{"id":3271,"date":"2025-08-15T23:55:45","date_gmt":"2025-08-15T23:55:45","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=3271"},"modified":"2025-10-20T18:39:38","modified_gmt":"2025-10-20T18:39:38","slug":"graphing-in-polar-coordinates-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/graphing-in-polar-coordinates-apply-it\/","title":{"raw":"Graphing in Polar Coordinates: Apply It","rendered":"Graphing in Polar Coordinates: Apply It"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Test polar equations for symmetry.<\/li>\r\n \t<li>Graph polar equations by plotting points.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<p class=\"whitespace-normal break-words\"><img class=\"alignright\" src=\"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/7\/77\/Spirograph_2018-11-30_to_3rd_CC0_Rev_Daniel_Elis_Axelrod_2021.png\/960px-Spirograph_2018-11-30_to_3rd_CC0_Rev_Daniel_Elis_Axelrod_2021.png?20211007163855\" alt=\"File:Spirograph 2018-11-30 to 3rd CC0 Rev Daniel Elis Axelrod 2021.png\" width=\"390\" height=\"308\" \/>Have you ever used a Spirograph toy to create intricate, mesmerizing designs? These mechanical drawing toys produce patterns that are actually polar curves! Artists, designers, and mathematicians use polar equations to create everything from decorative patterns to gear designs. Understanding how to graph polar equations allows us to predict and control these beautiful mathematical patterns.<\/p>\r\n\r\n<section class=\"textbox recall\" aria-label=\"Recall\">\r\n<p class=\"whitespace-normal break-words\">Different polar equations create distinct, recognizable shapes:<\/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-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Circles: [latex]r = a\\cos\\theta[\/latex] or [latex]r = a\\sin\\theta[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Cardioids: [latex]r = a \\pm b\\cos\\theta[\/latex] where [latex]a = b[\/latex] (heart-shaped)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Rose curves: [latex]r = a\\cos(n\\theta)[\/latex] or [latex]r = a\\sin(n\\theta)[\/latex] (petal patterns)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Lemniscates: [latex]r^2 = a^2\\cos(2\\theta)[\/latex] or [latex]r^2 = a^2\\sin(2\\theta)[\/latex] (figure-eight shapes)<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To:<\/strong> Graphing a Polar Equation by Plotting Points\r\n<ol>\r\n \t<li>Test for symmetry to reduce the number of points needed<\/li>\r\n \t<li>Find zeros by setting [latex]r = 0[\/latex] and solving for [latex]\\theta[\/latex]<\/li>\r\n \t<li>Find maximum values by determining when the trig function reaches its maximum<\/li>\r\n \t<li>Create a table of [latex]\\theta[\/latex] and [latex]r[\/latex] values<\/li>\r\n \t<li>Plot points and connect them smoothly, using symmetry to complete the graph<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox youChoose\" aria-label=\"You Choose\">[choosedataset divId=\"tnh-choose-dataset\" title=\"Choose Your Own Graph\" label=\"\" default=\"Choose a Graph\"]\r\n[datasetoption]\r\n[displayname]Flower[\/displayname]\r\n[ohmid]41514[\/ohmid]\r\n[\/datasetoption][datasetoption]\r\n[displayname]Snowflake[\/displayname]\r\n[ohmid]41515[\/ohmid]\r\n[\/datasetoption][datasetoption]\r\n[displayname]Butterfly[\/displayname]\r\n[ohmid]41516[\/ohmid]\r\n[\/datasetoption][\/choosedataset]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Test polar equations for symmetry.<\/li>\n<li>Graph polar equations by plotting points.<\/li>\n<\/ul>\n<\/section>\n<p class=\"whitespace-normal break-words\"><img loading=\"lazy\" decoding=\"async\" class=\"alignright\" src=\"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/7\/77\/Spirograph_2018-11-30_to_3rd_CC0_Rev_Daniel_Elis_Axelrod_2021.png\/960px-Spirograph_2018-11-30_to_3rd_CC0_Rev_Daniel_Elis_Axelrod_2021.png?20211007163855\" alt=\"File:Spirograph 2018-11-30 to 3rd CC0 Rev Daniel Elis Axelrod 2021.png\" width=\"390\" height=\"308\" \/>Have you ever used a Spirograph toy to create intricate, mesmerizing designs? These mechanical drawing toys produce patterns that are actually polar curves! Artists, designers, and mathematicians use polar equations to create everything from decorative patterns to gear designs. Understanding how to graph polar equations allows us to predict and control these beautiful mathematical patterns.<\/p>\n<section class=\"textbox recall\" aria-label=\"Recall\">\n<p class=\"whitespace-normal break-words\">Different polar equations create distinct, recognizable shapes:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Circles: [latex]r = a\\cos\\theta[\/latex] or [latex]r = a\\sin\\theta[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Cardioids: [latex]r = a \\pm b\\cos\\theta[\/latex] where [latex]a = b[\/latex] (heart-shaped)<\/li>\n<li class=\"whitespace-normal break-words\">Rose curves: [latex]r = a\\cos(n\\theta)[\/latex] or [latex]r = a\\sin(n\\theta)[\/latex] (petal patterns)<\/li>\n<li class=\"whitespace-normal break-words\">Lemniscates: [latex]r^2 = a^2\\cos(2\\theta)[\/latex] or [latex]r^2 = a^2\\sin(2\\theta)[\/latex] (figure-eight shapes)<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To:<\/strong> Graphing a Polar Equation by Plotting Points<\/p>\n<ol>\n<li>Test for symmetry to reduce the number of points needed<\/li>\n<li>Find zeros by setting [latex]r = 0[\/latex] and solving for [latex]\\theta[\/latex]<\/li>\n<li>Find maximum values by determining when the trig function reaches its maximum<\/li>\n<li>Create a table of [latex]\\theta[\/latex] and [latex]r[\/latex] values<\/li>\n<li>Plot points and connect them smoothly, using symmetry to complete the graph<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox youChoose\" aria-label=\"You Choose\">\n<div id=\"tnh-choose-dataset\" class=\"chooseDataset\">\n<h3>Choose Your Own Graph<\/h3>\n<form><select name=\"dataset\"><option value=\"\">Choose a Graph<\/option><option value=\"41514\">Flower<\/option><option value=\"41515\">Snowflake<\/option><option value=\"41516\">Butterfly<\/option><\/select><\/form>\n<div class=\"ohmContainer\"><\/div>\n<\/p><\/div>\n<\/section>\n","protected":false},"author":67,"menu_order":17,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":247,"module-header":"apply_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":[{"divId":"tnh-choose-dataset","title":"Choose Your Own Graph","label":"","default":"Choose a Graph","try_it_collection":[{"displayName":"Flower","value":"41514"},{"displayName":"Snowflake","value":"41515"},{"displayName":"Butterfly","value":"41516"}]}],"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3271"}],"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":6,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3271\/revisions"}],"predecessor-version":[{"id":4769,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3271\/revisions\/4769"}],"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\/3271\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=3271"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=3271"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=3271"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=3271"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}