{"id":1574,"date":"2025-07-25T03:52:01","date_gmt":"2025-07-25T03:52:01","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1574"},"modified":"2026-03-12T07:05:19","modified_gmt":"2026-03-12T07:05:19","slug":"polar-form-of-complex-numbers-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/polar-form-of-complex-numbers-fresh-take\/","title":{"raw":"Polar Form of Complex Numbers: Fresh Take","rendered":"Polar Form of Complex Numbers: Fresh Take"},"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<h2>Plotting Complex Numbers in the Complex Plane<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n\r\nComplex numbers combine a <strong data-start=\"100\" data-end=\"113\">real part<\/strong> and an <strong data-start=\"121\" data-end=\"139\">imaginary part<\/strong>. The <strong data-start=\"145\" data-end=\"162\">complex plane<\/strong> is a coordinate system where the horizontal axis represents the real part and the vertical axis represents the imaginary part. Plotting a complex number [latex]a+bi[\/latex] works just like plotting a point [latex](x,y)[\/latex] in the rectangular plane, except the vertical axis is labeled \u201cImaginary\u201d instead of \u201cy.\u201d\r\n\r\n<strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Plotting Complex Numbers<\/strong>\r\n<ol>\r\n \t<li data-start=\"532\" data-end=\"668\">\r\n<p data-start=\"535\" data-end=\"559\"><strong data-start=\"535\" data-end=\"557\">Identify the Parts<\/strong><\/p>\r\n\r\n<ul data-start=\"563\" data-end=\"668\">\r\n \t<li data-start=\"563\" data-end=\"611\">\r\n<p data-start=\"565\" data-end=\"611\">Real part = [latex]a[\/latex] (x-coordinate).<\/p>\r\n<\/li>\r\n \t<li data-start=\"615\" data-end=\"668\">\r\n<p data-start=\"617\" data-end=\"668\">Imaginary part = [latex]b[\/latex] (y-coordinate).<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"670\" data-end=\"772\">\r\n<p data-start=\"673\" data-end=\"694\"><strong data-start=\"673\" data-end=\"692\">Plot as a Point<\/strong><\/p>\r\n\r\n<ul data-start=\"1166\" data-end=\"1490\">\r\n \t<li data-start=\"698\" data-end=\"772\">\r\n<p data-start=\"700\" data-end=\"772\">Plot [latex]a+bi[\/latex] at [latex](a,b)[\/latex] on the complex plane.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div><section class=\"textbox example\">\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Plot each complex number on the complex plane.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">a) [latex]3 + 4i[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">b) [latex]-2 - 5i[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">c) [latex]4 - 3i[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">d) [latex]-1 + 2i[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">[reveal-answer q=\"664048\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"664048\"]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">a) [latex]3 + 4i[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The real part is [latex]a = 3[\/latex] and the imaginary part is [latex]b = 4[\/latex].<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><img class=\"alignnone wp-image-5662\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/13182754\/Screenshot-2026-02-13-at-11.22.24%E2%80%AFAM.png\" alt=\"The graph shows the complex plane with the horizontal axis labeled real and the vertical axis labeled imaginary. A single point is plotted at (3, 4) This represents the complex number 3 + 4i\" width=\"271\" height=\"271\" \/><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">b) [latex]-2 - 5i[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The real part is [latex]a = -2[\/latex] and the imaginary part is [latex]b = -5[\/latex].<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><img class=\"alignnone wp-image-5661\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/13182753\/Screenshot-2026-02-13-at-11.22.39%E2%80%AFAM.png\" alt=\"The graph shows the complex plane with the horizontal axis labeled real and the vertical axis labeled imaginary. A single point is plotted at (-2, -5) This represents the complex number -2 - 5i.\" width=\"270\" height=\"267\" \/><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">c) [latex]4 - 3i[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The real part is [latex]a = 4[\/latex] and the imaginary part is [latex]b = -3[\/latex].<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><img class=\"alignnone wp-image-5660\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/13182752\/Screenshot-2026-02-13-at-11.22.51%E2%80%AFAM.png\" alt=\"The graph shows the complex plane with the horizontal axis labeled real and the vertical axis labeled imaginary. A single point is plotted at (4, -3) This represents the complex number 4 - 3i.\" width=\"279\" height=\"278\" \/><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">d) [latex]-1 + 2i[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The real part is [latex]a = -1[\/latex] and the imaginary part is [latex]b = 2[\/latex].<\/p>\r\n<img class=\"alignnone wp-image-5659\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/13182751\/Screenshot-2026-02-13-at-11.23.04%E2%80%AFAM.png\" alt=\"The graph shows the complex plane with the horizontal axis labeled real and the vertical axis labeled imaginary. A single point is plotted at (-1, 2) This represents the complex number -1 + 2i.\" width=\"281\" height=\"280\" \/>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">[\/hidden-answer]<\/p>\r\n\r\n<\/section><\/div>\r\n<div><section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-dbfagdfd-kGzXIbauGQk\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/kGzXIbauGQk?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-dbfagdfd-kGzXIbauGQk\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12845026&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-dbfagdfd-kGzXIbauGQk&vembed=0&video_id=kGzXIbauGQk&video_target=tpm-plugin-dbfagdfd-kGzXIbauGQk'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Plotting+complex+numbers+on+the+complex+plane+%7C+Precalculus+%7C+Khan+Academy_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cPlotting complex numbers on the complex plane | Precalculus | Khan Academy\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><\/div>\r\n<h2>Writing Complex Numbers in Polar Form<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<p data-start=\"66\" data-end=\"411\">Complex numbers can be expressed not only in rectangular form [latex]a+bi[\/latex], but also in <strong data-start=\"161\" data-end=\"175\">polar form<\/strong>, which uses a magnitude (distance from the origin) and an angle (direction from the positive real axis). This form highlights the geometric meaning of complex numbers and is especially useful for multiplication, division, and powers.<\/p>\r\n<p data-start=\"413\" data-end=\"453\">The polar form of a complex number is:<\/p>\r\n<p data-start=\"455\" data-end=\"503\">[latex]z = r(\\cos\\theta + i\\sin\\theta)[\/latex]<\/p>\r\n<p data-start=\"505\" data-end=\"513\">where:<\/p>\r\n\r\n<ul data-start=\"514\" data-end=\"694\">\r\n \t<li data-start=\"514\" data-end=\"583\">\r\n<p data-start=\"516\" data-end=\"583\">[latex]r = \\sqrt{a^{2}+b^{2}}[\/latex] is the magnitude (modulus).<\/p>\r\n<\/li>\r\n \t<li data-start=\"584\" data-end=\"694\">\r\n<p data-start=\"586\" data-end=\"694\">[latex]\\theta = \\tan^{-1}\\left(\\dfrac{b}{a}\\right)[\/latex] is the argument (angle), adjusted for quadrant.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Converting to Polar Form<\/strong>\r\n<ol>\r\n \t<li data-start=\"745\" data-end=\"896\">\r\n<p data-start=\"748\" data-end=\"772\"><strong data-start=\"748\" data-end=\"770\">Find the Magnitude<\/strong><\/p>\r\n\r\n<ul data-start=\"776\" data-end=\"896\">\r\n \t<li data-start=\"776\" data-end=\"816\">\r\n<p data-start=\"778\" data-end=\"816\">[latex]r=\\sqrt{a^{2}+b^{2}}[\/latex].