{"id":9164,"date":"2012-05-20T22:14:55","date_gmt":"2012-05-20T21:14:55","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=9164"},"modified":"2021-12-29T02:50:14","modified_gmt":"2021-12-29T02:50:14","slug":"calcule-o-produto-na-forma-trigonometrica","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=9164","title":{"rendered":"Calcule o produto na forma trigonom\u00e9trica"},"content":{"rendered":"<p><ul id='GTTabs_ul_9164' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_9164' class='GTTabs_curr'><a  id=\"9164_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_9164' ><a  id=\"9164_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_9164'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Sendo $$\\begin{array}{*{20}{l}}<br \/>\n{{z_1} = 3\\operatorname{cis} \\left( { &#8211; \\frac{\\pi }{3}} \\right)}&amp;{\\text{;}}&amp;{{z_2} = \\sqrt 2 \\operatorname{cis} \\frac{\\pi }{6}}&amp;{\\text{e}}&amp;{{z_3} = \\operatorname{cis} \\frac{{5\\pi }}{6}}<br \/>\n\\end{array}$$ calcule:<\/p>\n<ol>\n<li>${z_1}.{z_2}$<\/li>\n<li>${z_2}.{z_3}$<\/li>\n<li>${z_1}.{z_2}.{z_3}$<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_9164' onClick='GTTabs_show(1,9164)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_9164'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<blockquote>\n<p><strong>Forma trigonom\u00e9trica do produto<\/strong>:<\/p>\n<p style=\"text-align: center;\">Se ${z_1} = {\\rho _1}\\operatorname{cis} {\\theta _1}$ e ${z_2} = {\\rho _2}\\operatorname{cis} {\\theta _2}$ s\u00e3o dois complexos n\u00e3o nulos, ent\u00e3o $${z_1}.{z_2} = {\\rho _1}{\\rho _2}\\operatorname{cis} \\left( {{\\theta _1} + {\\theta _2}} \\right)$$<\/p>\n<\/blockquote>\n<ol>\n<li>Ora,<br \/>\n$$\\begin{array}{*{20}{l}}<br \/>\n{{z_1}.{z_2}}&amp; = &amp;{3\\operatorname{cis} \\left( { &#8211; \\frac{\\pi }{3}} \\right) \\times \\sqrt 2 \\operatorname{cis} \\frac{\\pi }{6}} \\\\<br \/>\n{}&amp; = &amp;{3\\sqrt 2 \\operatorname{cis} \\left( { &#8211; \\frac{\\pi }{3} + \\frac{\\pi }{6}} \\right)} \\\\<br \/>\n{}&amp; = &amp;{3\\sqrt 2 \\operatorname{cis} \\left( { &#8211; \\frac{\\pi }{6}} \\right)}<br \/>\n\\end{array}$$<\/li>\n<li>Ora,<br \/>\n$$\\begin{array}{*{20}{l}}<br \/>\n{{z_2}.{z_3}}&amp; = &amp;{\\sqrt 2 \\operatorname{cis} \\frac{\\pi }{6} \\times \\operatorname{cis} \\frac{{5\\pi }}{6}} \\\\<br \/>\n{}&amp; = &amp;{\\sqrt 2 \\operatorname{cis} \\left( {\\frac{\\pi }{6} + \\frac{{5\\pi }}{6}} \\right)} \\\\<br \/>\n{}&amp; = &amp;{\\sqrt 2 \\operatorname{cis} \\pi }<br \/>\n\\end{array}$$<\/li>\n<li>Ora,<br \/>\n$$\\begin{array}{*{20}{l}}<br \/>\n{{z_1}.{z_2}.{z_3}}&amp; = &amp;{3\\operatorname{cis} \\left( { &#8211; \\frac{\\pi }{3}} \\right) \\times \\sqrt 2 \\operatorname{cis} \\frac{\\pi }{6} \\times \\operatorname{cis} \\frac{{5\\pi }}{6}} \\\\<br \/>\n{}&amp; = &amp;{3\\sqrt 2 \\operatorname{cis} \\left( { &#8211; \\frac{\\pi }{3} + \\frac{\\pi }{6} + \\frac{{5\\pi }}{6}} \\right)} \\\\<br \/>\n{}&amp; = &amp;{3\\sqrt 2 \\operatorname{cis} \\frac{{2\\pi }}{3}}<br \/>\n\\end{array}$$<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_9164' onClick='GTTabs_show(0,9164)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Sendo $$\\begin{array}{*{20}{l}} {{z_1} = 3\\operatorname{cis} \\left( { &#8211; \\frac{\\pi }{3}} \\right)}&amp;{\\text{;}}&amp;{{z_2} = \\sqrt 2 \\operatorname{cis} \\frac{\\pi }{6}}&amp;{\\text{e}}&amp;{{z_3} = \\operatorname{cis} \\frac{{5\\pi }}{6}} \\end{array}$$ calcule: ${z_1}.{z_2}$ ${z_2}.{z_3}$ ${z_1}.{z_2}.{z_3}$ Resolu\u00e7\u00e3o &gt;&gt; Resolu\u00e7\u00e3o &lt;&lt; Enunciado<\/p>\n","protected":false},"author":1,"featured_media":19600,"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-9164","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":2324,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2021\/12\/Mat197.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/9164","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=9164"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/9164\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=9164"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=9164"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=9164"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=9164"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}