{"id":1363,"date":"2023-06-05T14:50:58","date_gmt":"2023-06-05T14:50:58","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/chapter\/angles\/"},"modified":"2023-06-15T17:59:48","modified_gmt":"2023-06-15T17:59:48","slug":"angles","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/chapter\/angles\/","title":{"raw":"Angles","rendered":"Angles"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Draw angles in standard position.<\/li>\r\n \t<li>Convert between degrees and radians.<\/li>\r\n \t<li>Convert between degrees\/radians and Degrees Minutes Seconds (DMS)<\/li>\r\n \t<li>Find coterminal angles.<\/li>\r\n \t<li>Find the length of a circular arc.<\/li>\r\n \t<li>Find the area of a sector of a circle.<\/li>\r\n \t<li>Use linear and angular speed to describe motion on a circular path.<\/li>\r\n<\/ul>\r\n<\/div>\r\nA golfer swings to hit a ball over a sand trap and onto the green. An airline pilot maneuvers a plane toward a narrow runway. A dress designer creates the latest fashion. What do they all have in common? They all work with angles, and so do all of us at one time or another. Sometimes we need to measure angles exactly with instruments. Other times we estimate them or judge them by eye. Either way, the proper angle can make the difference between success and failure in many undertakings. In this section, we will examine properties of angles.\r\n<h2>Draw angles in standard position<\/h2>\r\n<div id=\"post-10770\" class=\"standard post-10770 chapter type-chapter status-publish hentry\">\r\n<div class=\"entry-content\">\r\n\r\nProperly defining an angle first requires that we define a ray. A\u00a0<strong>ray<\/strong>\u00a0consists of one point on a line and all points extending in one direction from that point. The first point is called the\u00a0<strong>endpoint<\/strong>\u00a0of the ray. We can refer to a specific ray by stating its endpoint and any other point on it. The ray in Figure 1\u00a0can be named as ray EF, or in symbolic form [latex]\\overrightarrow{EF}[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/923\/2015\/04\/25180159\/CNX_Precalc_Figure_05_01_0012.jpg\" alt=\"Illustration of Ray EF, with point F and endpoint E.\" width=\"487\" height=\"173\" \/> <b>Figure 1<\/b>[\/caption]\r\n\r\nAn\u00a0<strong>angle<\/strong>\u00a0is the union of two rays having a common endpoint. The endpoint is called the\u00a0<strong>vertex<\/strong>\u00a0of the angle, and the two rays are the sides of the angle. The angle in Figure 2\u00a0is formed from\u00a0[latex]\\overrightarrow{ED}[\/latex] and\u00a0[latex]\\overrightarrow{EF}[\/latex]. Angles can be named using a point on each ray and the vertex, such as angle [latex]{DEF}[\/latex], or in symbol form [latex]\\angle{DEF}[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/923\/2015\/04\/25180201\/CNX_Precalc_Figure_05_01_0022.jpg\" alt=\"Illustration of Angle DEF, with vertex E and points D and F.\" width=\"487\" height=\"246\" \/> <b>Figure 2<\/b>[\/caption]\r\n\r\nGreek letters are often used as variables for the measure of an angle. The table below\u00a0is a list of Greek letters commonly used to represent angles, and a sample angle is shown in Figure 2.\r\n<table id=\"Table_05_01_01\" style=\"height: 66px;\" summary=\"Two rows and five columns. First row shows symbols for theta, phi, alpha, beta, and gamma. Second row spells out name for each symbol.\">\r\n<tbody>\r\n<tr style=\"height: 53px;\">\r\n<td style=\"height: 53px; width: 130.313px;\"><span class=\"katex-html\"><span class=\"base textstyle uncramped\"><span class=\"reset-textstyle displaystyle textstyle uncramped\"><span class=\"mord mathit\">[latex]\\theta[\/latex]<\/span><\/span><\/span><\/span><\/td>\r\n<td style=\"height: 53px; width: 180.313px;\"><span class=\"katex-html\"><span class=\"base textstyle uncramped\"><span class=\"reset-textstyle displaystyle textstyle uncramped\"><span class=\"mord mathit\">[latex]\\phi \\text{ or }\\varphi[\/latex]<\/span><\/span><\/span><\/span><\/td>\r\n<td style=\"height: 53px; width: 106.676px;\"><span class=\"katex-html\"><span class=\"base textstyle uncramped\"><span class=\"reset-textstyle displaystyle textstyle uncramped\"><span class=\"mord