{"id":3275,"date":"2025-08-15T23:56:49","date_gmt":"2025-08-15T23:56:49","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=3275"},"modified":"2025-10-20T20:53:34","modified_gmt":"2025-10-20T20:53:34","slug":"polar-form-of-complex-numbers-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/polar-form-of-complex-numbers-apply-it\/","title":{"raw":"Polar Form of Complex Numbers: Apply It","rendered":"Polar Form of Complex Numbers: Apply It"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Plot complex numbers in the complex plane.<\/li>\r\n \t<li>Write complex numbers in polar form.<\/li>\r\n \t<li>Convert a complex number from polar to rectangular form.<\/li>\r\n \t<li>Find products and quotients of complex numbers in polar form.<\/li>\r\n \t<li>Find powers and roots of complex numbers in polar form.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<div>\r\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 !gap-3.5\"><\/div>\r\n<\/div>\r\n<div class=\"h-8\">\r\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 !gap-3.5\">\r\n<h3 class=\"text-lg font-bold text-text-100 mt-1 -mb-1.5\">De Moivre's Theorem for Powers<\/h3>\r\n<section class=\"textbox recall\" aria-label=\"Recall\">\r\n<p class=\"whitespace-normal break-words\">If [latex]z = r(\\cos\\theta + i\\sin\\theta)[\/latex], then:<\/p>\r\n<p class=\"whitespace-normal break-words\">[latex]z^n = r^n(\\cos(n\\theta) + i\\sin(n\\theta))[\/latex]<\/p>\r\n\r\n<\/section>\r\n<p class=\"whitespace-normal break-words\">This theorem also works in reverse to find <strong>nth roots<\/strong> of complex numbers, which helps us solve equations like [latex]x^n = c[\/latex] where [latex]c[\/latex] is any complex number.<\/p>\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"text-lg font-bold text-text-100 mt-1 -mb-1.5\">Solve [latex]x^3=8[\/latex]<\/p>\r\n\r\n[reveal-answer q=\"958344\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"958344\"]\r\n<p class=\"whitespace-normal break-words\">While we know [latex]x = 2[\/latex] is one solution, there are actually <strong>three<\/strong> complex solutions to this equation. Let's find all of them using De Moivre's Theorem.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Step 1:<\/strong> Write 8 in polar form<\/p>\r\n<p class=\"whitespace-normal break-words\">[latex]8 = 8 + 0i = 8(\\cos(0\u00b0) + i\\sin(0\u00b0))[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Step 2:<\/strong> Apply the nth root formula with [latex]n = 3[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">[latex]x = 8^{1\/3}\\left[\\cos\\left(\\frac{0\u00b0 + 360\u00b0k}{3}\\right) + i\\sin\\left(\\frac{0\u00b0 + 360\u00b0k}{3}\\right)\\right][\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">[latex]x = 2\\left[\\cos\\left(\\frac{360\u00b0k}{3}\\right) + i\\sin\\left(\\frac{360\u00b0k}{3}\\right)\\right][\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Step 3:<\/strong> Find all three roots using [latex]k = 0, 1, 2[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">For [latex]k = 0[\/latex]: [latex]x_1 = 2[\\cos(0\u00b0) + i\\sin(0\u00b0)] = 2[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">For [latex]k = 1[\/latex]: [latex]x_2 = 2[\\cos(120\u00b0) + i\\sin(120\u00b0)] = 2\\left(-\\frac{1}{2} + i\\frac{\\sqrt{3}}{2}\\right) = -1 + i\\sqrt{3}[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">For [latex]k = 2[\/latex]: [latex]x_3 = 2[\\cos(240\u00b0) + i\\sin(240\u00b0)] = 2\\left(-\\frac{1}{2} - i\\frac{\\sqrt{3}}{2}\\right) = -1 - i\\sqrt{3}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<div>\r\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 !gap-3.5\">\r\n<p class=\"whitespace-normal break-words\">Solve the equation [latex]x^5 = 32[\/latex] by finding all five complex roots.<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">[reveal-answer q=\"tryit3\"]Show Solution[\/reveal-answer] [hidden-answer a=\"tryit3\"] Write [latex]32[\/latex] in polar form: [latex]32 = 32(\\cos(0\u00b0) + i\\sin(0\u00b0))[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Apply the fifth root formula: [latex]x = 32^{1\/5}\\left[\\cos\\left(\\frac{0\u00b0 + 360\u00b0k}{5}\\right) + i\\sin\\left(\\frac{0\u00b0 + 360\u00b0k}{5}\\right)\\right][\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">[latex]x = 2\\left[\\cos(72\u00b0k) + i\\sin(72\u00b0k)\\right][\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">For [latex]k = 0[\/latex]: [latex]x_1 = 2(\\cos(0\u00b0) + i\\sin(0\u00b0)) = 2[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">For [latex]k = 1[\/latex]: [latex]x_2 = 2(\\cos(72\u00b0) + i\\sin(72\u00b0)) \\approx 0.62 + 1.90i[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">For [latex]k = 2[\/latex]: [latex]x_3 = 2(\\cos(144\u00b0) + i\\sin(144\u00b0)) \\approx -1.62 + 1.18i[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">For [latex]k = 3[\/latex]: [latex]x_4 = 2(\\cos(216\u00b0) + i\\sin(216\u00b0)) \\approx -1.62 - 1.18i[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">For [latex]k = 4[\/latex]: [latex]x_5 = 2(\\cos(288\u00b0) + i\\sin(288\u00b0)) \\approx 0.62 - 1.90i[\/latex] [\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Plot complex numbers in the complex plane.<\/li>\n<li>Write complex numbers in polar form.<\/li>\n<li>Convert a complex number from polar to rectangular form.<\/li>\n<li>Find products and quotients of complex numbers in polar form.