<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"898\" data-end=\"1155\">\r\n<p data-start=\"901\" data-end=\"921\"><strong data-start=\"901\" data-end=\"919\">Find the Angle<\/strong><\/p>\r\n\r\n<ul data-start=\"925\" data-end=\"1155\">\r\n \t<li data-start=\"925\" data-end=\"988\">\r\n<p data-start=\"927\" data-end=\"988\">[latex]\\theta = \\tan^{-1}\\left(\\dfrac{b}{a}\\right)[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"992\" data-end=\"1050\">\r\n<p data-start=\"994\" data-end=\"1050\">Adjust [latex]\\theta[\/latex] for the correct quadrant.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1157\" data-end=\"1264\">\r\n<p data-start=\"1160\" data-end=\"1185\"><strong data-start=\"1160\" data-end=\"1183\">Write in Polar Form<\/strong><\/p>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div><section class=\"textbox example\">\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Write [latex]-1 + i[\/latex] in polar form.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">[reveal-answer q=\"89333\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"89333\"]<\/p>\r\nStep 1: Find the magnitude [latex]r[\/latex]\r\n\r\n[latex] \\begin{aligned} r &amp;= \\sqrt{(-1)^2 + 1^2} \\\\ &amp;= \\sqrt{1 + 1} \\\\ &amp;= \\sqrt{2} \\end{aligned} [\/latex]\r\n\r\nStep 2: Find the angle [latex]\\theta[\/latex]\r\n\r\n[latex] \\begin{aligned} \\tan\\theta &amp;= \\frac{1}{-1} = -1 \\\\ \\theta &amp;= \\tan^{-1}(-1) \\\\ \\theta &amp;= -45\u00b0 \\quad \\text{or} \\quad -\\frac{\\pi}{4} \\end{aligned} [\/latex]\r\n\r\nIMPORTANT: The point [latex](-1, 1)[\/latex] is in Quadrant II, but [latex]\\tan^{-1}(-1)[\/latex] gives us an angle in Quadrant IV.\r\n\r\nAdjust for the correct quadrant:\r\n\r\n[latex] \\begin{aligned} \\theta &amp;= 180\u00b0 + (-45\u00b0) = 135\u00b0 \\\\ \\text{or} \\quad \\theta &amp;= \\pi + \\left(-\\frac{\\pi}{4}\\right) = \\frac{3\\pi}{4} \\end{aligned} [\/latex]\r\n\r\nStep 3: Write in polar form\r\n\r\n[latex] -1 + i = \\sqrt{2}\\left(\\cos 135\u00b0 + i\\sin 135\u00b0\\right) [\/latex]\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">[\/hidden-answer]<\/p>\r\n\r\n<\/section><\/div>\r\n<div><section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-gefhbabh-ncXI47FIgP8\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/ncXI47FIgP8?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-gefhbabh-ncXI47FIgP8\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=14661428&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-gefhbabh-ncXI47FIgP8&vembed=0&video_id=ncXI47FIgP8&video_target=tpm-plugin-gefhbabh-ncXI47FIgP8'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/How+to+Write+a+Complex+Number+in+Polar+Form%2C+Example+with+3+%2B+3i_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow to Write a Complex Number in Polar Form, Example with 3 + 3i\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><\/div>\r\n<h2>Converting Complex Numbers from Polar to Rectangular Form<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n\r\nPolar form expresses a complex number as [latex]z = r(\\cos\\theta + i\\sin\\theta)[\/latex]. To return to rectangular form [latex]a+bi[\/latex], we simply evaluate the cosine and sine at the given angle and multiply by [latex]r[\/latex]. This lets us move seamlessly between the two representations: geometric (polar) and algebraic (rectangular).\r\n\r\n<strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Polar \u2192 Rectangular<\/strong>\r\n<ol data-start=\"474\" data-end=\"1673\">\r\n \t<li data-start=\"474\" data-end=\"635\">\r\n<p data-start=\"477\" data-end=\"506\"><strong data-start=\"477\" data-end=\"504\">Recall the Relationship<\/strong><\/p>\r\n\r\n<ul data-start=\"510\" data-end=\"635\">\r\n \t<li data-start=\"510\" data-end=\"561\">\r\n<p data-start=\"512\" data-end=\"561\">[latex]z = r(\\cos\\theta + i\\sin\\theta)[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"565\" data-end=\"635\">\r\n<p data-start=\"567\" data-end=\"635\">So [latex]a = r\\cos\\theta[\/latex], [latex]b = r\\sin\\theta[\/latex].<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1355\" data-end=\"1530\">\r\n<p data-start=\"1358\" data-end=\"1391\"><strong data-start=\"1358\" data-end=\"1389\">Use Calculators When Needed<\/strong><\/p>\r\n\r\n<ul data-start=\"1395\" data-end=\"1530\">\r\n \t<li data-start=\"1395\" data-end=\"1530\">\r\n<p data-start=\"1397\" data-end=\"1530\">If [latex]\\theta[\/latex] is not a special angle, compute [latex]\\cos\\theta[\/latex] and [latex]\\sin\\theta[\/latex] with a calculator.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1532\" data-end=\"1673\">\r\n<p data-start=\"1535\" data-end=\"1560\"><strong data-start=\"1535\" data-end=\"1558\">Check Your Quadrant<\/strong><\/p>\r\n\r\n<ul data-start=\"1564\" data-end=\"1673\">\r\n \t<li data-start=\"1564\" data-end=\"1673\">\r\n<p data-start=\"1566\" data-end=\"1673\">Make sure the signs of [latex]a[\/latex] and [latex]b[\/latex] match the quadrant of [latex]\\theta[\/latex].<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n&nbsp;\r\n<div><section class=\"textbox example\">\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Convert each complex number from polar form to rectangular form.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">a) [latex]z = 5(\\cos 60\u00b0 + i\\sin 60\u00b0)[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">b) [latex]z = 4\\left(\\cos\\frac{3\\pi}{4} + i\\sin\\frac{3\\pi}{4}\\right)[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">[reveal-answer q=\"861094\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"861094\"]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">a) [latex]z = 5(\\cos 60\u00b0 + i\\sin 60\u00b0)[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">Use [latex]a = r\\cos\\theta[\/latex] and [latex]b = r\\sin\\theta[\/latex]:<\/p>\r\n[latex] \\begin{aligned} a &amp;= 5\\cos 60\u00b0 \\\\ &amp;= 5 \\cdot \\frac{1}{2} \\\\ &amp;= \\frac{5}{2} = 2.5 \\end{aligned} [\/latex]\r\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">[latex] \\begin{aligned} b &amp;= 5\\sin 60\u00b0 \\\\ &amp;= 5 \\cdot \\frac{\\sqrt{3}}{2} \\\\ &amp;= \\frac{5\\sqrt{3}}{2} \\end{aligned} [\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Rectangular form: [latex]z = \\frac{5}{2} + \\frac{5\\sqrt{3}}{2}i[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">b) [latex]z = 4\\left(\\cos\\frac{3\\pi}{4} + i\\sin\\frac{3\\pi}{4}\\right)[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">[latex] \\begin{aligned} a &amp;= 4\\cos\\frac{3\\pi}{4} \\\\ &amp;= 4 \\cdot \\left(-\\frac{\\sqrt{2}}{2}\\right) \\\\ &amp;= -2\\sqrt{2} \\end{aligned} [\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">[latex] \\begin{aligned} b &amp;= 4\\sin\\frac{3\\pi}{4} \\\\ &amp;= 4 \\cdot \\frac{\\sqrt{2}}{2} \\\\ &amp;= 2\\sqrt{2} \\end{aligned} [\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Rectangular form: [latex]z = -2\\sqrt{2} + 2\\sqrt{2}i[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">[\/hidden-answer]<\/p>\r\n\r\n<\/section><\/div>\r\n<div><section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-cbhgcacd-auywa7dydAk\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/auywa7dydAk?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-cbhgcacd-auywa7dydAk\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=14661429&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-cbhgcacd-auywa7dydAk&vembed=0&video_id=auywa7dydAk&video_target=tpm-plugin-cbhgcacd-auywa7dydAk'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Converting+a+complex+number+from+polar+to+rectangular+form+%7C+Precalculus+%7C+Khan+Academy_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cConverting a complex number from polar to rectangular form | Precalculus | Khan Academy\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><\/div>\r\n<h2>Products and Quotients of Complex Numbers in Polar Form<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<p data-start=\"84\" data-end=\"484\">Polar form makes multiplying and dividing complex numbers much easier than working in rectangular form. Instead of expanding and simplifying, we use the <strong data-start=\"237\" data-end=\"251\">magnitudes<\/strong> and <strong data-start=\"256\" data-end=\"266\">angles<\/strong> directly. Multiplication means multiplying magnitudes and adding angles, while division means dividing magnitudes and subtracting angles. This is a direct application of trigonometric identities and Euler\u2019s formula.