{"id":9944,"date":"2026-07-03T17:39:42","date_gmt":"2026-07-03T17:39:42","guid":{"rendered":"https:\/\/kapdec.com\/help\/?p=9944"},"modified":"2026-07-03T17:39:42","modified_gmt":"2026-07-03T17:39:42","slug":"linking-angles-radii-and-polar-graphs","status":"publish","type":"post","link":"https:\/\/kapdec.com\/help\/linking-angles-radii-and-polar-graphs\/","title":{"rendered":"Linking Angles, Radii, And Polar Graphs"},"content":{"rendered":"<div class=\"article-watermark-wrapper\">\n<div style=\"position: relative; z-index: 1;\">\n<p style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 9pt; color: #444444;\">KAPDEC&reg; | Elite STEM Learning Platform | <a href=\"https:\/\/kapdec.com\" target=\"_blank\" rel=\"noopener noreferrer\" style=\"color: #444444; text-decoration: underline;\">https:\/\/kapdec.com<\/a><\/p>\n<hr \/>\n<h2><strong>Unit: <\/strong><strong>Trigonometric &amp; Polar Functions<\/strong><\/h2>\n<h3><strong>Chapter: <\/strong><strong>Linking Angles, Radii, and Polar Graphs<\/strong><\/h3>\n<p><em>Reference: &#8211; Introduction, Calculus with polar functions, Polar graphing technology, Graphing Techniques, Polar functions, Symmetry, Plotting Points &amp; Axes, Polar &amp; Cartesian Coordinates, Conversion between polar &amp; Cartesian Coordinates, Angle &amp; Radius uses in AP Calculus.<\/em><\/p>\n<p><strong>After studying this chapter, you should be able to:<\/strong><\/p>\n<ul>\n<li>Introduction to Calculus with Polar Functions<\/li>\n<li>Graphing techniques, Symmetry &amp; Plotting Points<\/li>\n<li>Polar &amp; Cartesian Coordinates, Uses of Angle &amp; Radius<\/li>\n<\/ul>\n<p><strong><u>Introduction to Calculus with Polar Functions<\/u><\/strong><strong> <\/strong><\/p>\n<p>The word \u2018trigonometry\u2019 is derived from the Greek words \u2018trigon\u2019 and \u2018metron\u2019 and it means \u2018measuring the sides of a triangle\u2019. The subject was originally developed to solve geometric problems involving triangles. It was studied by sea captains for navigation, surveyor to map out the new lands, by engineers and others. Currently, trigonometry is used in many areas such as the science of seismology, designing electric circuits, describing the state of an atom, predicting the heights of tides in the ocean, analysing a musical tone and in many other areas.<\/p>\n<p>In earlier classes, we have studied the trigonometric ratios of acute angles as the ratio of the sides of a right angled triangle. We have also studied the trigonometric identities and application of trigonometric ratios in solving the problems related to heights and distances.<\/p>\n<p><strong>Angles<\/strong><\/p>\n<p>Angle is a measure of rotation of a given ray about its initial point. The original ray is called the initial side and the final position of the ray after rotation is called the terminal side of the angle. The point of rotation is called the vertex. If the direction of rotation is anticlockwise, the angle is said to be positive and if the direction of rotation is clockwise, then the angle is negative.<\/p>\n<p><div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img decoding=\"async\" alt=\"\" src=\"file:\/\/\/C:\/Users\/vinit\/AppData\/Local\/Temp\/msohtmlclip1\/01\/clip_image002.png\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p> The definition of an angle suggests a unit, viz. one complete revolution from the position of the initial side as indicated below:<\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img decoding=\"async\" alt=\"\" src=\"file:\/\/\/C:\/Users\/vinit\/AppData\/Local\/Temp\/msohtmlclip1\/01\/clip_image003.png\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p><strong>Degree measure:<\/strong><\/p>\n<p>The measure of an angle is determined by the amount of rotation from the initial side to the terminal side.\u00a0 One way to measure an angle is in terms of\u00a0<strong>degrees<\/strong>.