{"id":1020,"date":"2025-06-20T17:29:07","date_gmt":"2025-06-20T17:29:07","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=1020"},"modified":"2025-09-29T13:02:03","modified_gmt":"2025-09-29T13:02:03","slug":"understanding-polar-coordinates-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/understanding-polar-coordinates-apply-it\/","title":{"raw":"Understanding Polar Coordinates: Apply It","rendered":"Understanding Polar Coordinates: Apply It"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Plot points using polar coordinates (<em>r,\u03b8<\/em>)<\/li>\r\n \t<li>Switch back and forth between polar and rectangular (<em>x,y<\/em>) coordinates<\/li>\r\n \t<li>Draw polar curves from their equations<\/li>\r\n \t<li>Identify when polar curves have symmetry<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Applications of Polar Coordinates: Natural Spirals<\/h2>\r\nThe chambered nautilus is a fascinating creature. This animal feeds on hermit crabs, fish, and other crustaceans. It has a hard outer shell with many chambers connected in a spiral fashion, and it can retract into its shell to avoid predators. When part of the shell is cut away, a perfect spiral is revealed, with chambers inside that are somewhat similar to growth rings in a tree.\r\n<figure id=\"CNX_Calc_Figure_11_00_001\" class=\"splash\"><figcaption><\/figcaption>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"300\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11234553\/CNX_Calc_Figure_11_00_001.jpg\" alt=\"A photo of a cross section of a seashell that spirals from big chambers to smaller and smaller ones.\" width=\"300\" height=\"286\" data-media-type=\"image\/jpeg\" \/> The chambered nautilus is a marine animal that lives in the tropical Pacific Ocean. Scientists think they have existed mostly unchanged for about 500 million years.(credit: modification of work by Jitze Couperus, Flickr)[\/caption]<\/figure>\r\n<div id=\"fs-id1167794047296\" data-type=\"problem\">\r\n<p id=\"fs-id1167794047301\">This creature displays a spiral when half the outer shell is cut away. It is possible to describe a spiral using rectangular coordinates. Figure 1 below shows a spiral in rectangular coordinates. How can we describe this curve mathematically?<\/p>\r\n\r\n<figure id=\"CNX_Calc_Figure_11_03_010\"><figcaption><\/figcaption>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"417\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11234834\/CNX_Calc_Figure_11_03_010.jpg\" alt=\"A spiral starting at the origin and continually increasing its radius to a point P(x, y).\" width=\"417\" height=\"422\" data-media-type=\"image\/jpeg\" \/> Figure 1. How can we describe a spiral graph mathematically?[\/caption]<\/figure>\r\n<\/div>\r\nAs the point [latex]P[\/latex] travels around the spiral in a counterclockwise direction, its distance <em data-effect=\"italics\">d<\/em> from the origin increases. Assume that the distance [latex]d[\/latex] is a constant multiple [latex]k[\/latex] of the angle [latex]\\theta [\/latex] that the line segment [latex]OP[\/latex] makes with the positive [latex]x[\/latex]-axis. Therefore [latex]d\\left(P,O\\right)=k\\theta [\/latex], where [latex]O[\/latex] is the origin. Now use the distance formula and some trigonometry:\r\n<div id=\"fs-id1167794332290\" data-type=\"solution\">\r\n<div id=\"fs-id1167794333799\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{ccc}\\hfill d\\left(P,O\\right)&amp; =\\hfill &amp; k\\theta \\hfill \\\\ \\hfill \\sqrt{{\\left(x - 0\\right)}^{2}+{\\left(y - 0\\right)}^{2}}&amp; =\\hfill &amp; k\\text{arctan}\\left(\\frac{y}{x}\\right)\\hfill \\\\ \\hfill \\sqrt{{x}^{2}+{y}^{2}}&amp; =\\hfill &amp; k\\text{arctan}\\left(\\frac{y}{x}\\right)\\hfill \\\\ \\hfill \\text{arctan}\\left(\\frac{y}{x}\\right)&amp; =\\hfill &amp; \\frac{\\sqrt{{x}^{2}+{y}^{2}}}{k}\\hfill \\\\ \\hfill y&amp; =\\hfill &amp; x\\tan\\left(\\frac{\\sqrt{{x}^{2}+{y}^{2}}}{k}\\right).