<\/p>\r\n<p data-start=\"486\" data-end=\"497\">Formulas:<\/p>\r\n\r\n<ul data-start=\"499\" data-end=\"776\">\r\n \t<li data-start=\"499\" data-end=\"626\">\r\n<p data-start=\"501\" data-end=\"626\"><strong data-start=\"501\" data-end=\"513\">Product: <\/strong>[latex]z_{1}z_{2} = r_{1}r_{2}\\big(\\cos(\\theta_{1}+\\theta_{2}) + i\\sin(\\theta_{1}+\\theta_{2})\\big)[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"628\" data-end=\"776\">\r\n<p data-start=\"630\" data-end=\"776\"><strong data-start=\"630\" data-end=\"643\">Quotient: <\/strong>[latex]\\dfrac{z_{1}}{z_{2}} = \\dfrac{r_{1}}{r_{2}}\\big(\\cos(\\theta_{1}-\\theta_{2}) + i\\sin(\\theta_{1}-\\theta_{2})\\big)[\/latex]<\/p>\r\n<\/li>\r\n<\/ul>\r\n<strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Multiplying and Dividing in Polar Form<\/strong>\r\n<ol>\r\n \t<li data-start=\"841\" data-end=\"989\">\r\n<p data-start=\"844\" data-end=\"869\"><strong data-start=\"844\" data-end=\"867\">Multiplication Rule<\/strong><\/p>\r\n\r\n<ul data-start=\"873\" data-end=\"989\">\r\n \t<li data-start=\"873\" data-end=\"928\">\r\n<p data-start=\"875\" data-end=\"928\">Multiply the magnitudes: [latex]r_{1}r_{2}[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"932\" data-end=\"989\">\r\n<p data-start=\"934\" data-end=\"989\">Add the angles: [latex]\\theta_{1}+\\theta_{2}[\/latex].<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"991\" data-end=\"1146\">\r\n<p data-start=\"994\" data-end=\"1013\"><strong data-start=\"994\" data-end=\"1011\">Division Rule<\/strong><\/p>\r\n\r\n<ul data-start=\"1017\" data-end=\"1146\">\r\n \t<li data-start=\"1017\" data-end=\"1080\">\r\n<p data-start=\"1019\" data-end=\"1080\">Divide the magnitudes: [latex]\\dfrac{r_{1}}{r_{2}}[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"1084\" data-end=\"1146\">\r\n<p data-start=\"1086\" data-end=\"1146\">Subtract the angles: [latex]\\theta_{1}-\\theta_{2}[\/latex].<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1869\" data-end=\"2064\">\r\n<p data-start=\"1872\" data-end=\"1904\"><strong data-start=\"1872\" data-end=\"1902\">Keep Results in Polar Form<\/strong><\/p>\r\n\r\n<ul data-start=\"1908\" data-end=\"2064\">\r\n \t<li data-start=\"1908\" data-end=\"1956\">\r\n<p data-start=\"1910\" data-end=\"1956\">These operations are simplest in polar form.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1960\" data-end=\"2064\">\r\n<p data-start=\"1962\" data-end=\"2064\">If rectangular form is needed, convert at the end using [latex]x=r\\cos\\theta, y=r\\sin\\theta[\/latex].<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div><section class=\"textbox example\">\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Find the product of [latex]z_1 = 3(\\cos 45\u00b0 + i\\sin 45\u00b0)[\/latex] and [latex]z_2 = 2(\\cos 60\u00b0 + i\\sin 60\u00b0)[\/latex].<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">[reveal-answer q=\"398368\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"398368\"]<\/p>\r\nUse the product rule: Multiply magnitudes and add angles.\r\n\r\nStep 1: Multiply the magnitudes [latex] r = r_1 \\cdot r_2 = 3 \\cdot 2 = 6 [\/latex]\r\n\r\nStep 2: Add the angles [latex] \\theta = \\theta_1 + \\theta_2 = 45\u00b0 + 60\u00b0 = 105\u00b0 [\/latex]\r\n\r\nStep 3: Write the result in polar form [latex] z_1 \\cdot z_2 = 6(\\cos 105\u00b0 + i\\sin 105\u00b0) [\/latex]\r\n\r\nIf we want the rectangular form:\r\n\r\n[latex] \\begin{aligned} z &amp;= 6\\cos 105\u00b0 + 6i\\sin 105\u00b0 \\\\ &amp;\\approx 6(-0.2588) + 6i(0.9659) \\ &amp;\\approx -1.55 + 5.80i \\end{aligned} [\/latex]\r\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">[\/hidden-answer]<\/p>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Find the quotient [latex]\\frac{z_1}{z_2}[\/latex] where [latex]z_1 = 10(\\cos 150\u00b0 + i\\sin 150\u00b0)[\/latex] and [latex]z_2 = 2(\\cos 30\u00b0 + i\\sin 30\u00b0)[\/latex].<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">[reveal-answer q=\"36148\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"36148\"]<\/p>\r\nUse the quotient rule: Divide magnitudes and subtract angles.\r\n\r\nStep 1: Divide the magnitudes [latex] r = \\frac{r_1}{r_2} = \\frac{10}{2} = 5 [\/latex]\r\n\r\nStep 2: Subtract the angles [latex] \\theta = \\theta_1 - \\theta_2 = 150\u00b0 - 30\u00b0 = 120\u00b0 [\/latex]\r\n\r\nStep 3: Write the result in polar form [latex] \\frac{z_1}{z_2} = 5(\\cos 120\u00b0 + i\\sin 120\u00b0) [\/latex]\r\n\r\nConverting to rectangular form:\r\n\r\n[latex] \\begin{aligned} z &amp;= 5\\cos 120\u00b0 + 5i\\sin 120\u00b0 \\\\ &amp;= 5\\left(-\\frac{1}{2}\\right) + 5i\\left(\\frac{\\sqrt{3}}{2}\\right) \\\\ &amp;= -\\frac{5}{2} + \\frac{5\\sqrt{3}}{2}i \\end{aligned} [\/latex]\r\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">[\/hidden-answer]<\/p>\r\n\r\n<\/section><\/div>\r\n<div><section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-behhcfhc-NbyPwLEiShQ\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/NbyPwLEiShQ?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-behhcfhc-NbyPwLEiShQ\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=14661430&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-behhcfhc-NbyPwLEiShQ&vembed=0&video_id=NbyPwLEiShQ&video_target=tpm-plugin-behhcfhc-NbyPwLEiShQ'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Polar+form+Multiplication+and+division+of+complex+numbers_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cPolar form Multiplication and division of complex numbers\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><\/div>\r\n<h2>Powers and Roots of Complex Numbers in Polar Form<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<p data-start=\"78\" data-end=\"270\">Polar form makes it especially easy to compute powers and roots of complex numbers using <strong data-start=\"167\" data-end=\"190\">De Moivre\u2019s Theorem<\/strong>. Instead of expanding binomials, we work directly with magnitudes and angles.<\/p>\r\n\r\n<ul data-start=\"272\" data-end=\"621\">\r\n \t<li data-start=\"272\" data-end=\"426\">\r\n<p data-start=\"274\" data-end=\"426\"><strong data-start=\"274\" data-end=\"307\">De Moivre\u2019s Theorem (Powers):<\/strong><br data-start=\"307\" data-end=\"310\" \/>[latex]z^{n} = \\big(r(\\cos\\theta + i\\sin\\theta)\\big)^{n} = r^{n}\\big(\\cos(n\\theta) + i\\sin(n\\theta)\\big)[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"428\" data-end=\"621\">\r\n<p data-start=\"430\" data-end=\"621\"><strong data-start=\"430\" data-end=\"464\">nth Roots of a Complex Number:<\/strong><br data-start=\"464\" data-end=\"467\" \/>[latex]z_{k} = r^{\\tfrac{1}{n}}\\Big(\\cos\\Big(\\dfrac{\\theta+2k\\pi}{n}\\Big) + i\\sin\\Big(\\dfrac{\\theta+2k\\pi}{n}\\Big)\\Big), \\quad k=0,1,\\dots,n-1[\/latex]<\/p>\r\n<\/li>\r\n<\/ul>\r\n<p data-start=\"623\" data-end=\"769\">This guarantees that powers give a single answer, while roots give <strong data-start=\"690\" data-end=\"714\">n distinct solutions<\/strong>, evenly spaced around a circle in the complex plane.<\/p>\r\n<strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Powers and Roots<\/strong>\r\n<ol>\r\n \t<li data-start=\"812\" data-end=\"929\">\r\n<p data-start=\"815\" data-end=\"852\"><strong data-start=\"815\" data-end=\"850\">Powers with De Moivre\u2019s Theorem<\/strong><\/p>\r\n\r\n<ul data-start=\"856\" data-end=\"929\">\r\n \t<li data-start=\"856\" data-end=\"897\">\r\n<p data-start=\"858\" data-end=\"897\">Raise the magnitude to the nth power.<\/p>\r\n<\/li>\r\n \t<li data-start=\"901\" data-end=\"929\">\r\n<p data-start=\"903\" data-end=\"929\">Multiply the angle by n.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"931\" data-end=\"1272\">\r\n<p data-start=\"934\" data-end=\"999\"><strong data-start=\"934\" data-end=\"961\">Work an Example (Power): <\/strong>Find [latex](1+i)^{4}[\/latex].