} \\right)^4}$$<\/li>\n<li>$${\\left[ {3\\operatorname{cis} \\left( { &#8211; \\frac{{4\\pi }}{3}} \\right)} \\right]^5}$$<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_9193' onClick='GTTabs_show(1,9193)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_9193'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<blockquote>\n<p><strong>Forma trigonom\u00e9trica da pot\u00eancia (F\u00f3rmula de Moivre)<\/strong>:<\/p>\n<p style=\"text-align: center;\">Se $z = \\rho \\operatorname{cis} \\theta $ \u00e9 um n\u00famero complexo n\u00e3o nulo, ent\u00e3o $${z^n} = {\\rho ^n}\\operatorname{cis} \\left( {n\\theta } \\right)$$<\/p>\n<\/blockquote>\n<ol>\n<li>Ora,<br \/>\n$$\\begin{array}{*{20}{l}}<br \/>\n{{{\\left( { &#8211; 1 &#8211; \\sqrt 3 i} \\right)}^6}}&amp; = &amp;{{{\\left[ {2\\left( { &#8211; \\frac{1}{2} &#8211; \\frac{{\\sqrt 3 }}{2}i} \\right)} \\right]}^6}} \\\\<br \/>\n{}&amp; = &amp;{{{\\left( {2\\operatorname{cis} \\frac{{4\\pi }}{3}} \\right)}^6}} \\\\<br \/>\n{}&amp; = &amp;{{2^6}\\operatorname{cis} \\left( {6 \\times \\frac{{4\\pi }}{3}} \\right)} \\\\<br \/>\n{}&amp; = &amp;{64\\operatorname{cis} \\left( {8\\pi } \\right)} \\\\<br \/>\n{}&amp; = &amp;{64\\operatorname{cis} \\left( 0 \\right)}<br \/>\n\\end{array}$$<\/li>\n<li>Ora,<br \/>\n$$\\begin{array}{*{20}{l}}<br \/>\n{{{\\left( {\\frac{{2 + 2i}}{{2 &#8211; 2i}}} \\right)}^4}}&amp; = &amp;{{{\\left( {\\frac{{2\\sqrt 2 \\operatorname{cis} \\frac{\\pi }{4}}}{{2\\sqrt 2 \\operatorname{cis} \\left( { &#8211; \\frac{\\pi }{4}} \\right)}}} \\right)}^4}} \\\\<br \/>\n{}&amp; = &amp;{{{\\left( {\\frac{{2\\sqrt 2 }}{{2\\sqrt 2 }}\\operatorname{cis} \\left( {\\frac{\\pi }{4} + \\frac{\\pi }{4}} \\right)} \\right)}^4}} \\\\<br \/>\n{}&amp; = &amp;{{1^4}\\operatorname{cis} \\left( {4 \\times \\frac{\\pi }{2}} \\right)} \\\\<br \/>\n{}&amp; = &amp;{\\operatorname{cis} \\left( {2\\pi } \\right)} \\\\<br \/>\n{}&amp; = &amp;{\\operatorname{cis} \\left( 0 \\right)}<br \/>\n\\end{array}$$<\/li>\n<li>Ora,<br \/>\n$$\\begin{array}{*{20}{l}}<br \/>\n{{{\\left[ {3\\operatorname{cis} \\left( { &#8211; \\frac{{4\\pi }}{3}} \\right)} \\right]}^5}}&amp; = &amp;{{3^5}\\operatorname{cis} \\left( {5 \\times \\left( { &#8211; \\frac{{4\\pi }}{3}} \\right)} \\right)} \\\\<br \/>\n{}&amp; = &amp;{243\\operatorname{cis} \\left( { &#8211; \\frac{{20\\pi }}{3} + 8\\pi } \\right)} \\\\<br \/>\n{}&amp; = &amp;{243\\operatorname{cis} \\frac{{4\\pi }}{3}}<br \/>\n\\end{array}$$<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_9193' onClick='GTTabs_show(0,9193)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Calcule: $${\\left( { &#8211; 1 &#8211; \\sqrt 3 i} \\right)^6}$$ $${\\left( {\\frac{{2 + 2i}}{{2 &#8211; 2i}}} \\right)^4}$$ $${\\left[ {3\\operatorname{cis} \\left( { &#8211; \\frac{{4\\pi }}{3}} \\right)} \\right]^5}$$ Resolu\u00e7\u00e3o &gt;&gt; Resolu\u00e7\u00e3o &lt;&lt; Enunciado<\/p>\n","protected":false},"author":1,"featured_media":19561,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[226,97,310],"tags":[427,314,18],"series":[],"class_list":["post-9193","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-forma-trigonometrica","tag-numeros-complexos"],"views":1915,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2021\/12\/Mat180.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/9193","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=9193"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/9193\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19561"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=9193"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=9193"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=9193"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=9193"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}