\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1167794214981\">Although this equation describes the spiral, it is not possible to solve it directly for either [latex]x[\/latex] or [latex]y[\/latex]. However, if we use polar coordinates, the equation becomes much simpler. In particular, [latex]d\\left(P,O\\right)=r[\/latex], and [latex]\\theta [\/latex] is the second coordinate. Therefore the equation for the spiral becomes [latex]r=k\\theta [\/latex]. Note that when [latex]\\theta =0[\/latex] we also have [latex]r=0[\/latex], so the spiral emanates from the origin. We can remove this restriction by adding a constant to the equation. Then the equation for the spiral becomes [latex]r=a+k\\theta [\/latex] for arbitrary constants [latex]a[\/latex] and [latex]k[\/latex]. This is referred to as an <span class=\"no-emphasis\" data-type=\"term\">Archimedean spiral<\/span>, after the Greek mathematician Archimedes.<\/p>\r\n<p id=\"fs-id1167794175932\">Another type of spiral is the logarithmic spiral, described by the function [latex]r=a\\cdot {b}^{\\theta }[\/latex]. A graph of the function [latex]r=1.2\\left({1.25}^{\\theta }\\right)[\/latex] is given in Figure 2. This spiral describes the shell shape of the chambered nautilus.<\/p>\r\n\r\n<figure id=\"CNX_Calc_Figure_11_03_011\"><figcaption><\/figcaption>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"858\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11234837\/CNX_Calc_Figure_11_03_011.jpg\" alt=\"This figure has two figures. The first is a shell with many chambers that increase in size from the center out. The second is a spiral with equation r = 1.2(1.25\u03b8).\" width=\"858\" height=\"422\" data-media-type=\"image\/jpeg\" \/> Figure 2. A logarithmic spiral is similar to the shape of the chambered nautilus shell. (credit: modification of work by Jitze Couperus, Flickr)[\/caption]<\/figure>\r\n<\/div>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Plot points using polar coordinates (<em>r,\u03b8<\/em>)<\/li>\n<li>Switch back and forth between polar and rectangular (<em>x,y<\/em>) coordinates<\/li>\n<li>Draw polar curves from their equations<\/li>\n<li>Identify when polar curves have symmetry<\/li>\n<\/ul>\n<\/section>\n<h2>Applications of Polar Coordinates: Natural Spirals<\/h2>\n<p>The chambered nautilus is a fascinating creature. This animal feeds on hermit crabs, fish, and other crustaceans. It has a hard outer shell with many chambers connected in a spiral fashion, and it can retract into its shell to avoid predators. When part of the shell is cut away, a perfect spiral is revealed, with chambers inside that are somewhat similar to growth rings in a tree.<\/p>\n<figure id=\"CNX_Calc_Figure_11_00_001\" class=\"splash\"><figcaption><\/figcaption><figure style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11234553\/CNX_Calc_Figure_11_00_001.jpg\" alt=\"A photo of a cross section of a seashell that spirals from big chambers to smaller and smaller ones.\" width=\"300\" height=\"286\" data-media-type=\"image\/jpeg\" \/><figcaption class=\"wp-caption-text\">The chambered nautilus is a marine animal that lives in the tropical Pacific Ocean. Scientists think they have existed mostly unchanged for about 500 million years.(credit: modification of work by Jitze Couperus, Flickr)<\/figcaption><\/figure>\n<\/figure>\n<div id=\"fs-id1167794047296\" data-type=\"problem\">\n<p id=\"fs-id1167794047301\">This creature displays a spiral when half the outer shell is cut away. It is possible to describe a spiral using rectangular coordinates. Figure 1 below shows a spiral in rectangular coordinates. How can we describe this curve mathematically?<\/p>\n<figure id=\"CNX_Calc_Figure_11_03_010\"><figcaption><\/figcaption><figure style=\"width: 417px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11234834\/CNX_Calc_Figure_11_03_010.jpg\" alt=\"A spiral starting at the origin and continually increasing its radius to a point P(x, y).\" width=\"417\" height=\"422\" data-media-type=\"image\/jpeg\" \/><figcaption class=\"wp-caption-text\">Figure 1. How can we describe a spiral graph mathematically?