{"id":1568,"date":"2025-07-25T03:50:06","date_gmt":"2025-07-25T03:50:06","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1568"},"modified":"2026-03-12T06:55:23","modified_gmt":"2026-03-12T06:55:23","slug":"polar-coordinates-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/polar-coordinates-fresh-take\/","title":{"raw":"Polar Coordinates: Fresh Take","rendered":"Polar Coordinates: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Plot points using polar coordinates.<\/li>\r\n \t<li>Convert between polar coordinates and rectangular coordinates.<\/li>\r\n \t<li>Transform equations between polar and rectangular forms.<\/li>\r\n \t<li>Identify and graph polar equations by converting to rectangular equations.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Defining Polar Coordinates<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"whitespace-normal break-words\">Imagine you're giving directions to a friend. Instead of saying \"go 3 blocks east and 4 blocks north,\" you could say \"walk 5 blocks in the northeast direction.\" That's essentially what polar coordinates do\u2014they describe where a point is using distance and direction rather than horizontal and vertical measurements.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>The core concept:<\/strong> Every point can be described using two pieces of information:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]r[\/latex] = how far you are from the origin (like \"5 blocks away\")<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\theta[\/latex] = what angle you're at from the positive x-axis (like \"northeast direction\")<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\"><strong>The conversion formulas you need:<\/strong><\/p>\r\n\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Polar to Rectangular:<\/strong> [latex]x = r\\cos\\theta[\/latex] and [latex]y = r\\sin\\theta[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Rectangular to Polar:<\/strong> [latex]r^2 = x^2 + y^2[\/latex] and [latex]\\tan\\theta = \\frac{y}{x}[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\">Think of the right triangle formed by dropping a perpendicular from your point to the [latex]x[\/latex]-axis. The hypotenuse is [latex]r[\/latex], and basic trig (SohCahToa) gives you the relationships: cosine for the [latex]x[\/latex]-component, sine for the [latex]y[\/latex]-component.<\/p>\r\nWatch out for the angle calculation. Don't just use [latex]\\theta = \\tan^{-1}(y\/x)[\/latex] blindly! The inverse tangent function only gives angles in Quadrants I and IV. For points in Quadrants II or III, you need to add [latex]\\pi[\/latex] to get the correct angle.\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<div id=\"fs-id1167793901809\" data-type=\"problem\">\r\n<p id=\"fs-id1167794066199\">Convert [latex]\\left(-8,-8\\right)[\/latex] into polar coordinates and [latex]\\left(4,\\frac{2\\pi }{3}\\right)[\/latex] into rectangular coordinates.<\/p>\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558897\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"44558897\"]\r\n<div id=\"fs-id1167794064210\" data-type=\"commentary\" data-element-type=\"hint\">\r\n<p id=\"fs-id1167793262229\">Use both equations from the theorem. Make sure to check the quadrant when calculating [latex]\\theta [\/latex].<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"44558898\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558898\"]\r\n<div id=\"fs-id1167793805544\" data-type=\"solution\">\r\n<p id=\"fs-id1167793270874\" style=\"text-align: center;\">[latex]\\left(8\\sqrt{2},\\frac{5\\pi }{4}\\right)[\/latex] and [latex]\\left(-2,2\\sqrt{3}\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-dbfhhfde-IbxuHmH6D2c\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/IbxuHmH6D2c?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-dbfhhfde-IbxuHmH6D2c\" 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=14661414&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-dbfhhfde-IbxuHmH6D2c&vembed=0&video_id=IbxuHmH6D2c&video_target=tpm-plugin-dbfhhfde-IbxuHmH6D2c'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Converting+polar+coordinates+into+cartesian+coordinates_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cConverting polar coordinates into cartesian coordinates\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ghehbbgh-Qa5unxRmG7Y\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/Qa5unxRmG7Y?