{"id":251786,"date":"2025-07-10T20:01:26","date_gmt":"2025-07-10T20:01:26","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=251786"},"modified":"2025-07-10T20:01:29","modified_gmt":"2025-07-10T20:01:29","slug":"for-the-rectangular-coordinates","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/10\/for-the-rectangular-coordinates\/","title":{"rendered":"For the rectangular coordinates"},"content":{"rendered":"\n<p><br>For the rectangular coordinates ((5sqrt2)\/(2),-(5sqrt2)\/(2)), (a) plot the point and (b) give two pairs of polar coordinates for the point, where 0^(@) &lt;= theta &lt; 360^(@). (a) Choose the correct plot of ((5sqrt2)\/(2),-(5sqrt2)\/(2)) below. A. B. c. D. (b) Give two pairs of polar coordinates for the point, where 0^(@) &lt;= theta &lt; 360^(@). (Type an ordered pair. Simplify your answer, including any radicals. Use integers or fractions for any numbers in the text.<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>Let&#8217;s solve this step by step:<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">(a) Plotting the point (522,\u2212522)(\\frac{5\\sqrt{2}}{2}, -\\frac{5\\sqrt{2}}{2})(252\u200b\u200b,\u2212252\u200b\u200b) in rectangular coordinates:<\/h3>\n\n\n\n<p>The given point is in rectangular coordinates (x,y)=(522,\u2212522)(x, y) = \\left(\\frac{5\\sqrt{2}}{2}, -\\frac{5\\sqrt{2}}{2}\\right)(x,y)=(252\u200b\u200b,\u2212252\u200b\u200b). Here&#8217;s how we can plot it:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The xxx-coordinate is positive, and the yyy-coordinate is negative, meaning the point lies in the fourth quadrant of the coordinate plane.<\/li>\n\n\n\n<li>The magnitude of both coordinates is the same (52\/25\\sqrt{2}\/252\u200b\/2), so the point is equidistant from the xxx- and yyy-axes.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">(b) Converting to polar coordinates:<\/h3>\n\n\n\n<p>Polar coordinates (r,\u03b8)(r, \\theta)(r,\u03b8) are given by:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>r=x2+y2r = \\sqrt{x^2 + y^2}r=x2+y2\u200b, the radial distance from the origin.<\/li>\n\n\n\n<li>\u03b8=tan\u2061\u22121(yx)\\theta = \\tan^{-1}\\left(\\frac{y}{x}\\right)\u03b8=tan\u22121(xy\u200b), the angle with respect to the positive xxx-axis.<\/li>\n<\/ul>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Finding rrr:<\/strong><\/li>\n<\/ol>\n\n\n\n<p>r=(522)2+(\u2212522)2r = \\sqrt{\\left(\\frac{5\\sqrt{2}}{2}\\right)^2 + \\left(-\\frac{5\\sqrt{2}}{2}\\right)^2}r=(252\u200b\u200b)2+(\u2212252\u200b\u200b)2\u200b r=(52)24+(52)24=504+504=1004=25=5r = \\sqrt{\\frac{(5\\sqrt{2})^2}{4} + \\frac{(5\\sqrt{2})^2}{4}} = \\sqrt{\\frac{50}{4} + \\frac{50}{4}} = \\sqrt{\\frac{100}{4}} = \\sqrt{25} = 5r=4(52\u200b)2\u200b+4(52\u200b)2\u200b\u200b=450\u200b+450\u200b\u200b=4100\u200b\u200b=25\u200b=5<\/p>\n\n\n\n<p>So, r=5r = 5r=5.