<\/p>\r\n\r\n<ul data-start=\"1003\" data-end=\"1272\">\r\n \t<li data-start=\"1003\" data-end=\"1080\">\r\n<p data-start=\"1005\" data-end=\"1080\">Convert: [latex]r=\\sqrt{1^{2}+1^{2}}=\\sqrt{2}, \\ \\theta=45^\\circ[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"1084\" data-end=\"1197\">\r\n<p data-start=\"1086\" data-end=\"1197\">Apply De Moivre: [latex]z^{4} = (\\sqrt{2})^{4}\\big(\\cos(4\\cdot 45^\\circ)+i\\sin(4\\cdot 45^\\circ)\\big)[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"1201\" data-end=\"1272\">\r\n<p data-start=\"1203\" data-end=\"1272\">[latex]= 4(\\cos 180^\\circ + i\\sin 180^\\circ) = 4(-1+0i)=-4[\/latex].<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1274\" data-end=\"1445\">\r\n<p data-start=\"1277\" data-end=\"1292\"><strong data-start=\"1277\" data-end=\"1290\">nth Roots<\/strong><\/p>\r\n\r\n<ul data-start=\"1296\" data-end=\"1445\">\r\n \t<li data-start=\"1296\" data-end=\"1359\">\r\n<p data-start=\"1298\" data-end=\"1359\">Take the nth root of the magnitude: [latex]r^{1\/n}[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"1363\" data-end=\"1445\">\r\n<p data-start=\"1365\" data-end=\"1445\">Divide the angle by n, then add [latex]\\dfrac{2k\\pi}{n}[\/latex] for each root.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1447\" data-end=\"1889\">\r\n<p data-start=\"1450\" data-end=\"1553\"><strong data-start=\"1450\" data-end=\"1477\">Work an Example (Roots): <\/strong>Find the cube roots of [latex]8(\\cos 0^\\circ+i\\sin 0^\\circ)[\/latex].<\/p>\r\n\r\n<ul data-start=\"1557\" data-end=\"1889\">\r\n \t<li data-start=\"1557\" data-end=\"1597\">\r\n<p data-start=\"1559\" data-end=\"1597\">Magnitude: [latex]8^{1\/3}=2[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"1601\" data-end=\"1670\">\r\n<p data-start=\"1603\" data-end=\"1670\">Angles: [latex]\\dfrac{0^\\circ+360^\\circ k}{3}, \\ k=0,1,2[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"1674\" data-end=\"1889\">\r\n<p data-start=\"1676\" data-end=\"1684\">Roots:<\/p>\r\n\r\n<ul data-start=\"1690\" data-end=\"1889\">\r\n \t<li data-start=\"1690\" data-end=\"1741\">\r\n<p data-start=\"1692\" data-end=\"1741\">[latex]2(\\cos 0^\\circ+i\\sin 0^\\circ)=2[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"1747\" data-end=\"1815\">\r\n<p data-start=\"1749\" data-end=\"1815\">[latex]2(\\cos 120^\\circ+i\\sin 120^\\circ) = -1+\\sqrt{3}i[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"1821\" data-end=\"1889\">\r\n<p data-start=\"1823\" data-end=\"1889\">[latex]2(\\cos 240^\\circ+i\\sin 240^\\circ) = -1-\\sqrt{3}i[\/latex].<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1891\" data-end=\"2069\">\r\n<p data-start=\"1894\" data-end=\"1921\"><strong data-start=\"1894\" data-end=\"1919\">Remember the Geometry<\/strong><\/p>\r\n\r\n<ul data-start=\"1925\" data-end=\"2069\">\r\n \t<li data-start=\"1925\" data-end=\"1971\">\r\n<p data-start=\"1927\" data-end=\"1971\">Powers \u201cstretch\u201d the vector and rotate it.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1975\" data-end=\"2069\">\r\n<p data-start=\"1977\" data-end=\"2069\">Roots divide the angle into equal arcs, producing evenly spaced solutions around a circle.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n&nbsp;\r\n<div><section class=\"textbox example\">\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Find [latex](1 + i)^4[\/latex].<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">[reveal-answer q=\"869236\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"869236\"]<\/p>\r\nStep 1: Convert to polar form\r\n\r\nFind the magnitude:\r\n[latex] r = \\sqrt{1^2 + 1^2} = \\sqrt{2} [\/latex]\r\n\r\nFind the angle:\r\n[latex] \\theta = \\tan^{-1}\\left(\\frac{1}{1}\\right) = 45\u00b0 = \\frac{\\pi}{4} [\/latex]\r\n\r\nSo [latex]1 + i = \\sqrt{2}\\left(\\cos 45\u00b0 + i\\sin 45\u00b0\\right)[\/latex]\r\n\r\nStep 2: Apply De Moivre's Theorem\r\n\r\n[latex]z^n = r^n(\\cos(n\\theta) + i\\sin(n\\theta))[\/latex]\r\n\r\n[latex] \\begin{aligned} (1 + i)^4 &amp;= (\\sqrt{2})^4(\\cos(4 \\cdot 45\u00b0) + i\\sin(4 \\cdot 45\u00b0)) \\\\ &amp;= 4(\\cos 180\u00b0 + i\\sin 180\u00b0) \\\\ &amp;= 4(-1 + 0i) \\ &amp;= -4 \\end{aligned} [\/latex]\r\n\r\nTherefore, [latex](1 + i)^4 = -4[\/latex].\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">[\/hidden-answer]<\/p>\r\n\r\n<\/section><\/div>\r\n<div><section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ffeeeabe-Kl8Vms2IO2s\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/Kl8Vms2IO2s?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-ffeeeabe-Kl8Vms2IO2s\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=14661431&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-ffeeeabe-Kl8Vms2IO2s&vembed=0&video_id=Kl8Vms2IO2s&video_target=tpm-plugin-ffeeeabe-Kl8Vms2IO2s'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Roots+of+Complex+Numbers+in+Polar+Form_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cRoots of Complex Numbers in Polar Form\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><\/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<h2>Plotting Complex Numbers in the Complex Plane<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p>Complex numbers combine a <strong data-start=\"100\" data-end=\"113\">real part<\/strong> and an <strong data-start=\"121\" data-end=\"139\">imaginary part<\/strong>. The <strong data-start=\"145\" data-end=\"162\">complex plane<\/strong> is a coordinate system where the horizontal axis represents the real part and the vertical axis represents the imaginary part. Plotting a complex number [latex]a+bi[\/latex] works just like plotting a point [latex](x,y)[\/latex] in the rectangular plane, except the vertical axis is labeled \u201cImaginary\u201d instead of \u201cy.\u201d<\/p>\n<p><strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Plotting Complex Numbers<\/strong><\/p>\n<ol>\n<li data-start=\"532\" data-end=\"668\">\n<p data-start=\"535\" data-end=\"559\"><strong data-start=\"535\" data-end=\"557\">Identify the Parts<\/strong><\/p>\n<ul data-start=\"563\" data-end=\"668\">\n<li data-start=\"563\" data-end=\"611\">\n<p data-start=\"565\" data-end=\"611\">Real part = [latex]a[\/latex] (x-coordinate).<\/p>\n<\/li>\n<li data-start=\"615\" data-end=\"668\">\n<p data-start=\"617\" data-end=\"668\">Imaginary part = [latex]b[\/latex] (y-coordinate).<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"670\" data-end=\"772\">\n<p data-start=\"673\" data-end=\"694\"><strong data-start=\"673\" data-end=\"692\">Plot as a Point<\/strong><\/p>\n<ul data-start=\"1166\" data-end=\"1490\">\n<li data-start=\"698\" data-end=\"772\">\n<p data-start=\"700\" data-end=\"772\">Plot [latex]a+bi[\/latex] at [latex](a,b)[\/latex] on the complex plane.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<div>\n<section class=\"textbox example\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Plot each complex number on the complex plane.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">a) [latex]3 + 4i[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">b) [latex]-2 - 5i[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">c) [latex]4 - 3i[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">d) [latex]-1 + 2i[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q664048\">Show Solution<\/button><\/p>\n<div id=\"q664048\" class=\"hidden-answer\" style=\"display: none\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">a) [latex]3 + 4i[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">The real part is [latex]a = 3[\/latex] and the imaginary part is [latex]b = 4[\/latex].<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-5662\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/13182754\/Screenshot-2026-02-13-at-11.22.24%E2%80%AFAM.png\" alt=\"The graph shows the complex plane with the horizontal axis labeled real and the vertical axis labeled imaginary. A single point is plotted at (3, 4) This represents the complex number 3 + 4i\" width=\"271\" height=\"271\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/13182754\/Screenshot-2026-02-13-at-11.22.24%E2%80%AFAM.png 594w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/13182754\/Screenshot-2026-02-13-at-11.22.24%E2%80%AFAM-300x300.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/13182754\/Screenshot-2026-02-13-at-11.22.24%E2%80%AFAM-150x150.png 150w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/13182754\/Screenshot-2026-02-13-at-11.22.24%E2%80%AFAM-65x65.