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-ghehbbgh-Qa5unxRmG7Y\" 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=14661415&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-ghehbbgh-Qa5unxRmG7Y&vembed=0&video_id=Qa5unxRmG7Y&video_target=tpm-plugin-ghehbbgh-Qa5unxRmG7Y'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Converting+cartesian+coordinates+into+polar+coordinates_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cConverting cartesian coordinates into polar coordinates\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Plotting Points in the Polar Plane<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"whitespace-normal break-words\">In polar coordinates, the same point can be represented by infinitely many coordinate pairs. Unlike rectangular coordinates where every point has exactly one (x,y) pair, polar coordinates allow multiple representations for the same location.<\/p>\r\n<p class=\"whitespace-normal break-words\">This happens because:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">You can add or subtract full rotations ([latex]2\\pi[\/latex]) to any angle<\/li>\r\n \t<li class=\"whitespace-normal break-words\">You can use negative [latex]r[\/latex] values by pointing in the opposite direction<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\"><strong>Problem-Solving Strategy:<\/strong><\/p>\r\n\r\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>Start with the angle<\/strong> [latex]\\theta[\/latex]: Rotate counterclockwise from the positive x-axis (polar axis) if positive, clockwise if negative<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Handle the distance<\/strong> [latex]r[\/latex]:\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">If [latex]r &gt; 0[\/latex]: Move that distance along your angle's direction<\/li>\r\n \t<li class=\"whitespace-normal break-words\">If [latex]r &lt; 0[\/latex]: Move that distance in the <strong>opposite<\/strong> direction from your angle<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\nWhen [latex]r[\/latex] is negative, you move in the direction opposite to your angle. For [latex](-3, \\frac{2\\pi}{3})[\/latex], you first rotate to angle [latex]\\frac{2\\pi}{3}[\/latex], then move 3 units in the opposite direction.\r\n\r\nThe <strong>polar plane<\/strong> consists of concentric circles representing constant distances ([latex]r = 1, r = 2[\/latex], etc.) and straight lines radiating from the pole (origin) representing constant angles ([latex]\\theta = \\frac{\\pi}{4}, \\theta = \\frac{\\pi}{2}[\/latex], etc.).\r\n\r\nTo check your work, convert your polar coordinates to rectangular using [latex]x = r\\cos\\theta[\/latex] and [latex]y = r\\sin\\theta[\/latex]. The point should land where you expect it in the xy-plane.\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<div id=\"fs-id1167794292524\" data-type=\"problem\">\r\n<p id=\"fs-id1167794330185\">Plot [latex]\\left(4,\\frac{5\\pi }{3}\\right)[\/latex] and [latex]\\left(-3,-\\frac{7\\pi }{2}\\right)[\/latex] on the polar plane.<\/p>\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558894\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"44558894\"]\r\n<div id=\"fs-id1167794039158\" data-type=\"commentary\" data-element-type=\"hint\">\r\n<p id=\"fs-id1167794060098\">Start with [latex]\\theta [\/latex], then use [latex]r[\/latex].<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"44558895\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558895\"]\r\n<div id=\"fs-id1167794039142\" data-type=\"solution\">\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11234815\/CNX_Calc_Figure_11_03_004.jpg\" alt=\"Two points are marked on a polar coordinate plane, specifically (\u22123, \u22127\u03c0\/2) on the y-axis and (4, 5\u03c0\/3) in the fourth quadrant.\" width=\"417\" height=\"417\" data-media-type=\"image\/jpeg\" \/>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-fdbabhee-oWNFXLjWouQ\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/oWNFXLjWouQ?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-fdbabhee-oWNFXLjWouQ\" 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=14661416&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-fdbabhee-oWNFXLjWouQ&vembed=0&video_id=oWNFXLjWouQ&video_target=tpm-plugin-fdbabhee-oWNFXLjWouQ'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Plotting+polar+points_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cPlotting polar points\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h3>Transform Equations Between Polar and Rectangular Forms<\/h3>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Convert each polar equation to rectangular form.