\u00a0 A measure of one degree (\u00a01\u00b0\u00a0) is equivalent to a rotation of\u00a0<em>1<\/em><em>360<\/em><\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img decoding=\"async\" alt=\"\" src=\"file:\/\/\/C:\/Users\/vinit\/AppData\/Local\/Temp\/msohtmlclip1\/01\/clip_image004.png\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p> \u00a0of a complete revolution.<\/p>\n<p>To measure angles, it is convenient to mark degrees on the\u00a0<a href=\"https:\/\/www.varsitytutors.com\/hotmath\/hotmath_help\/topics\/circumference.html\" target=\"_blank\" rel=\"noopener\">circumference\u00a0<\/a>of a\u00a0<a href=\"https:\/\/www.varsitytutors.com\/hotmath\/hotmath_help\/topics\/circle.html\" target=\"_blank\" rel=\"noopener\">circle\u00a0<\/a>.\u00a0 Thus, a complete revolution is\u00a0360\u00b0, half a revolution is\u00a0180\u00b0, a quarter of a revolution is\u00a090\u00b0\u00a0and so forth.<\/p>\n<p>If a rotation from the initial side to terminal side is <em>1<\/em><em>360<\/em><em>th<\/em><\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img decoding=\"async\" alt=\"\" src=\"file:\/\/\/C:\/Users\/vinit\/AppData\/Local\/Temp\/msohtmlclip1\/01\/clip_image006.png\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p> \u00a0a revolution,<br \/>\nthe angle is said to have a measure of one degree, written as 1\u00b0.<br \/>\nA degree is divided into 60 minutes, and a minute is divided into<br \/>\n60 seconds. One sixtieth of a degree is called a minute, written as 1\u2019, and one sixtieth of a minute is called a second, written as 1\u201d.<\/p>\n<p>Thus, \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 1\u00b0 = 60\u2019, 1\u2019 = 60\u201d<\/p>\n<p>Some of the angles whose measures are 360\u00b0, 180\u00b0, 270\u00b0, 420\u00b0, \u2013 30\u00b0,<br \/>\n\u2013 420\u00b0 are shown below:<\/p>\n<p><strong>Radian Measure: <\/strong><\/p>\n<p>There is another unit for measurement of an angle, called the radian measure. Angle subtended at the centre by an arc of length 1 unit<br \/>\nin a unit circle (circle of radius 1 unit) is said to have a measure of<br \/>\n1 radian. In the figures given below, OA is the initial side and OB is<br \/>\nthe terminal side. The figures show the angles whose measures are<br \/>\n1 radian, \u20131 radian, <em>1<\/em><em>1<\/em><em>2<\/em><\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img decoding=\"async\" alt=\"\" src=\"file:\/\/\/C:\/Users\/vinit\/AppData\/Local\/Temp\/msohtmlclip1\/01\/clip_image009.png\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p> radian and \u2013<em>1<\/em><em>1<\/em><em>2<\/em><\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img decoding=\"async\" alt=\"\" src=\"file:\/\/\/C:\/Users\/vinit\/AppData\/Local\/Temp\/msohtmlclip1\/01\/clip_image009.png\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p> \u00a0radian.<\/p>\n<p><div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img decoding=\"async\" alt=\"\" src=\"file:\/\/\/C:\/Users\/vinit\/AppData\/Local\/Temp\/msohtmlclip1\/01\/clip_image010.png\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p>We know that the circumference of a circle of radius 1 unit is 2p. Thus, one complete revolution of the initial side subtends an angle of 2pradian.<\/p>\n<p><strong>Note: <\/strong>If in a circle of radius r, an arc of length <em>l <\/em>subtends an angle qradian at the centre, we have<em> \u03b8=<\/em><em>l<\/em><em>r<\/em><em>or l=r \u03b8<\/em><\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img decoding=\"async\" alt=\"\" src=\"file:\/\/\/C:\/Users\/vinit\/AppData\/Local\/Temp\/msohtmlclip1\/01\/clip_image011.png\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p> .