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/13182754\/Screenshot-2026-02-13-at-11.22.24%E2%80%AFAM-225x225.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/13182754\/Screenshot-2026-02-13-at-11.22.24%E2%80%AFAM-350x350.png 350w\" sizes=\"(max-width: 271px) 100vw, 271px\" \/><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">b) [latex]-2 - 5i[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">The real part is [latex]a = -2[\/latex] and the imaginary part is [latex]b = -5[\/latex].<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-5661\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/13182753\/Screenshot-2026-02-13-at-11.22.39%E2%80%AFAM.png\" alt=\"The graph shows the complex plane with the horizontal axis labeled real and the vertical axis labeled imaginary. A single point is plotted at (-2, -5) This represents the complex number -2 - 5i.\" width=\"270\" height=\"267\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/13182753\/Screenshot-2026-02-13-at-11.22.39%E2%80%AFAM.png 596w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/13182753\/Screenshot-2026-02-13-at-11.22.39%E2%80%AFAM-300x297.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/13182753\/Screenshot-2026-02-13-at-11.22.39%E2%80%AFAM-65x64.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/13182753\/Screenshot-2026-02-13-at-11.22.39%E2%80%AFAM-225x223.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/13182753\/Screenshot-2026-02-13-at-11.22.39%E2%80%AFAM-350x346.png 350w\" sizes=\"(max-width: 270px) 100vw, 270px\" \/><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">c) [latex]4 - 3i[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">The real part is [latex]a = 4[\/latex] and the imaginary part is [latex]b = -3[\/latex].<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-5660\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/13182752\/Screenshot-2026-02-13-at-11.22.51%E2%80%AFAM.png\" alt=\"The graph shows the complex plane with the horizontal axis labeled real and the vertical axis labeled imaginary. A single point is plotted at (4, -3) This represents the complex number 4 - 3i.\" width=\"279\" height=\"278\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/13182752\/Screenshot-2026-02-13-at-11.22.51%E2%80%AFAM.png 594w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/13182752\/Screenshot-2026-02-13-at-11.22.51%E2%80%AFAM-300x300.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/13182752\/Screenshot-2026-02-13-at-11.22.51%E2%80%AFAM-150x150.png 150w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/13182752\/Screenshot-2026-02-13-at-11.22.51%E2%80%AFAM-65x65.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/13182752\/Screenshot-2026-02-13-at-11.22.51%E2%80%AFAM-225x224.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/13182752\/Screenshot-2026-02-13-at-11.22.51%E2%80%AFAM-350x349.png 350w\" sizes=\"(max-width: 279px) 100vw, 279px\" \/><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">d) [latex]-1 + 2i[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">The real part is [latex]a = -1[\/latex] and the imaginary part is [latex]b = 2[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-5659\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/13182751\/Screenshot-2026-02-13-at-11.23.04%E2%80%AFAM.png\" alt=\"The graph shows the complex plane with the horizontal axis labeled real and the vertical axis labeled imaginary. A single point is plotted at (-1, 2) This represents the complex number -1 + 2i.\" width=\"281\" height=\"280\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/13182751\/Screenshot-2026-02-13-at-11.23.04%E2%80%AFAM.png 596w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/13182751\/Screenshot-2026-02-13-at-11.23.04%E2%80%AFAM-300x300.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/13182751\/Screenshot-2026-02-13-at-11.23.04%E2%80%AFAM-150x150.png 150w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/13182751\/Screenshot-2026-02-13-at-11.23.04%E2%80%AFAM-65x65.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/13182751\/Screenshot-2026-02-13-at-11.23.04%E2%80%AFAM-225x224.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/13182751\/Screenshot-2026-02-13-at-11.23.04%E2%80%AFAM-350x349.png 350w\" sizes=\"(max-width: 281px) 100vw, 281px\" \/><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\"><\/div>\n<\/div>\n<\/section>\n<\/div>\n<div>\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-dbfagdfd-kGzXIbauGQk\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/kGzXIbauGQk?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-dbfagdfd-kGzXIbauGQk\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=12845026&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-dbfagdfd-kGzXIbauGQk&#38;vembed=0&#38;video_id=kGzXIbauGQk&#38;video_target=tpm-plugin-dbfagdfd-kGzXIbauGQk\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Plotting+complex+numbers+on+the+complex+plane+%7C+Precalculus+%7C+Khan+Academy_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cPlotting complex numbers on the complex plane | Precalculus | Khan Academy\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/div>\n<h2>Writing Complex Numbers in Polar Form<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p data-start=\"66\" data-end=\"411\">Complex numbers can be expressed not only in rectangular form [latex]a+bi[\/latex], but also in <strong data-start=\"161\" data-end=\"175\">polar form<\/strong>, which uses a magnitude (distance from the origin) and an angle (direction from the positive real axis). This form highlights the geometric meaning of complex numbers and is especially useful for multiplication, division, and powers.<\/p>\n<p data-start=\"413\" data-end=\"453\">The polar form of a complex number is:<\/p>\n<p data-start=\"455\" data-end=\"503\">[latex]z = r(\\cos\\theta + i\\sin\\theta)[\/latex]<\/p>\n<p data-start=\"505\" data-end=\"513\">where:<\/p>\n<ul data-start=\"514\" data-end=\"694\">\n<li data-start=\"514\" data-end=\"583\">\n<p data-start=\"516\" data-end=\"583\">[latex]r = \\sqrt{a^{2}+b^{2}}[\/latex] is the magnitude (modulus).<\/p>\n<\/li>\n<li data-start=\"584\" data-end=\"694\">\n<p data-start=\"586\" data-end=\"694\">[latex]\\theta = \\tan^{-1}\\left(\\dfrac{b}{a}\\right)[\/latex] is the argument (angle), adjusted for quadrant.<\/p>\n<\/li>\n<\/ul>\n<p><strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Converting to Polar Form<\/strong><\/p>\n<ol>\n<li data-start=\"745\" data-end=\"896\">\n<p data-start=\"748\" data-end=\"772\"><strong data-start=\"748\" data-end=\"770\">Find the Magnitude<\/strong><\/p>\n<ul data-start=\"776\" data-end=\"896\">\n<li data-start=\"776\" data-end=\"816\">\n<p data-start=\"778\" data-end=\"816\">[latex]r=\\sqrt{a^{2}+b^{2}}[\/latex].<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"898\" data-end=\"1155\">\n<p data-start=\"901\" data-end=\"921\"><strong data-start=\"901\" data-end=\"919\">Find the Angle<\/strong><\/p>\n<ul data-start=\"925\" data-end=\"1155\">\n<li data-start=\"925\" data-end=\"988\">\n<p data-start=\"927\" data-end=\"988\">[latex]\\theta = \\tan^{-1}\\left(\\dfrac{b}{a}\\right)[\/latex].<\/p>\n<\/li>\n<li data-start=\"992\" data-end=\"1050\">\n<p data-start=\"994\" data-end=\"1050\">Adjust [latex]\\theta[\/latex] for the correct quadrant.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1157\" data-end=\"1264\">\n<p data-start=\"1160\" data-end=\"1185\"><strong data-start=\"1160\" data-end=\"1183\">Write in Polar Form<\/strong><\/p>\n<\/li>\n<\/ol>\n<\/div>\n<div>\n<section class=\"textbox example\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Write [latex]-1 + i[\/latex] in polar form.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q89333\">Show Solution<\/button><\/p>\n<div id=\"q89333\" class=\"hidden-answer\" style=\"display: none\">\n<p>Step 1: Find the magnitude [latex]r[\/latex]<\/p>\n<p>[latex]\\begin{aligned} r &= \\sqrt{(-1)^2 + 1^2} \\\\ &= \\sqrt{1 + 1} \\\\ &= \\sqrt{2} \\end{aligned}[\/latex]<\/p>\n<p>Step 2: Find the angle [latex]\\theta[\/latex]<\/p>\n<p>[latex]\\begin{aligned} \\tan\\theta &= \\frac{1}{-1} = -1 \\\\ \\theta &= \\tan^{-1}(-1) \\\\ \\theta &= -45\u00b0 \\quad \\text{or} \\quad -\\frac{\\pi}{4} \\end{aligned}[\/latex]<\/p>\n<p>IMPORTANT: The point [latex](-1, 1)[\/latex] is in Quadrant II, but [latex]\\tan^{-1}(-1)[\/latex] gives us an angle in Quadrant IV.