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">a) [latex]r = 5[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">b) [latex]\\theta = \\frac{\\pi}{3}[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">c) [latex]r = 4\\cos\\theta[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">[reveal-answer q=\"753370\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"753370\"]<\/p>\r\na) [latex]r = 5[\/latex]\r\n\r\nUse the relationship [latex]r^2 = x^2 + y^2[\/latex]:\r\n\r\n[latex] \\begin{aligned} r &amp;= 5 \\\\ r^2 &amp;= 25 \\quad \\text{(square both sides)} \\\\ x^2 + y^2 &amp;= 25 \\end{aligned} [\/latex]\r\n\r\nThis is a circle centered at the origin with radius 5.\r\n\r\nb) [latex]\\theta = \\frac{\\pi}{3}[\/latex]\r\n\r\nUse the relationship [latex]\\tan\\theta = \\frac{y}{x}[\/latex]:\r\n\r\n[latex] \\begin{aligned} \\theta &amp;= \\frac{\\pi}{3} \\\\ \\tan\\theta &amp;= \\tan\\left(\\frac{\\pi}{3}\\right) \\\\ \\frac{y}{x} &amp;= \\sqrt{3} \\\\ y &amp;= \\sqrt{3}x \\quad \\text{(for } x \\geq 0\\text{)} \\end{aligned} [\/latex]\r\n\r\nThis is a line through the origin with slope [latex]\\sqrt{3}[\/latex], but only the ray where [latex]x \\geq 0[\/latex].\r\n\r\nc) [latex]r = 4\\cos\\theta[\/latex]\r\n\r\nUse the substitution [latex]\\cos\\theta = \\frac{x}{r}[\/latex]:\r\n\r\n[latex] \\begin{aligned} r &amp;= 4\\cos\\theta \\\\ r &amp;= 4 \\cdot \\frac{x}{r} \\\\ r^2 &amp;= 4x \\quad \\text{(multiply both sides by } r\\text{)} \\\\ x^2 + y^2 &amp;= 4x \\quad \\text{(substitute } r^2 = x^2 + y^2\\text{)} \\end{aligned} [\/latex]\r\n\r\nThis is a circle with center [latex](2, 0)[\/latex] and radius 2.\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">[\/hidden-answer]<\/p>\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-feeeagab-lh9Alon3i44\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/lh9Alon3i44?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-feeeagab-lh9Alon3i44\" 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=14661417&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-feeeagab-lh9Alon3i44&vembed=0&video_id=lh9Alon3i44&video_target=tpm-plugin-feeeagab-lh9Alon3i44'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Converting+Polar+Equations+to+and+from+Rectangular+Equations_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cConverting Polar Equations to and from Rectangular Equations\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Plot points using polar coordinates.<\/li>\n<li>Convert between polar coordinates and rectangular coordinates.<\/li>\n<li>Transform equations between polar and rectangular forms.<\/li>\n<li>Identify and graph polar equations by converting to rectangular equations.<\/li>\n<\/ul>\n<\/section>\n<h2>Defining Polar Coordinates<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Imagine you&#8217;re giving directions to a friend. Instead of saying &#8220;go 3 blocks east and 4 blocks north,&#8221; you could say &#8220;walk 5 blocks in the northeast direction.&#8221; That&#8217;s essentially what polar coordinates do\u2014they describe where a point is using distance and direction rather than horizontal and vertical measurements.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>The core concept:<\/strong> Every point can be described using two pieces of information:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">[latex]r[\/latex] = how far you are from the origin (like &#8220;5 blocks away&#8221;)<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\theta[\/latex] = what angle you&#8217;re at from the positive x-axis (like &#8220;northeast direction&#8221;)<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\"><strong>The conversion formulas you need:<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\"><strong>Polar to Rectangular:<\/strong> [latex]x = r\\cos\\theta[\/latex] and [latex]y = r\\sin\\theta[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Rectangular to Polar:<\/strong> [latex]r^2 = x^2 + y^2[\/latex] and [latex]\\tan\\theta = \\frac{y}{x}[\/latex]<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\">Think of the right triangle formed by dropping a perpendicular from your point to the [latex]x[\/latex]-axis. The hypotenuse is [latex]r[\/latex], and basic trig (SohCahToa) gives you the relationships: cosine for the [latex]x[\/latex]-component, sine for the [latex]y[\/latex]-component.<\/p>\n<p>Watch out for the angle calculation. Don&#8217;t just use [latex]\\theta = \\tan^{-1}(y\/x)[\/latex] blindly! The inverse tangent function only gives angles in Quadrants I and IV. For points in Quadrants II or III, you need to add [latex]\\pi[\/latex] to get the correct angle.