<\/li>\n<li>Find powers and roots of complex numbers in polar form.<\/li>\n<\/ul>\n<\/section>\n<div>\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 !gap-3.5\"><\/div>\n<\/div>\n<div class=\"h-8\">\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 !gap-3.5\">\n<h3 class=\"text-lg font-bold text-text-100 mt-1 -mb-1.5\">De Moivre&#8217;s Theorem for Powers<\/h3>\n<section class=\"textbox recall\" aria-label=\"Recall\">\n<p class=\"whitespace-normal break-words\">If [latex]z = r(\\cos\\theta + i\\sin\\theta)[\/latex], then:<\/p>\n<p class=\"whitespace-normal break-words\">[latex]z^n = r^n(\\cos(n\\theta) + i\\sin(n\\theta))[\/latex]<\/p>\n<\/section>\n<p class=\"whitespace-normal break-words\">This theorem also works in reverse to find <strong>nth roots<\/strong> of complex numbers, which helps us solve equations like [latex]x^n = c[\/latex] where [latex]c[\/latex] is any complex number.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"text-lg font-bold text-text-100 mt-1 -mb-1.5\">Solve [latex]x^3=8[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q958344\">Show Solution<\/button><\/p>\n<div id=\"q958344\" class=\"hidden-answer\" style=\"display: none\">\n<p class=\"whitespace-normal break-words\">While we know [latex]x = 2[\/latex] is one solution, there are actually <strong>three<\/strong> complex solutions to this equation. Let&#8217;s find all of them using De Moivre&#8217;s Theorem.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Step 1:<\/strong> Write 8 in polar form<\/p>\n<p class=\"whitespace-normal break-words\">[latex]8 = 8 + 0i = 8(\\cos(0\u00b0) + i\\sin(0\u00b0))[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Step 2:<\/strong> Apply the nth root formula with [latex]n = 3[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">[latex]x = 8^{1\/3}\\left[\\cos\\left(\\frac{0\u00b0 + 360\u00b0k}{3}\\right) + i\\sin\\left(\\frac{0\u00b0 + 360\u00b0k}{3}\\right)\\right][\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">[latex]x = 2\\left[\\cos\\left(\\frac{360\u00b0k}{3}\\right) + i\\sin\\left(\\frac{360\u00b0k}{3}\\right)\\right][\/latex]<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Step 3:<\/strong> Find all three roots using [latex]k = 0, 1, 2[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">For [latex]k = 0[\/latex]: [latex]x_1 = 2[\\cos(0\u00b0) + i\\sin(0\u00b0)] = 2[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">For [latex]k = 1[\/latex]: [latex]x_2 = 2[\\cos(120\u00b0) + i\\sin(120\u00b0)] = 2\\left(-\\frac{1}{2} + i\\frac{\\sqrt{3}}{2}\\right) = -1 + i\\sqrt{3}[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">For [latex]k = 2[\/latex]: [latex]x_3 = 2[\\cos(240\u00b0) + i\\sin(240\u00b0)] = 2\\left(-\\frac{1}{2} - i\\frac{\\sqrt{3}}{2}\\right) = -1 - i\\sqrt{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<div>\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 !gap-3.5\">\n<p class=\"whitespace-normal break-words\">Solve the equation [latex]x^5 = 32[\/latex] by finding all five complex roots.<\/p>\n<p class=\"whitespace-pre-wrap break-words\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qtryit3\">Show Solution<\/button> <\/p>\n<div id=\"qtryit3\" class=\"hidden-answer\" style=\"display: none\"> Write [latex]32[\/latex] in polar form: [latex]32 = 32(\\cos(0\u00b0) + i\\sin(0\u00b0))[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">Apply the fifth root formula: [latex]x = 32^{1\/5}\\left[\\cos\\left(\\frac{0\u00b0 + 360\u00b0k}{5}\\right) + i\\sin\\left(\\frac{0\u00b0 + 360\u00b0k}{5}\\right)\\right][\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">[latex]x = 2\\left[\\cos(72\u00b0k) + i\\sin(72\u00b0k)\\right][\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">For [latex]k = 0[\/latex]: [latex]x_1 = 2(\\cos(0\u00b0) + i\\sin(0\u00b0)) = 2[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">For [latex]k = 1[\/latex]: [latex]x_2 = 2(\\cos(72\u00b0) + i\\sin(72\u00b0)) \\approx 0.62 + 1.90i[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">For [latex]k = 2[\/latex]: [latex]x_3 = 2(\\cos(144\u00b0) + i\\sin(144\u00b0)) \\approx -1.62 + 1.18i[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">For [latex]k = 3[\/latex]: [latex]x_4 = 2(\\cos(216\u00b0) + i\\sin(216\u00b0)) \\approx -1.62 - 1.18i[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">For [latex]k = 4[\/latex]: [latex]x_5 = 2(\\cos(288\u00b0) + i\\sin(288\u00b0)) \\approx 0.62 - 1.90i[\/latex] <\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n","protected":false},"author":67,"menu_order":24,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":247,"module-header":"apply_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3275"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":4,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3275\/revisions"}],"predecessor-version":[{"id":4772,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3275\/revisions\/4772"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/247"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3275\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=3275"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=3275"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=3275"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=3275"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}