<\/p>\n<p>Adjust for the correct quadrant:<\/p>\n<p>[latex]\\begin{aligned} \\theta &= 180\u00b0 + (-45\u00b0) = 135\u00b0 \\\\ \\text{or} \\quad \\theta &= \\pi + \\left(-\\frac{\\pi}{4}\\right) = \\frac{3\\pi}{4} \\end{aligned}[\/latex]<\/p>\n<p>Step 3: Write in polar form<\/p>\n<p>[latex]-1 + i = \\sqrt{2}\\left(\\cos 135\u00b0 + i\\sin 135\u00b0\\right)[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\"><\/div>\n<\/div>\n<\/section>\n<\/div>\n<div>\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-gefhbabh-ncXI47FIgP8\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/ncXI47FIgP8?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-gefhbabh-ncXI47FIgP8\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=14661428&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-gefhbabh-ncXI47FIgP8&#38;vembed=0&#38;video_id=ncXI47FIgP8&#38;video_target=tpm-plugin-gefhbabh-ncXI47FIgP8\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/How+to+Write+a+Complex+Number+in+Polar+Form%2C+Example+with+3+%2B+3i_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow to Write a Complex Number in Polar Form, Example with 3 + 3i\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/div>\n<h2>Converting Complex Numbers from Polar to Rectangular Form<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p>Polar form expresses a complex number as [latex]z = r(\\cos\\theta + i\\sin\\theta)[\/latex]. To return to rectangular form [latex]a+bi[\/latex], we simply evaluate the cosine and sine at the given angle and multiply by [latex]r[\/latex]. This lets us move seamlessly between the two representations: geometric (polar) and algebraic (rectangular).<\/p>\n<p><strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Polar \u2192 Rectangular<\/strong><\/p>\n<ol data-start=\"474\" data-end=\"1673\">\n<li data-start=\"474\" data-end=\"635\">\n<p data-start=\"477\" data-end=\"506\"><strong data-start=\"477\" data-end=\"504\">Recall the Relationship<\/strong><\/p>\n<ul data-start=\"510\" data-end=\"635\">\n<li data-start=\"510\" data-end=\"561\">\n<p data-start=\"512\" data-end=\"561\">[latex]z = r(\\cos\\theta + i\\sin\\theta)[\/latex].<\/p>\n<\/li>\n<li data-start=\"565\" data-end=\"635\">\n<p data-start=\"567\" data-end=\"635\">So [latex]a = r\\cos\\theta[\/latex], [latex]b = r\\sin\\theta[\/latex].<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1355\" data-end=\"1530\">\n<p data-start=\"1358\" data-end=\"1391\"><strong data-start=\"1358\" data-end=\"1389\">Use Calculators When Needed<\/strong><\/p>\n<ul data-start=\"1395\" data-end=\"1530\">\n<li data-start=\"1395\" data-end=\"1530\">\n<p data-start=\"1397\" data-end=\"1530\">If [latex]\\theta[\/latex] is not a special angle, compute [latex]\\cos\\theta[\/latex] and [latex]\\sin\\theta[\/latex] with a calculator.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1532\" data-end=\"1673\">\n<p data-start=\"1535\" data-end=\"1560\"><strong data-start=\"1535\" data-end=\"1558\">Check Your Quadrant<\/strong><\/p>\n<ul data-start=\"1564\" data-end=\"1673\">\n<li data-start=\"1564\" data-end=\"1673\">\n<p data-start=\"1566\" data-end=\"1673\">Make sure the signs of [latex]a[\/latex] and [latex]b[\/latex] match the quadrant of [latex]\\theta[\/latex].<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\n<div>\n<section class=\"textbox example\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Convert each complex number from polar form to rectangular form.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">a) [latex]z = 5(\\cos 60\u00b0 + i\\sin 60\u00b0)[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">b) [latex]z = 4\\left(\\cos\\frac{3\\pi}{4} + i\\sin\\frac{3\\pi}{4}\\right)[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q861094\">Show Solution<\/button><\/p>\n<div id=\"q861094\" class=\"hidden-answer\" style=\"display: none\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">a) [latex]z = 5(\\cos 60\u00b0 + i\\sin 60\u00b0)[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-&#091;1.7&#093;\">Use [latex]a = r\\cos\\theta[\/latex] and [latex]b = r\\sin\\theta[\/latex]:<\/p>\n<p>[latex]\\begin{aligned} a &= 5\\cos 60\u00b0 \\\\ &= 5 \\cdot \\frac{1}{2} \\\\ &= \\frac{5}{2} = 2.5 \\end{aligned}[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-&#091;1.7&#093;\">[latex]\\begin{aligned} b &= 5\\sin 60\u00b0 \\\\ &= 5 \\cdot \\frac{\\sqrt{3}}{2} \\\\ &= \\frac{5\\sqrt{3}}{2} \\end{aligned}[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">Rectangular form: [latex]z = \\frac{5}{2} + \\frac{5\\sqrt{3}}{2}i[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">b) [latex]z = 4\\left(\\cos\\frac{3\\pi}{4} + i\\sin\\frac{3\\pi}{4}\\right)[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-&#091;1.7&#093;\">[latex]\\begin{aligned} a &= 4\\cos\\frac{3\\pi}{4} \\\\ &= 4 \\cdot \\left(-\\frac{\\sqrt{2}}{2}\\right) \\\\ &= -2\\sqrt{2} \\end{aligned}[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-&#091;1.7&#093;\">[latex]\\begin{aligned} b &= 4\\sin\\frac{3\\pi}{4} \\\\ &= 4 \\cdot \\frac{\\sqrt{2}}{2} \\\\ &= 2\\sqrt{2} \\end{aligned}[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">Rectangular form: [latex]z = -2\\sqrt{2} + 2\\sqrt{2}i[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\"><\/div>\n<\/div>\n<\/section>\n<\/div>\n<div>\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-cbhgcacd-auywa7dydAk\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/auywa7dydAk?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-cbhgcacd-auywa7dydAk\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=14661429&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-cbhgcacd-auywa7dydAk&#38;vembed=0&#38;video_id=auywa7dydAk&#38;video_target=tpm-plugin-cbhgcacd-auywa7dydAk\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Converting+a+complex+number+from+polar+to+rectangular+form+%7C+Precalculus+%7C+Khan+Academy_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cConverting a complex number from polar to rectangular form | Precalculus | Khan Academy\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/div>\n<h2>Products and Quotients of Complex Numbers in Polar Form<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p data-start=\"84\" data-end=\"484\">Polar form makes multiplying and dividing complex numbers much easier than working in rectangular form. Instead of expanding and simplifying, we use the <strong data-start=\"237\" data-end=\"251\">magnitudes<\/strong> and <strong data-start=\"256\" data-end=\"266\">angles<\/strong> directly. Multiplication means multiplying magnitudes and adding angles, while division means dividing magnitudes and subtracting angles. This is a direct application of trigonometric identities and Euler\u2019s formula.<\/p>\n<p data-start=\"486\" data-end=\"497\">Formulas:<\/p>\n<ul data-start=\"499\" data-end=\"776\">\n<li data-start=\"499\" data-end=\"626\">\n<p data-start=\"501\" data-end=\"626\"><strong data-start=\"501\" data-end=\"513\">Product: <\/strong>[latex]z_{1}z_{2} = r_{1}r_{2}\\big(\\cos(\\theta_{1}+\\theta_{2}) + i\\sin(\\theta_{1}+\\theta_{2})\\big)[\/latex]<\/p>\n<\/li>\n<li data-start=\"628\" data-end=\"776\">\n<p data-start=\"630\" data-end=\"776\"><strong data-start=\"630\" data-end=\"643\">Quotient: <\/strong>[latex]\\dfrac{z_{1}}{z_{2}} = \\dfrac{r_{1}}{r_{2}}\\big(\\cos(\\theta_{1}-\\theta_{2}) + i\\sin(\\theta_{1}-\\theta_{2})\\big)[\/latex]<\/p>\n<\/li>\n<\/ul>\n<p><strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Multiplying and Dividing in Polar Form<\/strong><\/p>\n<ol>\n<li data-start=\"841\" data-end=\"989\">\n<p data-start=\"844\" data-end=\"869\"><strong data-start=\"844\" data-end=\"867\">Multiplication Rule<\/strong><\/p>\n<ul data-start=\"873\" data-end=\"989\">\n<li data-start=\"873\" data-end=\"928\">\n<p data-start=\"875\" data-end=\"928\">Multiply the magnitudes: [latex]r_{1}r_{2}[\/latex].