<\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<div id=\"fs-id1167793901809\" data-type=\"problem\">\n<p id=\"fs-id1167794066199\">Convert [latex]\\left(-8,-8\\right)[\/latex] into polar coordinates and [latex]\\left(4,\\frac{2\\pi }{3}\\right)[\/latex] into rectangular coordinates.<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558897\">Hint<\/button><\/p>\n<div id=\"q44558897\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1167794064210\" data-type=\"commentary\" data-element-type=\"hint\">\n<p id=\"fs-id1167793262229\">Use both equations from the theorem. Make sure to check the quadrant when calculating [latex]\\theta[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558898\">Show Solution<\/button><\/p>\n<div id=\"q44558898\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1167793805544\" data-type=\"solution\">\n<p id=\"fs-id1167793270874\" style=\"text-align: center;\">[latex]\\left(8\\sqrt{2},\\frac{5\\pi }{4}\\right)[\/latex] and [latex]\\left(-2,2\\sqrt{3}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-dbfhhfde-IbxuHmH6D2c\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/IbxuHmH6D2c?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-dbfhhfde-IbxuHmH6D2c\" 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=14661414&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-dbfhhfde-IbxuHmH6D2c&#38;vembed=0&#38;video_id=IbxuHmH6D2c&#38;video_target=tpm-plugin-dbfhhfde-IbxuHmH6D2c\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Converting+polar+coordinates+into+cartesian+coordinates_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cConverting polar coordinates into cartesian coordinates\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ghehbbgh-Qa5unxRmG7Y\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/Qa5unxRmG7Y?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-ghehbbgh-Qa5unxRmG7Y\" 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=14661415&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-ghehbbgh-Qa5unxRmG7Y&#38;vembed=0&#38;video_id=Qa5unxRmG7Y&#38;video_target=tpm-plugin-ghehbbgh-Qa5unxRmG7Y\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Converting+cartesian+coordinates+into+polar+coordinates_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cConverting cartesian coordinates into polar coordinates\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Plotting Points in the Polar Plane<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">In polar coordinates, the same point can be represented by infinitely many coordinate pairs. Unlike rectangular coordinates where every point has exactly one (x,y) pair, polar coordinates allow multiple representations for the same location.<\/p>\n<p class=\"whitespace-normal break-words\">This happens because:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">You can add or subtract full rotations ([latex]2\\pi[\/latex]) to any angle<\/li>\n<li class=\"whitespace-normal break-words\">You can use negative [latex]r[\/latex] values by pointing in the opposite direction<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\"><strong>Problem-Solving Strategy:<\/strong><\/p>\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\"><strong>Start with the angle<\/strong> [latex]\\theta[\/latex]: Rotate counterclockwise from the positive x-axis (polar axis) if positive, clockwise if negative<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Handle the distance<\/strong> [latex]r[\/latex]:\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">If [latex]r > 0[\/latex]: Move that distance along your angle&#8217;s direction<\/li>\n<li class=\"whitespace-normal break-words\">If [latex]r < 0[\/latex]: Move that distance in the <strong>opposite<\/strong> direction from your angle<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<p>When [latex]r[\/latex] is negative, you move in the direction opposite to your angle. For [latex](-3, \\frac{2\\pi}{3})[\/latex], you first rotate to angle [latex]\\frac{2\\pi}{3}[\/latex], then move 3 units in the opposite direction.<\/p>\n<p>The <strong>polar plane<\/strong> consists of concentric circles representing constant distances ([latex]r = 1, r = 2[\/latex], etc.) and straight lines radiating from the pole (origin) representing constant angles ([latex]\\theta = \\frac{\\pi}{4}, \\theta = \\frac{\\pi}{2}[\/latex], etc.).<\/p>\n<p>To check your work, convert your polar coordinates to rectangular using [latex]x = r\\cos\\theta[\/latex] and [latex]y = r\\sin\\theta[\/latex]. The point should land where you expect it in the xy-plane.