{"id":9174,"date":"2012-05-20T23:40:17","date_gmt":"2012-05-20T22:40:17","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=9174"},"modified":"2021-12-29T13:11:11","modified_gmt":"2021-12-29T13:11:11","slug":"represente-na-forma-trigonometrica-o-simetrico-e-o-inverso-dos-numeros-complexos","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=9174","title":{"rendered":"Represente na forma trigonom\u00e9trica o sim\u00e9trico e o inverso dos n\u00fameros complexos"},"content":{"rendered":"<p><ul id='GTTabs_ul_9174' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_9174' class='GTTabs_curr'><a  id=\"9174_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_9174' ><a  id=\"9174_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_9174'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p id=\"sub-title\">Represente na forma trigonom\u00e9trica o sim\u00e9trico e o inverso de cada um dos seguintes\u00a0n\u00fameros complexos:<\/p>\n<ol>\n<li>$z =\u00a0 &#8211; 3 + 3i$<\/li>\n<li>$z = 2\\operatorname{cis} \\left( { &#8211; \\pi } \\right)$<\/li>\n<li>$z = 2,3\\operatorname{cis} \\left( {\\frac{{5\\pi }}{4}} \\right)$<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_9174' onClick='GTTabs_show(1,9174)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_9174'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<blockquote>\n<p><strong>Forma trigonom\u00e9trica do sim\u00e9trico<\/strong>:<\/p>\n<p style=\"text-align: center;\">Se $z = \\rho \\operatorname{cis} \\theta $, ent\u00e3o $ &#8211; z = \\rho \\operatorname{cis} \\left( {\\pi\u00a0 + \\theta } \\right)$.<\/p>\n<\/blockquote>\n<blockquote>\n<p><strong>Forma trigonom\u00e9trica do inverso<\/strong>:<\/p>\n<p style=\"text-align: center;\">Se $z = \\rho \\operatorname{cis} \\theta $, ent\u00e3o $\\frac{1}{z} = \\frac{{\\bar z}}{{{{\\left| z \\right|}^2}}} = \\frac{1}{\\rho }\\operatorname{cis} \\left( { &#8211; \\theta } \\right)$.<\/p>\n<\/blockquote>\n<ol>\n<li>Como $z =\u00a0 &#8211; 3 + 3i = 3\\sqrt 2 \\operatorname{cis} \\frac{{3\\pi }}{4}$, ent\u00e3o:<br \/>\n$$\\begin{array}{*{20}{l}}<br \/>\n{ &#8211; z}&amp; = &amp;{3\\sqrt 2 \\operatorname{cis} \\left( {\\pi\u00a0 + \\frac{{3\\pi }}{4}} \\right)}&amp;{}&amp;{}&amp;{}&amp;{\\overline z }&amp; = &amp;{\\frac{1}{{3\\sqrt 2 }}\\operatorname{cis} \\left( { &#8211; \\frac{{3\\pi }}{4}} \\right)} \\\\<br \/>\n{}&amp; = &amp;{3\\sqrt 2 \\operatorname{cis} \\left( {\\frac{{7\\pi }}{4}} \\right)}&amp;{}&amp;{}&amp;{}&amp;{}&amp; = &amp;{\\frac{{\\sqrt 2 }}{6}\\operatorname{cis} \\frac{{5\\pi }}{4}}<br \/>\n\\end{array}$$<\/li>\n<li>Como $z = 2\\operatorname{cis} \\left( { &#8211; \\pi } \\right)$, ent\u00e3o:<br \/>\n$$\\begin{array}{*{20}{l}}<br \/>\n{ &#8211; z}&amp; = &amp;{3\\sqrt 2 \\operatorname{cis} \\left( {\\pi\u00a0 + \\frac{{3\\pi }}{4}} \\right)}&amp;{}&amp;{}&amp;{}&amp;{\\overline z }&amp; = &amp;{\\frac{1}{2}\\operatorname{cis} \\left( { &#8211; \\pi } \\right)} \\\\<br \/>\n{}&amp; = &amp;{3\\sqrt 2 \\operatorname{cis} \\left( {\\frac{{7\\pi }}{4}} \\right)}&amp;{}&amp;{}&amp;{}&amp;{}&amp; = &amp;{\\frac{1}{2}\\operatorname{cis} \\pi }<br \/>\n\\end{array}$$<\/li>\n<li>Como $z = 2,3\\operatorname{cis} \\left( {\\frac{{5\\pi }}{4}} \\right)$, ent\u00e3o:<br \/>\n$$\\begin{array}{*{20}{l}}<br \/>\n{ &#8211; z}&amp; = &amp;{2,3\\operatorname{cis} \\left( {\\pi\u00a0 + \\frac{{5\\pi }}{4}} \\right)}&amp;{}&amp;{}&amp;{}&amp;{\\overline z }&amp; = &amp;{\\frac{1}{{2,3}}\\operatorname{cis} \\left( { &#8211; \\frac{{5\\pi }}{4}} \\right)} \\\\<br \/>\n{}&amp; = &amp;{2,3\\operatorname{cis} \\left( {\\frac{\\pi }{4}} \\right)}&amp;{}&amp;{}&amp;{}&amp;{}&amp; = &amp;{\\frac{{10}}{{23}}\\operatorname{cis} \\frac{{3\\pi }}{4}}<br \/>\n\\end{array}$$<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_9174' onClick='GTTabs_show(0,9174)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Represente na forma trigonom\u00e9trica o sim\u00e9trico e o inverso de cada um dos seguintes\u00a0n\u00fameros complexos: $z =\u00a0 &#8211; 3 + 3i$ $z = 2\\operatorname{cis} \\left( { &#8211; \\pi } \\right)$ $z&#46;&#46;&#46;<\/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":[427,314,18],"series":[],"class_list":["post-9174","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":4120,"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\/9174","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=9174"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/9174\/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=9174"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=9174"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=9174"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=9174"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}