<\/p>\n<p><strong>Relation between Radian and Real Numbers:<\/strong><\/p>\n<p>Consider the unit circle with centre O. Let A be any point on the circle. Consider OA as initial side of an angle. Then the length of an arc of the circle will give the radian measure of the angle which the arc will subtend at the centre of the circle. Consider the line PAQ which is tangent to the circle at A. Let the point A represent the real number zero, AP represents positive real number and AQ represents negative real numbers. If we rope the line AP in the anticlockwise direction along the circle, and AQ in the clockwise direction, then every real number will correspond to a radian measure and conversely. Thus, radian measures and real numbers can be considered as one and the same.<\/p>\n<p><div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img decoding=\"async\" alt=\"\" src=\"file:\/\/\/C:\/Users\/vinit\/AppData\/Local\/Temp\/msohtmlclip1\/01\/clip_image012.png\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p><strong>Relation between Degree and Radian:<\/strong><\/p>\n<p>The size of a radian is determined by the requirement that there are 2<\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img decoding=\"async\" alt=\"\" src=\"file:\/\/\/C:\/Users\/vinit\/AppData\/Local\/Temp\/msohtmlclip1\/01\/clip_image013.gif\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p> \u00a0radians in a circle. Thus 2<\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img decoding=\"async\" alt=\"\" src=\"file:\/\/\/C:\/Users\/vinit\/AppData\/Local\/Temp\/msohtmlclip1\/01\/clip_image013.gif\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p> \u00a0radians equals 360 degrees. This means that 1 radian = 180\/<\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img decoding=\"async\" alt=\"\" src=\"file:\/\/\/C:\/Users\/vinit\/AppData\/Local\/Temp\/msohtmlclip1\/01\/clip_image013.gif\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p> \u00a0degrees, and 1 degree =\u00a0<\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img decoding=\"async\" alt=\"\" src=\"file:\/\/\/C:\/Users\/vinit\/AppData\/Local\/Temp\/msohtmlclip1\/01\/clip_image013.gif\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p> \/180 radians.<\/p>\n<p>1 radian = <em>180<\/em><em>0<\/em><em>\u03c0<\/em><\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img decoding=\"async\" alt=\"\" src=\"file:\/\/\/C:\/Users\/vinit\/AppData\/Local\/Temp\/msohtmlclip1\/01\/clip_image014.png\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p> = 57\u00b0 16\u2019 approximately.<\/p>\n<p>Also \u00a0 1\u00b0 = <em>\u03c0<\/em><em>180<\/em><\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img decoding=\"async\" alt=\"\" src=\"file:\/\/\/C:\/Users\/vinit\/AppData\/Local\/Temp\/msohtmlclip1\/01\/clip_image015.png\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p> \u00a0radian = 0.01746 radian approximately.<strong> <\/strong><\/p>\n<p><strong>Notational Convention<\/strong><\/p>\n<p>Since angles are measured either in degrees or in radians, we adopt the convention that whenever we write angle q\u00b0, we mean the angle whose degree measure is qand whenever we write angle b, we mean the angle whose radian measure is b.<\/p>\n<p>Note that when an angle is expressed in radians, the word \u2018radian\u2019 is frequently omitted. Thus, p = 180<sup>0<\/sup> and <em>\u03c0<\/em><em>4<\/em><\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img decoding=\"async\" alt=\"\" src=\"file:\/\/\/C:\/Users\/vinit\/AppData\/Local\/Temp\/msohtmlclip1\/01\/clip_image017.png\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p> \u00a0= 45\u00b0 are written with the understanding that pand <em>\u03c0<\/em><em>4<\/em><\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img decoding=\"async\" alt=\"\" src=\"file:\/\/\/C:\/Users\/vinit\/AppData\/Local\/Temp\/msohtmlclip1\/01\/clip_image017.png\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p> \u00a0are radian measures. Thus, we can say that<\/p>\n<p>Radian measure = <em>\u03c0<\/em><em>180<\/em><\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img decoding=\"async\" alt=\"\" src=\"file:\/\/\/C:\/Users\/vinit\/AppData\/Local\/Temp\/msohtmlclip1\/01\/clip_image015.png\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p> \u00d7 Degree measure<\/p>\n<p>Degree measure = <em>180<\/em><em>\u03c0<\/em><\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img decoding=\"async\" alt=\"\" src=\"file:\/\/\/C:\/Users\/vinit\/AppData\/Local\/Temp\/msohtmlclip1\/01\/clip_image018.png\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p> \u00d7 Radian measure<\/p>\n<p><strong>Example: <\/strong>Convert 30 degrees angle to radians.