<\/figcaption><\/figure>\n<\/figure>\n<\/div>\n<p>As the point [latex]P[\/latex] travels around the spiral in a counterclockwise direction, its distance <em data-effect=\"italics\">d<\/em> from the origin increases. Assume that the distance [latex]d[\/latex] is a constant multiple [latex]k[\/latex] of the angle [latex]\\theta[\/latex] that the line segment [latex]OP[\/latex] makes with the positive [latex]x[\/latex]-axis. Therefore [latex]d\\left(P,O\\right)=k\\theta[\/latex], where [latex]O[\/latex] is the origin. Now use the distance formula and some trigonometry:<\/p>\n<div id=\"fs-id1167794332290\" data-type=\"solution\">\n<div id=\"fs-id1167794333799\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{ccc}\\hfill d\\left(P,O\\right)& =\\hfill & k\\theta \\hfill \\\\ \\hfill \\sqrt{{\\left(x - 0\\right)}^{2}+{\\left(y - 0\\right)}^{2}}& =\\hfill & k\\text{arctan}\\left(\\frac{y}{x}\\right)\\hfill \\\\ \\hfill \\sqrt{{x}^{2}+{y}^{2}}& =\\hfill & k\\text{arctan}\\left(\\frac{y}{x}\\right)\\hfill \\\\ \\hfill \\text{arctan}\\left(\\frac{y}{x}\\right)& =\\hfill & \\frac{\\sqrt{{x}^{2}+{y}^{2}}}{k}\\hfill \\\\ \\hfill y& =\\hfill & x\\tan\\left(\\frac{\\sqrt{{x}^{2}+{y}^{2}}}{k}\\right).\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1167794214981\">Although this equation describes the spiral, it is not possible to solve it directly for either [latex]x[\/latex] or [latex]y[\/latex]. However, if we use polar coordinates, the equation becomes much simpler. In particular, [latex]d\\left(P,O\\right)=r[\/latex], and [latex]\\theta[\/latex] is the second coordinate. Therefore the equation for the spiral becomes [latex]r=k\\theta[\/latex]. Note that when [latex]\\theta =0[\/latex] we also have [latex]r=0[\/latex], so the spiral emanates from the origin. We can remove this restriction by adding a constant to the equation. Then the equation for the spiral becomes [latex]r=a+k\\theta[\/latex] for arbitrary constants [latex]a[\/latex] and [latex]k[\/latex]. This is referred to as an <span class=\"no-emphasis\" data-type=\"term\">Archimedean spiral<\/span>, after the Greek mathematician Archimedes.<\/p>\n<p id=\"fs-id1167794175932\">Another type of spiral is the logarithmic spiral, described by the function [latex]r=a\\cdot {b}^{\\theta }[\/latex]. A graph of the function [latex]r=1.2\\left({1.25}^{\\theta }\\right)[\/latex] is given in Figure 2. This spiral describes the shell shape of the chambered nautilus.<\/p>\n<figure id=\"CNX_Calc_Figure_11_03_011\"><figcaption><\/figcaption><figure style=\"width: 858px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11234837\/CNX_Calc_Figure_11_03_011.jpg\" alt=\"This figure has two figures. The first is a shell with many chambers that increase in size from the center out. The second is a spiral with equation r = 1.2(1.25\u03b8).\" width=\"858\" height=\"422\" data-media-type=\"image\/jpeg\" \/><figcaption class=\"wp-caption-text\">Figure 2. A logarithmic spiral is similar to the shape of the chambered nautilus shell. (credit: modification of work by Jitze Couperus, Flickr)<\/figcaption><\/figure>\n<\/figure>\n<\/div>\n","protected":false},"author":15,"menu_order":9,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":677,"module-header":"- Select Header -","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1020"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":9,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1020\/revisions"}],"predecessor-version":[{"id":2397,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1020\/revisions\/2397"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/677"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1020\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=1020"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=1020"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=1020"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=1020"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}