{"id":9263,"date":"2012-05-22T00:02:33","date_gmt":"2012-05-21T23:02:33","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=9263"},"modified":"2021-12-29T15:23:20","modified_gmt":"2021-12-29T15:23:20","slug":"considere-os-seguintes-numeros-complexos","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=9263","title":{"rendered":"Considere os seguintes n\u00fameros complexos"},"content":{"rendered":"<p><ul id='GTTabs_ul_9263' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_9263' class='GTTabs_curr'><a  id=\"9263_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_9263' ><a  id=\"9263_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_9263'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Considere $$\\begin{array}{*{20}{c}}<br \/>\n{{z_1} = \\frac{{\\sqrt 3 }}{2} &#8211; \\frac{1}{2}i}&amp;{\\text{e}}&amp;{{z_2} = \\operatorname{cis} \\frac{{3\\pi }}{4}}<br \/>\n\\end{array}$$<\/p>\n<ol>\n<li>Determine ${z_1}.{z_2}$, na forma trigonom\u00e9trica e na forma alg\u00e9brica.<\/li>\n<li>Utilizando os resultados obtidos na al\u00ednea anterior, deduza os valores exatos de $\\cos \\frac{{7\\pi }}{{12}}$ e $\\operatorname{sen} \\frac{{7\\pi }}{{12}}$.<\/li>\n<li>Obtenha os valores de $\\cos \\frac{{7\\pi }}{{12}}$ e $\\operatorname{sen} \\frac{{7\\pi }}{{12}}$ utilizando outro processo.<br \/>\n(Sugest\u00e3o: $\\frac{{7\\pi }}{{12}} = \\frac{\\pi }{4} + \\frac{\\pi }{3}$)<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_9263' onClick='GTTabs_show(1,9263)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_9263'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<blockquote>\n<p>$$\\begin{array}{*{20}{c}}<br \/>\n{{z_1} = \\frac{{\\sqrt 3 }}{2} &#8211; \\frac{1}{2}i}&amp;{\\text{e}}&amp;{{z_2} = \\operatorname{cis} \\frac{{3\\pi }}{4}}<br \/>\n\\end{array}$$<\/p>\n<\/blockquote>\n<ol>\n<li>Ora,<br \/>\n$$\\begin{array}{*{20}{l}}<br \/>\n{{z_1}.{z_2}}&amp; = &amp;{\\left( {\\frac{{\\sqrt 3 }}{2} &#8211; \\frac{1}{2}i} \\right) \\times \\operatorname{cis} \\frac{{3\\pi }}{4}} \\\\<br \/>\n{}&amp; = &amp;{\\operatorname{cis} \\left( { &#8211; \\frac{\\pi }{6}} \\right) \\times \\operatorname{cis} \\frac{{3\\pi }}{4}} \\\\<br \/>\n{}&amp; = &amp;{\\operatorname{cis} \\left( { &#8211; \\frac{\\pi }{6} + \\frac{{3\\pi }}{4}} \\right)} \\\\<br \/>\n{}&amp; = &amp;{\\operatorname{cis} \\left( {\\frac{{7\\pi }}{{12}}} \\right)}<br \/>\n\\end{array}$$<\/p>\n<p>$$\\begin{array}{*{20}{l}}<br \/>\n{{z_1}.{z_2}}&amp; = &amp;{\\left( {\\frac{{\\sqrt 3 }}{2} &#8211; \\frac{1}{2}i} \\right) \\times \\operatorname{cis} \\frac{{3\\pi }}{4}} \\\\<br \/>\n{}&amp; = &amp;{\\left( {\\frac{{\\sqrt 3 }}{2} &#8211; \\frac{1}{2}i} \\right) \\times \\left( { &#8211; \\frac{{\\sqrt 2 }}{2} + \\frac{{\\sqrt 2 }}{2}i} \\right)} \\\\<br \/>\n{}&amp; = &amp;{ &#8211; \\frac{{\\sqrt 6 }}{4} + \\frac{{\\sqrt 6 }}{4}i + \\frac{{\\sqrt 2 }}{4}i + \\frac{{\\sqrt 2 }}{4}} \\\\<br \/>\n{}&amp; = &amp;{\\frac{{ &#8211; \\sqrt 6\u00a0 + \\sqrt 2 }}{4} + \\frac{{\\sqrt 6\u00a0 + \\sqrt 2 }}{4}i}<br \/>\n\\end{array}$$<\/p>\n<\/li>\n<li>Como $${z_1}.