<\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<div id=\"fs-id1167794292524\" data-type=\"problem\">\n<p id=\"fs-id1167794330185\">Plot [latex]\\left(4,\\frac{5\\pi }{3}\\right)[\/latex] and [latex]\\left(-3,-\\frac{7\\pi }{2}\\right)[\/latex] on the polar plane.<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558894\">Hint<\/button><\/p>\n<div id=\"q44558894\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1167794039158\" data-type=\"commentary\" data-element-type=\"hint\">\n<p id=\"fs-id1167794060098\">Start with [latex]\\theta[\/latex], then use [latex]r[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558895\">Show Solution<\/button><\/p>\n<div id=\"q44558895\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1167794039142\" data-type=\"solution\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11234815\/CNX_Calc_Figure_11_03_004.jpg\" alt=\"Two points are marked on a polar coordinate plane, specifically (\u22123, \u22127\u03c0\/2) on the y-axis and (4, 5\u03c0\/3) in the fourth quadrant.\" width=\"417\" height=\"417\" data-media-type=\"image\/jpeg\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-fdbabhee-oWNFXLjWouQ\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/oWNFXLjWouQ?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-fdbabhee-oWNFXLjWouQ\" 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=14661416&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-fdbabhee-oWNFXLjWouQ&#38;vembed=0&#38;video_id=oWNFXLjWouQ&#38;video_target=tpm-plugin-fdbabhee-oWNFXLjWouQ\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Plotting+polar+points_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cPlotting polar points\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h3>Transform Equations Between Polar and Rectangular Forms<\/h3>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Convert each polar equation to rectangular form.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">a) [latex]r = 5[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">b) [latex]\\theta = \\frac{\\pi}{3}[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">c) [latex]r = 4\\cos\\theta[\/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=\"q753370\">Show Solution<\/button><\/p>\n<div id=\"q753370\" class=\"hidden-answer\" style=\"display: none\">\n<p>a) [latex]r = 5[\/latex]<\/p>\n<p>Use the relationship [latex]r^2 = x^2 + y^2[\/latex]:<\/p>\n<p>[latex]\\begin{aligned} r &= 5 \\\\ r^2 &= 25 \\quad \\text{(square both sides)} \\\\ x^2 + y^2 &= 25 \\end{aligned}[\/latex]<\/p>\n<p>This is a circle centered at the origin with radius 5.<\/p>\n<p>b) [latex]\\theta = \\frac{\\pi}{3}[\/latex]<\/p>\n<p>Use the relationship [latex]\\tan\\theta = \\frac{y}{x}[\/latex]:<\/p>\n<p>[latex]\\begin{aligned} \\theta &= \\frac{\\pi}{3} \\\\ \\tan\\theta &= \\tan\\left(\\frac{\\pi}{3}\\right) \\\\ \\frac{y}{x} &= \\sqrt{3} \\\\ y &= \\sqrt{3}x \\quad \\text{(for } x \\geq 0\\text{)} \\end{aligned}[\/latex]<\/p>\n<p>This is a line through the origin with slope [latex]\\sqrt{3}[\/latex], but only the ray where [latex]x \\geq 0[\/latex].<\/p>\n<p>c) [latex]r = 4\\cos\\theta[\/latex]<\/p>\n<p>Use the substitution [latex]\\cos\\theta = \\frac{x}{r}[\/latex]:<\/p>\n<p>[latex]\\begin{aligned} r &= 4\\cos\\theta \\\\ r &= 4 \\cdot \\frac{x}{r} \\\\ r^2 &= 4x \\quad \\text{(multiply both sides by } r\\text{)} \\\\ x^2 + y^2 &= 4x \\quad \\text{(substitute } r^2 = x^2 + y^2\\text{)} \\end{aligned}[\/latex]<\/p>\n<p>This is a circle with center [latex](2, 0)[\/latex] and radius 2.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\"><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-feeeagab-lh9Alon3i44\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/lh9Alon3i44?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-feeeagab-lh9Alon3i44\" 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=14661417&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-feeeagab-lh9Alon3i44&#38;vembed=0&#38;video_id=lh9Alon3i44&#38;video_target=tpm-plugin-feeeagab-lh9Alon3i44\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Converting+Polar+Equations+to+and+from+Rectangular+Equations_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cConverting Polar Equations to and from Rectangular Equations\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n","protected":false},"author":67,"menu_order":10,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Converting polar coordinates into cartesian 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