<\/p>\n<p><strong>Solution: <\/strong>We know 180\u00b0 = pradian.<\/p>\n<p>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 30\u00b0= <em>\u03c0<\/em><em>180<\/em><em>\u00d7<\/em><em>30<\/em><\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img decoding=\"async\" alt=\"\" src=\"file:\/\/\/C:\/Users\/vinit\/AppData\/Local\/Temp\/msohtmlclip1\/01\/clip_image019.png\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p> \u00a0radian = <em>\u03c0<\/em><em>6<\/em><\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img decoding=\"async\" alt=\"\" src=\"file:\/\/\/C:\/Users\/vinit\/AppData\/Local\/Temp\/msohtmlclip1\/01\/clip_image020.png\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p> \u00a0radian.<\/p>\n<p><strong>Example: <\/strong>Convert 520\u00b0 degrees angle to radians.<\/p>\n<p><strong>Solution: <\/strong>We know that 180\u00b0 = \u03c0 radian<\/p>\n<p>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 520<sup>0 = <\/sup><em>\u03c0<\/em><em>180<\/em><em>\u00d7<\/em><em>520<\/em><\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img decoding=\"async\" alt=\"\" src=\"file:\/\/\/C:\/Users\/vinit\/AppData\/Local\/Temp\/msohtmlclip1\/01\/clip_image021.png\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p> \u00a0radian = <em>26\u03c0<\/em><em>9<\/em><\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img decoding=\"async\" alt=\"\" src=\"file:\/\/\/C:\/Users\/vinit\/AppData\/Local\/Temp\/msohtmlclip1\/01\/clip_image022.png\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p> \u00a0radian<\/p>\n<p><strong>Example: <\/strong>Convert 6 radians into degree measure.<\/p>\n<p><strong>Solution: <\/strong>We know that pradian = 180\u00b0.<\/p>\n<p>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 6 radians = <em>180<\/em><em>\u03c0<\/em><em>\u00d76<\/em><\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img decoding=\"async\" alt=\"\" src=\"file:\/\/\/C:\/Users\/vinit\/AppData\/Local\/Temp\/msohtmlclip1\/01\/clip_image023.png\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p> \u00a0degree = <em>1080\u00d77<\/em><em>22<\/em><\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img decoding=\"async\" alt=\"\" src=\"file:\/\/\/C:\/Users\/vinit\/AppData\/Local\/Temp\/msohtmlclip1\/01\/clip_image024.png\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p> \u00a0degree<\/p>\n<p>\u00a0\u00a0\u00a0 = 343<em>7<\/em><em>11<\/em><\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img decoding=\"async\" alt=\"\" src=\"file:\/\/\/C:\/Users\/vinit\/AppData\/Local\/Temp\/msohtmlclip1\/01\/clip_image025.png\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p> \u00a0degree = 343\u00b0 + <em>7\u00d760<\/em><em>11<\/em><\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img decoding=\"async\" alt=\"\" src=\"file:\/\/\/C:\/Users\/vinit\/AppData\/Local\/Temp\/msohtmlclip1\/01\/clip_image026.png\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p> \u00a0minute [as 1\u00b0 = 60\u2019]<\/p>\n<p>\u00a0\u00a0\u00a0 = 343\u00b0 + 38\u2019 + <em>2<\/em><em>11<\/em><\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img decoding=\"async\" alt=\"\" src=\"file:\/\/\/C:\/Users\/vinit\/AppData\/Local\/Temp\/msohtmlclip1\/01\/clip_image027.png\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p> \u00a0minute\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 [as 1\u2019 = 60\u2019\u2019]<\/p>\n<p>\u00a0\u00a0\u00a0 = 343\u00b0 + 38\u2019 + 10.9\u201d= 343\u00b038\u2019 11\u201dapproximately.<\/p>\n<p>Hence, 6 radians = 343\u00b0 38\u2019 11\u201d approximately.