<\/p>\n<\/li>\n<li data-start=\"932\" data-end=\"989\">\n<p data-start=\"934\" data-end=\"989\">Add the angles: [latex]\\theta_{1}+\\theta_{2}[\/latex].<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"991\" data-end=\"1146\">\n<p data-start=\"994\" data-end=\"1013\"><strong data-start=\"994\" data-end=\"1011\">Division Rule<\/strong><\/p>\n<ul data-start=\"1017\" data-end=\"1146\">\n<li data-start=\"1017\" data-end=\"1080\">\n<p data-start=\"1019\" data-end=\"1080\">Divide the magnitudes: [latex]\\dfrac{r_{1}}{r_{2}}[\/latex].<\/p>\n<\/li>\n<li data-start=\"1084\" data-end=\"1146\">\n<p data-start=\"1086\" data-end=\"1146\">Subtract the angles: [latex]\\theta_{1}-\\theta_{2}[\/latex].<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1869\" data-end=\"2064\">\n<p data-start=\"1872\" data-end=\"1904\"><strong data-start=\"1872\" data-end=\"1902\">Keep Results in Polar Form<\/strong><\/p>\n<ul data-start=\"1908\" data-end=\"2064\">\n<li data-start=\"1908\" data-end=\"1956\">\n<p data-start=\"1910\" data-end=\"1956\">These operations are simplest in polar form.<\/p>\n<\/li>\n<li data-start=\"1960\" data-end=\"2064\">\n<p data-start=\"1962\" data-end=\"2064\">If rectangular form is needed, convert at the end using [latex]x=r\\cos\\theta, y=r\\sin\\theta[\/latex].<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<div>\n<section class=\"textbox example\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Find the product of [latex]z_1 = 3(\\cos 45\u00b0 + i\\sin 45\u00b0)[\/latex] and [latex]z_2 = 2(\\cos 60\u00b0 + i\\sin 60\u00b0)[\/latex].<\/p>\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q398368\">Show Solution<\/button><\/p>\n<div id=\"q398368\" class=\"hidden-answer\" style=\"display: none\">\n<p>Use the product rule: Multiply magnitudes and add angles.<\/p>\n<p>Step 1: Multiply the magnitudes [latex]r = r_1 \\cdot r_2 = 3 \\cdot 2 = 6[\/latex]<\/p>\n<p>Step 2: Add the angles [latex]\\theta = \\theta_1 + \\theta_2 = 45\u00b0 + 60\u00b0 = 105\u00b0[\/latex]<\/p>\n<p>Step 3: Write the result in polar form [latex]z_1 \\cdot z_2 = 6(\\cos 105\u00b0 + i\\sin 105\u00b0)[\/latex]<\/p>\n<p>If we want the rectangular form:<\/p>\n<p>[latex]\\begin{aligned} z &= 6\\cos 105\u00b0 + 6i\\sin 105\u00b0 \\\\ &\\approx 6(-0.2588) + 6i(0.9659) \\ &\\approx -1.55 + 5.80i \\end{aligned}[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-&#091;1.7&#093;\"><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Find the quotient [latex]\\frac{z_1}{z_2}[\/latex] where [latex]z_1 = 10(\\cos 150\u00b0 + i\\sin 150\u00b0)[\/latex] and [latex]z_2 = 2(\\cos 30\u00b0 + i\\sin 30\u00b0)[\/latex].<\/p>\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q36148\">Show Solution<\/button><\/p>\n<div id=\"q36148\" class=\"hidden-answer\" style=\"display: none\">\n<p>Use the quotient rule: Divide magnitudes and subtract angles.<\/p>\n<p>Step 1: Divide the magnitudes [latex]r = \\frac{r_1}{r_2} = \\frac{10}{2} = 5[\/latex]<\/p>\n<p>Step 2: Subtract the angles [latex]\\theta = \\theta_1 - \\theta_2 = 150\u00b0 - 30\u00b0 = 120\u00b0[\/latex]<\/p>\n<p>Step 3: Write the result in polar form [latex]\\frac{z_1}{z_2} = 5(\\cos 120\u00b0 + i\\sin 120\u00b0)[\/latex]<\/p>\n<p>Converting to rectangular form:<\/p>\n<p>[latex]\\begin{aligned} z &= 5\\cos 120\u00b0 + 5i\\sin 120\u00b0 \\\\ &= 5\\left(-\\frac{1}{2}\\right) + 5i\\left(\\frac{\\sqrt{3}}{2}\\right) \\\\ &= -\\frac{5}{2} + \\frac{5\\sqrt{3}}{2}i \\end{aligned}[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-&#091;1.7&#093;\"><\/div>\n<\/div>\n<\/section>\n<\/div>\n<div>\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-behhcfhc-NbyPwLEiShQ\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/NbyPwLEiShQ?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-behhcfhc-NbyPwLEiShQ\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=14661430&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-behhcfhc-NbyPwLEiShQ&#38;vembed=0&#38;video_id=NbyPwLEiShQ&#38;video_target=tpm-plugin-behhcfhc-NbyPwLEiShQ\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Polar+form+Multiplication+and+division+of+complex+numbers_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cPolar form Multiplication and division of complex numbers\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/div>\n<h2>Powers and Roots of Complex Numbers in Polar Form<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p data-start=\"78\" data-end=\"270\">Polar form makes it especially easy to compute powers and roots of complex numbers using <strong data-start=\"167\" data-end=\"190\">De Moivre\u2019s Theorem<\/strong>. Instead of expanding binomials, we work directly with magnitudes and angles.<\/p>\n<ul data-start=\"272\" data-end=\"621\">\n<li data-start=\"272\" data-end=\"426\">\n<p data-start=\"274\" data-end=\"426\"><strong data-start=\"274\" data-end=\"307\">De Moivre\u2019s Theorem (Powers):<\/strong><br data-start=\"307\" data-end=\"310\" \/>[latex]z^{n} = \\big(r(\\cos\\theta + i\\sin\\theta)\\big)^{n} = r^{n}\\big(\\cos(n\\theta) + i\\sin(n\\theta)\\big)[\/latex]<\/p>\n<\/li>\n<li data-start=\"428\" data-end=\"621\">\n<p data-start=\"430\" data-end=\"621\"><strong data-start=\"430\" data-end=\"464\">nth Roots of a Complex Number:<\/strong><br data-start=\"464\" data-end=\"467\" \/>[latex]z_{k} = r^{\\tfrac{1}{n}}\\Big(\\cos\\Big(\\dfrac{\\theta+2k\\pi}{n}\\Big) + i\\sin\\Big(\\dfrac{\\theta+2k\\pi}{n}\\Big)\\Big), \\quad k=0,1,\\dots,n-1[\/latex]<\/p>\n<\/li>\n<\/ul>\n<p data-start=\"623\" data-end=\"769\">This guarantees that powers give a single answer, while roots give <strong data-start=\"690\" data-end=\"714\">n distinct solutions<\/strong>, evenly spaced around a circle in the complex plane.<\/p>\n<p><strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Powers and Roots<\/strong><\/p>\n<ol>\n<li data-start=\"812\" data-end=\"929\">\n<p data-start=\"815\" data-end=\"852\"><strong data-start=\"815\" data-end=\"850\">Powers with De Moivre\u2019s Theorem<\/strong><\/p>\n<ul data-start=\"856\" data-end=\"929\">\n<li data-start=\"856\" data-end=\"897\">\n<p data-start=\"858\" data-end=\"897\">Raise the magnitude to the nth power.<\/p>\n<\/li>\n<li data-start=\"901\" data-end=\"929\">\n<p data-start=\"903\" data-end=\"929\">Multiply the angle by n.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"931\" data-end=\"1272\">\n<p data-start=\"934\" data-end=\"999\"><strong data-start=\"934\" data-end=\"961\">Work an Example (Power): <\/strong>Find [latex](1+i)^{4}[\/latex].<\/p>\n<ul data-start=\"1003\" data-end=\"1272\">\n<li data-start=\"1003\" data-end=\"1080\">\n<p data-start=\"1005\" data-end=\"1080\">Convert: [latex]r=\\sqrt{1^{2}+1^{2}}=\\sqrt{2}, \\ \\theta=45^\\circ[\/latex].<\/p>\n<\/li>\n<li data-start=\"1084\" data-end=\"1197\">\n<p data-start=\"1086\" data-end=\"1197\">Apply De Moivre: [latex]z^{4} = (\\sqrt{2})^{4}\\big(\\cos(4\\cdot 45^\\circ)+i\\sin(4\\cdot 45^\\circ)\\big)[\/latex].<\/p>\n<\/li>\n<li data-start=\"1201\" data-end=\"1272\">\n<p data-start=\"1203\" data-end=\"1272\">[latex]= 4(\\cos 180^\\circ + i\\sin 180^\\circ) = 4(-1+0i)=-4[\/latex].<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1274\" data-end=\"1445\">\n<p data-start=\"1277\" data-end=\"1292\"><strong data-start=\"1277\" data-end=\"1290\">nth Roots<\/strong><\/p>\n<ul data-start=\"1296\" data-end=\"1445\">\n<li data-start=\"1296\" data-end=\"1359\">\n<p data-start=\"1298\" data-end=\"1359\">Take the nth root of the magnitude: [latex]r^{1\/n}[\/latex].<\/p>\n<\/li>\n<li data-start=\"1363\" data-end=\"1445\">\n<p data-start=\"1365\" data-end=\"1445\">Divide the angle by n, then add [latex]\\dfrac{2k\\pi}{n}[\/latex] for each root.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1447\" data-end=\"1889\">\n<p data-start=\"1450\" data-end=\"1553\"><strong data-start=\"1450\" data-end=\"1477\">Work an Example (Roots): <\/strong>Find the cube roots of [latex]8(\\cos 0^\\circ+i\\sin 0^\\circ)[\/latex].<\/p>\n<ul data-start=\"1557\" data-end=\"1889\">\n<li data-start=\"1557\" data-end=\"1597\">\n<p data-start=\"1559\" data-end=\"1597\">Magnitude: [latex]8^{1\/3}=2[\/latex].