{"id":717,"date":"2021-05-10T19:10:49","date_gmt":"2021-05-10T19:10:49","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=717"},"modified":"2021-11-17T23:48:52","modified_gmt":"2021-11-17T23:48:52","slug":"summary-of-polar-coordinates","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/summary-of-polar-coordinates\/","title":{"raw":"Summary of Polar Coordinates","rendered":"Summary of Polar Coordinates"},"content":{"raw":"<section id=\"fs-id1167794324566\" class=\"key-concepts\" data-depth=\"1\">\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul id=\"fs-id1167794324573\" data-bullet-style=\"bullet\">\r\n \t<li>The polar coordinate system provides an alternative way to locate points in the plane.<\/li>\r\n \t<li>Convert points between rectangular and polar coordinates using the formulas<span data-type=\"newline\">\r\n<\/span>\r\n<div id=\"fs-id1167794324587\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">[latex]x=r\\cos\\theta \\text{ and }y=r\\sin\\theta [\/latex]\r\nand<span data-type=\"newline\">\r\n<\/span><\/div>\r\n<div id=\"fs-id1167794324627\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">[latex]r^2={x}^{2}+{y}^{2}\\text{ and }\\tan\\theta =\\frac{y}{x}[\/latex].<\/div><\/li>\r\n \t<li>To sketch a polar curve from a given polar function, make a table of values and take advantage of periodic properties.<\/li>\r\n \t<li>Use the conversion formulas to convert equations between rectangular and polar coordinates.<\/li>\r\n \t<li>Identify symmetry in polar curves, which can occur through the pole, the horizontal axis, or the vertical axis.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section>\r\n<div data-type=\"glossary\">\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1167794188366\">\r\n \t<dt>angular coordinate<\/dt>\r\n \t<dd id=\"fs-id1167794188372\">[latex]\\theta [\/latex] the angle formed by a line segment connecting the origin to a point in the polar coordinate system with the positive radial (<em data-effect=\"italics\">x<\/em>) axis, measured counterclockwise<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167794188386\">\r\n \t<dt>cardioid<\/dt>\r\n \t<dd id=\"fs-id1167794188392\">a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius; the equation of a cardioid is [latex]r=a\\left(1+\\sin\\theta \\right)[\/latex] or [latex]r=a\\left(1+\\cos\\theta \\right)[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167794188445\">\r\n \t<dt>lima\u00e7on<\/dt>\r\n \t<dd id=\"fs-id1167794188451\">the graph of the equation [latex]r=a+b\\sin\\theta [\/latex] or [latex]r=a+b\\cos\\theta [\/latex]. If [latex]a=b[\/latex] then the graph is a cardioid<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167794188504\">\r\n \t<dt>polar axis<\/dt>\r\n \t<dd id=\"fs-id1167794188509\">the horizontal axis in the polar coordinate system corresponding to [latex]r\\ge 0[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167794188522\">\r\n \t<dt>polar coordinate system<\/dt>\r\n \t<dd id=\"fs-id1167794188528\">a system for locating points in the plane. The coordinates are [latex]r[\/latex], the radial coordinate, and [latex]\\theta [\/latex], the angular coordinate<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167794188547\">\r\n \t<dt>polar equation<\/dt>\r\n \t<dd id=\"fs-id1167794188553\">an equation or function relating the radial coordinate to the angular coordinate in the polar coordinate system<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167794188558\">\r\n \t<dt>pole<\/dt>\r\n \t<dd id=\"fs-id1167794188563\">the central point of the polar coordinate system, equivalent to the origin of a Cartesian system<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167794188569\">\r\n \t<dt>radial coordinate<\/dt>\r\n \t<dd id=\"fs-id1167794188574\">[latex]r[\/latex] the coordinate in the polar coordinate system that measures the distance from a point in the plane to the pole<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167794188583\">\r\n \t<dt>rose<\/dt>\r\n \t<dd id=\"fs-id1167794188588\">graph of the polar equation [latex]r=a\\cos{n}\\theta [\/latex] or [latex]r=a\\sin{n}\\theta [\/latex] for a positive constant [latex]a[\/latex] and an integer [latex]n \\ge 2[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167794188629\">\r\n \t<dt>space-filling curve<\/dt>\r\n \t<dd id=\"fs-id1167794188635\">a curve that completely occupies a two-dimensional subset of the real plane<\/dd>\r\n<\/dl>\r\n<\/div>","rendered":"<section id=\"fs-id1167794324566\" class=\"key-concepts\" data-depth=\"1\">\n<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul id=\"fs-id1167794324573\" data-bullet-style=\"bullet\">\n<li>The polar coordinate system provides an alternative way to locate points in the plane.