<\/p>\n<p><strong>Example: <\/strong>Convert <em>11<\/em><em>16<\/em><\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img decoding=\"async\" alt=\"\" src=\"file:\/\/\/C:\/Users\/vinit\/AppData\/Local\/Temp\/msohtmlclip1\/01\/clip_image028.png\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p> \u00a0radians into degree measure.<\/p>\n<p><strong>Solution: <\/strong>We know that \u03c0 radian = 180\u00b0<\/p>\n<p><div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img decoding=\"async\" alt=\"\" src=\"file:\/\/\/C:\/Users\/vinit\/AppData\/Local\/Temp\/msohtmlclip1\/01\/clip_image029.gif\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p><strong>Example: <\/strong>Find the radius of the circle in which a central angle of<br \/>\n60\u00b0 intercepts an arc of length 37.4 cm (use p = <em>22<\/em><em>7<\/em><\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img decoding=\"async\" alt=\"\" src=\"file:\/\/\/C:\/Users\/vinit\/AppData\/Local\/Temp\/msohtmlclip1\/01\/clip_image030.png\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p> ).<\/p>\n<p><strong>Solution: <\/strong>Here <em>l<\/em>= 37.4 cm and q= 60\u00b0 = <em>60\u03c0<\/em><em>180<\/em><\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img decoding=\"async\" alt=\"\" src=\"file:\/\/\/C:\/Users\/vinit\/AppData\/Local\/Temp\/msohtmlclip1\/01\/clip_image031.png\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p> \u00a0radian = <em>\u03c0<\/em><em>3<\/em><\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img decoding=\"async\" alt=\"\" src=\"file:\/\/\/C:\/Users\/vinit\/AppData\/Local\/Temp\/msohtmlclip1\/01\/clip_image032.png\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p>Hence, \u00a0\u00a0\u00a0\u00a0\u00a0 by r = <em>l<\/em><em>\u03b8<\/em><\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img decoding=\"async\" alt=\"\" src=\"file:\/\/\/C:\/Users\/vinit\/AppData\/Local\/Temp\/msohtmlclip1\/01\/clip_image033.png\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p>\u00a0\u00a0\u00a0 r = <em>37.4\u00d73<\/em><em>\u03c0<\/em><em>=<\/em><em>37.4\u00d73\u00d77<\/em><em>22<\/em><\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img decoding=\"async\" alt=\"\" src=\"file:\/\/\/C:\/Users\/vinit\/AppData\/Local\/Temp\/msohtmlclip1\/01\/clip_image034.png\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p> \u00a0= 35.7 cm<\/p>\n<p><strong>Example: <\/strong>In a circle of diameter 40 cm, the length of a chord is 20 cm. Find the length of minor arc of the chord.<\/p>\n<p><strong>Solution: <\/strong>Diameter of the circle = 40 cm<\/p>\n<p>\u2234Radius (r) of the circle = <em>40<\/em><em>2<\/em><\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img decoding=\"async\" alt=\"\" src=\"file:\/\/\/C:\/Users\/vinit\/AppData\/Local\/Temp\/msohtmlclip1\/01\/clip_image035.png\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p> \u00a0cm = 20 cm<\/p>\n<p>Let AB be a chord (length = 20 cm) of the circle.<\/p>\n<p><div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img decoding=\"async\" alt=\"\" src=\"file:\/\/\/C:\/Users\/vinit\/AppData\/Local\/Temp\/msohtmlclip1\/01\/clip_image036.jpg\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p>In \u0394OAB, OA = OB = Radius of circle = 20 cm<\/p>\n<p>Also, AB = 20 cm<\/p>\n<p>Thus, \u0394OAB is an equilateral triangle.<\/p>\n<p>\u2234\u03b8 = 60\u00b0 = <em>\u03c0<\/em><em>3<\/em><\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img decoding=\"async\" alt=\"\" src=\"file:\/\/\/C:\/Users\/vinit\/AppData\/Local\/Temp\/msohtmlclip1\/01\/clip_image037.png\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p> \u00a0radian<\/p>\n<p>We know that in a circle of radius\u00a0r\u00a0unit, if an arc of length\u00a0l\u00a0unit subtends an angle\u00a0\u03b8\u00a0radian at the centre, then. <em>\u03b8<\/em><\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img decoding=\"async\" alt=\"\" src=\"file:\/\/\/C:\/Users\/vinit\/AppData\/Local\/Temp\/msohtmlclip1\/01\/clip_image038.png\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p> \u00a0=<em>l<\/em><em>r<\/em><\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img