{z_2} = \\operatorname{cis} \\left( {\\frac{{7\\pi }}{{12}}} \\right) = \\cos \\frac{{7\\pi }}{{12}} + i\\operatorname{sen} \\frac{{7\\pi }}{{12}}$$ ent\u00e3o $$\\cos \\frac{{7\\pi }}{{12}} = \\frac{{ &#8211; \\sqrt 6\u00a0 + \\sqrt 2 }}{4}$$ e $$\\operatorname{sen} \\frac{{7\\pi }}{{12}} = \\frac{{\\sqrt 6\u00a0 + \\sqrt 2 }}{4}$$<\/li>\n<li>Ora,<br \/>\n$$\\begin{array}{*{20}{l}}<br \/>\n{\\cos \\frac{{7\\pi }}{{12}}}&amp; = &amp;{\\cos \\left( {\\frac{\\pi }{4} + \\frac{\\pi }{3}} \\right)} \\\\<br \/>\n{}&amp; = &amp;{\\cos \\frac{\\pi }{4}\\cos \\frac{\\pi }{3} &#8211; \\operatorname{sen} \\frac{\\pi }{4}\\operatorname{sen} \\frac{\\pi }{3}} \\\\<br \/>\n{}&amp; = &amp;{\\frac{{\\sqrt 2 }}{2} \\times \\frac{1}{2} &#8211; \\frac{{\\sqrt 2 }}{2} \\times \\frac{{\\sqrt 3 }}{2}} \\\\<br \/>\n{}&amp; = &amp;{\\frac{{ &#8211; \\sqrt 6\u00a0 + \\sqrt 2 }}{4}}<br \/>\n\\end{array}$$<\/p>\n<p>$$\\begin{array}{*{20}{l}}<br \/>\n{\\operatorname{sen} \\frac{{7\\pi }}{{12}}}&amp; = &amp;{\\operatorname{sen} \\left( {\\frac{\\pi }{4} + \\frac{\\pi }{3}} \\right)} \\\\<br \/>\n{}&amp; = &amp;{\\operatorname{sen} \\frac{\\pi }{4}\\cos \\frac{\\pi }{3} + \\operatorname{sen} \\frac{\\pi }{3}\\cos \\frac{\\pi }{4}} \\\\<br \/>\n{}&amp; = &amp;{\\frac{{\\sqrt 2 }}{2} \\times \\frac{1}{2} + \\frac{{\\sqrt 3 }}{2} \\times \\frac{{\\sqrt 2 }}{2}} \\\\<br \/>\n{}&amp; = &amp;{\\frac{{\\sqrt 6\u00a0 + \\sqrt 2 }}{4}}<br \/>\n\\end{array}$$<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_9263' onClick='GTTabs_show(0,9263)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Considere $$\\begin{array}{*{20}{c}} {{z_1} = \\frac{{\\sqrt 3 }}{2} &#8211; \\frac{1}{2}i}&amp;{\\text{e}}&amp;{{z_2} = \\operatorname{cis} \\frac{{3\\pi }}{4}} \\end{array}$$ Determine ${z_1}.{z_2}$, na forma trigonom\u00e9trica e na forma alg\u00e9brica. Utilizando os resultados obtidos na al\u00ednea anterior, deduza&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":19177,"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-9263","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":1857,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2021\/12\/Mat68.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/9263","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=9263"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/9263\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=9263"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=9263"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=9263"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=9263"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}