<\/p>\n\n\n\n<ol start=\"2\" class=\"wp-block-list\">\n<li><strong>Finding \u03b8\\theta\u03b8:<\/strong><\/li>\n<\/ol>\n\n\n\n<p>\u03b8=tan\u2061\u22121(yx)=tan\u2061\u22121(\u2212522522)=tan\u2061\u22121(\u22121)\\theta = \\tan^{-1}\\left(\\frac{y}{x}\\right) = \\tan^{-1}\\left(\\frac{-\\frac{5\\sqrt{2}}{2}}{\\frac{5\\sqrt{2}}{2}}\\right) = \\tan^{-1}(-1)\u03b8=tan\u22121(xy\u200b)=tan\u22121(252\u200b\u200b\u2212252\u200b\u200b\u200b)=tan\u22121(\u22121)<\/p>\n\n\n\n<p>Since the point is in the fourth quadrant, the angle corresponding to tan\u2061\u22121(\u22121)\\tan^{-1}(-1)tan\u22121(\u22121) is \u03b8=\u221245\u2218\\theta = -45^\\circ\u03b8=\u221245\u2218, or equivalently, \u03b8=315\u2218\\theta = 315^\\circ\u03b8=315\u2218 in the range 0\u2218\u2264\u03b8&lt;360\u22180^\\circ \\leq \\theta &lt; 360^\\circ0\u2218\u2264\u03b8&lt;360\u2218.<\/p>\n\n\n\n<p>Thus, the first pair of polar coordinates is (r,\u03b8)=(5,315\u2218)(r, \\theta) = (5, 315^\\circ)(r,\u03b8)=(5,315\u2218).<\/p>\n\n\n\n<ol start=\"3\" class=\"wp-block-list\">\n<li><strong>Second pair of polar coordinates:<\/strong><br>In polar coordinates, we can also represent the point by adding 180\u2218180^\\circ180\u2218 to the angle (this is the same point, but with the opposite direction for the angle):<\/li>\n<\/ol>\n\n\n\n<p>\u03b8\u2032=315\u2218+180\u2218=495\u2218\\theta&#8217; = 315^\\circ + 180^\\circ = 495^\\circ\u03b8\u2032=315\u2218+180\u2218=495\u2218<\/p>\n\n\n\n<p>However, we want \u03b8\\theta\u03b8 to be in the range 0\u2218\u2264\u03b8&lt;360\u22180^\\circ \\leq \\theta &lt; 360^\\circ0\u2218\u2264\u03b8&lt;360\u2218, so we subtract 360\u2218360^\\circ360\u2218 from 495\u2218495^\\circ495\u2218: 495\u2218\u2212360\u2218=135\u2218495^\\circ &#8211; 360^\\circ = 135^\\circ495\u2218\u2212360\u2218=135\u2218<\/p>\n\n\n\n<p>Thus, the second pair of polar coordinates is (r,\u03b8)=(5,135\u2218)(r, \\theta) = (5, 135^\\circ)(r,\u03b8)=(5,135\u2218).<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Final Answer:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>(a)<\/strong> The point (522,\u2212522)\\left(\\frac{5\\sqrt{2}}{2}, -\\frac{5\\sqrt{2}}{2}\\right)(252\u200b\u200b,\u2212252\u200b\u200b) lies in the fourth quadrant.<\/li>\n\n\n\n<li><strong>(b)<\/strong> Two pairs of polar coordinates for the point are: (5,315\u2218)and(5,135\u2218)(5, 315^\\circ) \\quad \\text{and} \\quad (5, 135^\\circ)(5,315\u2218)and(5,135\u2218)<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/07\/learnexams-banner6-178.jpeg\" alt=\"\" class=\"wp-image-251800\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>For the rectangular coordinates ((5sqrt2)\/(2),-(5sqrt2)\/(2)), (a) plot the point and (b) give two pairs of polar coordinates for the point, where 0^(@) &lt;= theta &lt; 360^(@). (a) Choose the correct plot of ((5sqrt2)\/(2),-(5sqrt2)\/(2)) below. A. B. c. D. (b) Give two pairs of polar coordinates for the point, where 0^(@) &lt;= theta &lt; 360^(@). (Type [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-251786","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/251786","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=251786"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/251786\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=251786"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=251786"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=251786"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}