decoding=\"async\" alt=\"\" src=\"file:\/\/\/C:\/Users\/vinit\/AppData\/Local\/Temp\/msohtmlclip1\/01\/clip_image039.png\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p><div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img decoding=\"async\" alt=\"\" src=\"file:\/\/\/C:\/Users\/vinit\/AppData\/Local\/Temp\/msohtmlclip1\/01\/clip_image040.gif\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p>Thus, the length of the minor arc of the chord is. <em>20\u03c0<\/em><em>3<\/em><\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img decoding=\"async\" alt=\"\" src=\"file:\/\/\/C:\/Users\/vinit\/AppData\/Local\/Temp\/msohtmlclip1\/01\/clip_image041.png\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p> \u00a0cm<\/p>\n<p><strong><u>Key Points<\/u><\/strong><\/p>\n<ul>\n<li>Polar coordinates consist of an angle (\u03b8) and a radius (r) and are used to represent points in a plane.<\/li>\n<li>The angle \u03b8 is measured counterclockwise from the positive x-axis, typically in radians.<\/li>\n<li>The radius r represents the distance from the origin to the point and can be negative.<\/li>\n<li>The conversion from polar coordinates to Cartesian coordinates is given by x = r * cos(\u03b8) and y = r * sin(\u03b8).<\/li>\n<li>The conversion from Cartesian coordinates to polar coordinates is given by r = \u221a(x<sup>2<\/sup> + y<sup>2<\/sup>) and \u03b8 = arctan(y \/ x).<\/li>\n<li>Polar graphs are plotted using the angle \u03b8 as the independent variable and the radius r as the dependent variable.<\/li>\n<li>Symmetry in polar graphs can be determined by replacing \u03b8 with -\u03b8 or \u03b8 + \u03c0, and r with -r.<\/li>\n<li>Polar equations can have multiple representations due to periodicity, such as r = a sin(b\u03b8) and r = a cos(b\u03b8), where a and b are constants.<\/li>\n<li>The shape of a polar graph can be determined by analyzing the equation and identifying patterns related to the angle \u03b8.<\/li>\n<li>Key features of polar graphs include the number of petals, the presence of loops or cusps, and the behavior at the origin.<\/li>\n<li>When graphing polar equations, it is important to choose an appropriate range of \u03b8 to capture the desired portion of the curve.<\/li>\n<li>The graphing of polar equations can be facilitated using technology like graphing calculators or computer software.<\/li>\n<li>To find the slope of a tangent line to a polar curve, the derivative must be calculated using the chain rule and trigonometric identities.<\/li>\n<li>The area bounded by a polar curve can be found using integration and the formula for the area of a sector.<\/li>\n<li>Arc lengths of polar curves can be determined by integrating a differential arc length formula based on the Pythagorean theorem.<\/li>\n<\/ul>\n<p><!--kapdec-footer-start--><\/p>\n<style>.kapdec-article-footer{font-family:Arial,Helvetica,Calibri,sans-serif;color:#444;}.kapdec-footer-grid{display:flex;align-items:stretch;border:1px solid #e5e7eb;border-radius:6px;overflow:hidden;}.kapdec-footer-left,.kapdec-qr-block{flex:1 1 50%;width:50%;box-sizing:border-box;min-width:0;}.kapdec-footer-left{padding:22px 28px;border-right:1px solid #e5e7eb;}.kapdec-citation-block{line-height:1.6;font-size:9pt;color:#333;margin:0;}.kapdec-citation-block p{margin:0 0 10px 0;}.kapdec-citation-block a{color:#0066cc;text-decoration:underline;}.kapdec-copyright-block{margin-top:18px;padding-top:14px;border-top:1px solid #e5e7eb;font-size:7.5pt;color:#777;line-height:1.55;text-align:left;}.kapdec-copyright-block p{margin:0 0 5px 0;}.kapdec-qr-block{padding:22px 28px;display:flex;flex-direction:column;align-items:center;justify-content:center;text-align:center;}.kapdec-qr-label{margin:0 0 8px 0;font-size:8.5pt;font-weight:600;color:#444;line-height:1.35;letter-spacing:.02em;}.kapdec-qr-url{margin:0 0 14px 0;font-size:7.5pt;line-height:1.4;color:#777;word-break:break-word;max-width:100%;}.kapdec-qr-url a{color:#777;text-decoration:underline;}@media (max-width:640px){.kapdec-footer-grid{flex-direction:column;}.kapdec-footer-left,.kapdec-qr-block{width:100%;flex-basis:100%;border-right:none;}.kapdec-footer-left{border-bottom:1px solid #e5e7eb;}}<\/style>\n<div class=\"kapdec-article-footer\" style=\"margin-top: 28px; padding-top: 4px;\">\n<div class=\"kapdec-footer-grid\">\n<div class=\"kapdec-footer-left\">\n<div class=\"kapdec-citation-block\">\n<p>A Kapdec&reg; learning guide &#8211; Crafted by elite STEM mentors for ambitious learners.