<\/li>\n<li>Convert points between rectangular and polar coordinates using the formulas<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<div id=\"fs-id1167794324587\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">[latex]x=r\\cos\\theta \\text{ and }y=r\\sin\\theta[\/latex]<br \/>\nand<span data-type=\"newline\"><br \/>\n<\/span><\/div>\n<div id=\"fs-id1167794324627\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">[latex]r^2={x}^{2}+{y}^{2}\\text{ and }\\tan\\theta =\\frac{y}{x}[\/latex].<\/div>\n<\/li>\n<li>To sketch a polar curve from a given polar function, make a table of values and take advantage of periodic properties.<\/li>\n<li>Use the conversion formulas to convert equations between rectangular and polar coordinates.<\/li>\n<li>Identify symmetry in polar curves, which can occur through the pole, the horizontal axis, or the vertical axis.<\/li>\n<\/ul>\n<\/div>\n<\/section>\n<div data-type=\"glossary\">\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1167794188366\">\n<dt>angular coordinate<\/dt>\n<dd id=\"fs-id1167794188372\">[latex]\\theta[\/latex] the angle formed by a line segment connecting the origin to a point in the polar coordinate system with the positive radial (<em data-effect=\"italics\">x<\/em>) axis, measured counterclockwise<\/dd>\n<\/dl>\n<dl id=\"fs-id1167794188386\">\n<dt>cardioid<\/dt>\n<dd id=\"fs-id1167794188392\">a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius; the equation of a cardioid is [latex]r=a\\left(1+\\sin\\theta \\right)[\/latex] or [latex]r=a\\left(1+\\cos\\theta \\right)[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1167794188445\">\n<dt>lima\u00e7on<\/dt>\n<dd id=\"fs-id1167794188451\">the graph of the equation [latex]r=a+b\\sin\\theta[\/latex] or [latex]r=a+b\\cos\\theta[\/latex]. If [latex]a=b[\/latex] then the graph is a cardioid<\/dd>\n<\/dl>\n<dl id=\"fs-id1167794188504\">\n<dt>polar axis<\/dt>\n<dd id=\"fs-id1167794188509\">the horizontal axis in the polar coordinate system corresponding to [latex]r\\ge 0[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1167794188522\">\n<dt>polar coordinate system<\/dt>\n<dd id=\"fs-id1167794188528\">a system for locating points in the plane. The coordinates are [latex]r[\/latex], the radial coordinate, and [latex]\\theta[\/latex], the angular coordinate<\/dd>\n<\/dl>\n<dl id=\"fs-id1167794188547\">\n<dt>polar equation<\/dt>\n<dd id=\"fs-id1167794188553\">an equation or function relating the radial coordinate to the angular coordinate in the polar coordinate system<\/dd>\n<\/dl>\n<dl id=\"fs-id1167794188558\">\n<dt>pole<\/dt>\n<dd id=\"fs-id1167794188563\">the central point of the polar coordinate system, equivalent to the origin of a Cartesian system<\/dd>\n<\/dl>\n<dl id=\"fs-id1167794188569\">\n<dt>radial coordinate<\/dt>\n<dd id=\"fs-id1167794188574\">[latex]r[\/latex] the coordinate in the polar coordinate system that measures the distance from a point in the plane to the pole<\/dd>\n<\/dl>\n<dl id=\"fs-id1167794188583\">\n<dt>rose<\/dt>\n<dd id=\"fs-id1167794188588\">graph of the polar equation [latex]r=a\\cos{n}\\theta[\/latex] or [latex]r=a\\sin{n}\\theta[\/latex] for a positive constant [latex]a[\/latex] and an integer [latex]n \\ge 2[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1167794188629\">\n<dt>space-filling curve<\/dt>\n<dd id=\"fs-id1167794188635\">a curve that completely occupies a two-dimensional subset of the real plane<\/dd>\n<\/dl>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-717\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 2. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction\">https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":416434,"menu_order":13,"template":"","meta":{"_candela_citation":"{\"1\":{\"type\":\"cc\",\"description\":\"Calculus Volume 2\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction\"}}","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-717","chapter","type-chapter","status-publish","hentry"],"part":162,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/717","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/416434"}],"version-history":[{"count":9,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/717\/revisions"}],"predecessor-version":[{"id":1894,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/717\/revisions\/1894"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/162"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/717\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=717"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=717"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=717"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=717"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}