<\/p>\n<\/li>\n<li data-start=\"1601\" data-end=\"1670\">\n<p data-start=\"1603\" data-end=\"1670\">Angles: [latex]\\dfrac{0^\\circ+360^\\circ k}{3}, \\ k=0,1,2[\/latex].<\/p>\n<\/li>\n<li data-start=\"1674\" data-end=\"1889\">\n<p data-start=\"1676\" data-end=\"1684\">Roots:<\/p>\n<ul data-start=\"1690\" data-end=\"1889\">\n<li data-start=\"1690\" data-end=\"1741\">\n<p data-start=\"1692\" data-end=\"1741\">[latex]2(\\cos 0^\\circ+i\\sin 0^\\circ)=2[\/latex].<\/p>\n<\/li>\n<li data-start=\"1747\" data-end=\"1815\">\n<p data-start=\"1749\" data-end=\"1815\">[latex]2(\\cos 120^\\circ+i\\sin 120^\\circ) = -1+\\sqrt{3}i[\/latex].<\/p>\n<\/li>\n<li data-start=\"1821\" data-end=\"1889\">\n<p data-start=\"1823\" data-end=\"1889\">[latex]2(\\cos 240^\\circ+i\\sin 240^\\circ) = -1-\\sqrt{3}i[\/latex].<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1891\" data-end=\"2069\">\n<p data-start=\"1894\" data-end=\"1921\"><strong data-start=\"1894\" data-end=\"1919\">Remember the Geometry<\/strong><\/p>\n<ul data-start=\"1925\" data-end=\"2069\">\n<li data-start=\"1925\" data-end=\"1971\">\n<p data-start=\"1927\" data-end=\"1971\">Powers \u201cstretch\u201d the vector and rotate it.<\/p>\n<\/li>\n<li data-start=\"1975\" data-end=\"2069\">\n<p data-start=\"1977\" data-end=\"2069\">Roots divide the angle into equal arcs, producing evenly spaced solutions around a circle.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\n<div>\n<section class=\"textbox example\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Find [latex](1 + i)^4[\/latex].<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q869236\">Show Solution<\/button><\/p>\n<div id=\"q869236\" class=\"hidden-answer\" style=\"display: none\">\n<p>Step 1: Convert to polar form<\/p>\n<p>Find the magnitude:<br \/>\n[latex]r = \\sqrt{1^2 + 1^2} = \\sqrt{2}[\/latex]<\/p>\n<p>Find the angle:<br \/>\n[latex]\\theta = \\tan^{-1}\\left(\\frac{1}{1}\\right) = 45\u00b0 = \\frac{\\pi}{4}[\/latex]<\/p>\n<p>So [latex]1 + i = \\sqrt{2}\\left(\\cos 45\u00b0 + i\\sin 45\u00b0\\right)[\/latex]<\/p>\n<p>Step 2: Apply De Moivre&#8217;s Theorem<\/p>\n<p>[latex]z^n = r^n(\\cos(n\\theta) + i\\sin(n\\theta))[\/latex]<\/p>\n<p>[latex]\\begin{aligned} (1 + i)^4 &= (\\sqrt{2})^4(\\cos(4 \\cdot 45\u00b0) + i\\sin(4 \\cdot 45\u00b0)) \\\\ &= 4(\\cos 180\u00b0 + i\\sin 180\u00b0) \\\\ &= 4(-1 + 0i) \\ &= -4 \\end{aligned}[\/latex]<\/p>\n<p>Therefore, [latex](1 + i)^4 = -4[\/latex].<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\"><\/div>\n<\/div>\n<\/section>\n<\/div>\n<div>\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ffeeeabe-Kl8Vms2IO2s\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/Kl8Vms2IO2s?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-ffeeeabe-Kl8Vms2IO2s\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=14661431&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-ffeeeabe-Kl8Vms2IO2s&#38;vembed=0&#38;video_id=Kl8Vms2IO2s&#38;video_target=tpm-plugin-ffeeeabe-Kl8Vms2IO2s\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Roots+of+Complex+Numbers+in+Polar+Form_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cRoots of Complex Numbers in Polar Form\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/div>\n","protected":false},"author":67,"menu_order":25,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Plotting complex numbers on the complex plane | Precalculus | Khan Academy\",\"author\":\"\",\"organization\":\"Khan Academy\",\"url\":\"https:\/\/youtu.be\/kGzXIbauGQk\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"How to Write a Complex Number in Polar Form, Example with 3 + 3i\",\"author\":\"\",\"organization\":\"The Math Sorcerer\",\"url\":\"https:\/\/youtu.be\/ncXI47FIgP8\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Converting a complex number from polar to rectangular form | Precalculus | Khan Academy\",\"author\":\"\",\"organization\":\"Khan Academy\",\"url\":\"https:\/\/youtu.be\/auywa7dydAk\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Polar form Multiplication and division of complex numbers\",\"author\":\"\",\"organization\":\"Joel Speranza Math\",\"url\":\"https:\/\/youtu.be\/NbyPwLEiShQ\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Roots of Complex Numbers in Polar Form\",\"author\":\"Daniel Kopsas\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/Kl8Vms2IO2s\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":247,"module-header":"fresh_take","content_attributions":[{"type":"copyrighted_video","description":"Plotting complex numbers on the complex plane | Precalculus | Khan Academy","author":"","organization":"Khan Academy","url":"https:\/\/youtu.be\/kGzXIbauGQk","project":"","license":"arr","license_terms":"Standard YouTube License"},{"type":"copyrighted_video","description":"How to Write a Complex Number in Polar Form, Example with 3 + 3i","author":"","organization":"The Math Sorcerer","url":"https:\/\/youtu.be\/ncXI47FIgP8","project":"","license":"arr","license_terms":"Standard YouTube License"},{"type":"copyrighted_video","description":"Converting a complex number from polar to rectangular form | Precalculus | Khan Academy","author":"","organization":"Khan Academy","url":"https:\/\/youtu.be\/auywa7dydAk","project":"","license":"arr","license_terms":"Standard YouTube License"},{"type":"copyrighted_video","description":"Polar form Multiplication and division of complex numbers","author":"","organization":"Joel Speranza Math","url":"https:\/\/youtu.be\/NbyPwLEiShQ","project":"","license":"arr","license_terms":"Standard YouTube License"},{"type":"copyrighted_video","description":"Roots of Complex Numbers in Polar Form","author":"Daniel Kopsas","organization":"","url":"https:\/\/youtu.be\/Kl8Vms2IO2s","project":"","license":"arr","license_terms":"Standard YouTube License"}],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"<script type='text\/javascript' src='https:\/\/www.youtube.com\/iframe_api'><\/script><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12845026&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-dbfagdfd-kGzXIbauGQk&vembed=0&video_id=kGzXIbauGQk&video_target=tpm-plugin-dbfagdfd-kGzXIbauGQk'><\/script>\n<script type='text\/javascript' src='https:\/\/www.youtube.com\/iframe_api'><\/script><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=14661428&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-gefhbabh-ncXI47FIgP8&vembed=0&video_id=ncXI47FIgP8&video_target=tpm-plugin-gefhbabh-ncXI47FIgP8'><\/script>\n<script type='text\/javascript' src='https:\/\/www.youtube.com\/iframe_api'><\/script><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=14661429&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-cbhgcacd-auywa7dydAk&vembed=0&video_id=auywa7dydAk&video_target=tpm-plugin-cbhgcacd-auywa7dydAk'><\/script>\n<script type='text\/javascript' src='https:\/\/www.youtube.com\/iframe_api'><\/script><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=14661430&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-behhcfhc-NbyPwLEiShQ&vembed=0&video_id=NbyPwLEiShQ&video_target=tpm-plugin-behhcfhc-NbyPwLEiShQ'><\/script>\n<script type='text\/javascript' src='https:\/\/www.youtube.com\/iframe_api'><\/script><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=14661431&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-ffeeeabe-Kl8Vms2IO2s&vembed=0&video_id=Kl8Vms2IO2s&video_target=tpm-plugin-ffeeeabe-Kl8Vms2IO2s'><\/script>\n","media_targets":["tpm-plugin-dbfagdfd-kGzXIbauGQk","tpm-plugin-gefhbabh-ncXI47FIgP8","tpm-plugin-cbhgcacd-auywa7dydAk","tpm-plugin-behhcfhc-NbyPwLEiShQ","tpm-plugin-ffeeeabe-Kl8Vms2IO2s"]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1574"}],"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":15,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1574\/revisions"}],"predecessor-version":[{"id":5822,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1574\/revisions\/5822"}],"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\/1574\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=1574"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=1574"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=1574"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=1574"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}