<\/p>\n<p><a href=\"https:\/\/kapdec.com\" target=\"_blank\" rel=\"noopener noreferrer\">Learn more at https:\/\/kapdec.com<\/a><\/p>\n<\/div>\n<div class=\"kapdec-copyright-block\">\n<p>Author: Kapdec | Publisher: Kapdec | Copyright: &copy; Kapdec. All Rights Reserved.<\/p>\n<p>Unauthorized reproduction, distribution, or commercial use of this material is prohibited.<\/p>\n<\/div>\n<\/div>\n<div class=\"kapdec-qr-block\">\n<p class=\"kapdec-qr-label\">Scan to visit this resource online<\/p>\n<p class=\"kapdec-qr-url\"><a href=\"https:\/\/kapdec.com\/resources\/linking-angles-radii-and-polar-graphs\" target=\"_blank\" rel=\"noopener noreferrer\">https:\/\/kapdec.com\/resources\/linking-angles-radii-and-polar-graphs<\/a><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"data:image\/svg+xml;base64,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\" alt=\"QR code\" width=\"110\" height=\"110\" style=\"display: block; width: 110px; height: 110px; max-width: 110px; margin: 0 auto;\" \/><\/div>\n<\/div>\n<\/div>\n<p><!--kapdec-footer-end--><\/div>\n<div aria-hidden=\"true\" class=\"article-watermark-layer\" style=\"background-image:url(data:image\/svg+xml;base64,PD94bWwgdmVyc2lvbj0iMS4wIiBlbmNvZGluZz0iVVRGLTgiPz48c3ZnIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8yMDAwL3N2ZyIgd2lkdGg9Ijc1MCIgaGVpZ2h0PSI0NTAiPjx0ZXh0IHg9IjQwIiB5PSIyMzAiIHRyYW5zZm9ybT0icm90YXRlKC0zMiA0MCAyMzApIiBmb250LWZhbWlseT0iQXJpYWwsSGVsdmV0aWNhLENhbGlicmksc2Fucy1zZXJpZiIgZm9udC1zaXplPSIxOCIgZm9udC13ZWlnaHQ9IjQwMCIgdGV4dC1yZW5kZXJpbmc9Imdlb21ldHJpY1ByZWNpc2lvbiIgZmlsbD0iI2I1YjViNSIgZmlsbC1vcGFjaXR5PSIwLjMyIj5LQVBERUMmIzE3NDsgfCBFbGl0ZSBTVEVNIExlYXJuaW5nPC90ZXh0Pjwvc3ZnPg==);background-repeat:repeat;background-size:750px 450px;\"><\/div>\n<\/div>\n<style>.article-watermark-wrapper{position:relative;overflow:hidden;}.article-watermark-layer{position:absolute;inset:0;overflow:hidden;pointer-events:none;z-index:2;background-repeat:repeat;background-size:750px 450px;}@media print{.article-watermark-layer{position:fixed;inset:0;background-repeat:repeat!important;background-size:750px 450px!important;-webkit-print-color-adjust:exact;print-color-adjust:exact;}}<\/style>\n","protected":false},"excerpt":{"rendered":"<p>KAPDEC&reg; | Elite STEM Learning Platform | https:\/\/kapdec.com Unit: Trigonometric &amp; Polar Functions Chapter: Linking Angles, Radii, and Polar Graphs Reference: &#8211; Introduction, Calculus with polar functions, Polar graphing technology, Graphing Techniques, Polar functions, Symmetry, Plotting Points &amp; Axes, Polar &amp; Cartesian Coordinates, Conversion between polar &amp; Cartesian Coordinates, Angle &amp; Radius uses in AP [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[628],"tags":[],"class_list":["post-9944","post","type-post","status-publish","format-standard","hentry","category-ap-precalculus"],"_links":{"self":[{"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/posts\/9944","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/comments?post=9944"}],"version-history":[{"count":0,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/posts\/9944\/revisions"}],"wp:attachment":[{"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/media?parent=9944"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/categories?post=9944"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/tags?post=9944"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}