mathit\">[latex]\\alpha[\/latex]<\/span><\/span><\/span><\/span><\/td>\r\n<td style=\"height: 53px; width: 102.131px;\"><span class=\"katex-html\"><span class=\"base textstyle uncramped\"><span class=\"reset-textstyle displaystyle textstyle uncramped\"><span class=\"mord mathit\">[latex]\\beta[\/latex]<\/span><\/span><\/span><\/span><\/td>\r\n<td style=\"height: 53px; width: 122.131px;\"><span class=\"katex-html\"><span class=\"base textstyle uncramped\"><span class=\"reset-textstyle displaystyle textstyle uncramped\"><span class=\"mord mathit\">[latex]\\gamma[\/latex]<\/span><\/span><\/span><\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 13px;\">\r\n<td style=\"height: 13px; width: 130.313px;\">theta<\/td>\r\n<td style=\"height: 13px; width: 180.313px;\">phi<\/td>\r\n<td style=\"height: 13px; width: 106.676px;\">alpha<\/td>\r\n<td style=\"height: 13px; width: 102.131px;\">beta<\/td>\r\n<td style=\"height: 13px; width: 122.131px;\">gamma<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<figure id=\"Figure_05_01_003\" class=\"small\">\r\n<figure id=\"Figure_05_01_003\" class=\"small\"><span id=\"fs-id1165135192939\"> <img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/923\/2015\/04\/25180202\/CNX_Precalc_Figure_05_01_0032.jpg\" alt=\"Illustration of angle theta.\" \/><\/span><\/figure>\r\n<p style=\"text-align: center;\"><strong>Figure 3.\u00a0<\/strong>Angle theta, shown as [latex]\\angle \\theta [\/latex]<\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/923\/2015\/04\/25180203\/CNX_Precalc_Figure_05_01_0042.jpg\" alt=\"Illustration of an angle with labels for initial side, terminal side, and vertex.\" width=\"487\" height=\"247\" \/> <b>Figure 4<\/b>[\/caption]<\/figure>\r\nAngle creation is a dynamic process. We start with two rays lying on top of one another. We leave one fixed in place, and rotate the other. The fixed ray is the\u00a0<strong>initial side<\/strong>, and the rotated ray is the\u00a0<strong>terminal side<\/strong>. In order to identify the different sides, we indicate the rotation with a small arc and arrow close to the vertex as in Figure 4.<span id=\"fs-id1165137737991\">\r\n<\/span>\r\n\r\nThe following video provides an illustration of angles in standard position.\r\n\r\n[embed]https:\/\/youtube.com\/watch?v=hpIjaKLOo6o[\/embed]\r\n\r\nAs we discussed at the beginning of the section, there are many applications for angles, but in order to use them correctly, we must be able to measure them. The\u00a0<strong>measure of an angle<\/strong>\u00a0is the amount of rotation from the initial side to the terminal side. Probably the most familiar unit of angle measurement is the degree. One\u00a0<strong>degree<\/strong>\u00a0is [latex]\\frac{1}{360}[\/latex] of a circular rotation, so a complete circular rotation contains 360 degrees. An angle measured in degrees should always include the unit \u201cdegrees\u201d after the number, or include the degree symbol \u00b0. For example, 90 degrees = 90\u00b0.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/923\/2015\/04\/25180205\/CNX_Precalc_Figure_05_01_0052.jpg\" alt=\"Graph of an angle in standard position with labels for the initial side and terminal side.\" width=\"487\" height=\"417\" \/> <b>Figure 5<\/b>[\/caption]\r\n\r\nTo formalize our work, we will begin by drawing angles on an <em>x<\/em>-<em>y<\/em> coordinate plane. Angles can occur in any position on the coordinate plane, but for the purpose of comparison, the convention is to illustrate them in the same position whenever possible. An angle is in <strong>standard position<\/strong> if its vertex is located at the origin, and its initial side extends along the positive <em>x<\/em>-axis.\u00a0<span id=\"fs-id1165137804556\">\r\n<\/span>\r\n\r\nIf the angle is measured in a counterclockwise direction from the initial side to the terminal side, the angle is said to be a <strong>positive angle<\/strong>. If the angle is measured in a clockwise direction, the angle is said to be a <strong>negative angl<\/strong><strong style=\"font-size: 1em;\">e<\/strong><span style=\"font-size: 1em;\">.<\/span>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/923\/2015\/04\/25180207\/CNX_Precalc_Figure_05_01_0062.jpg\" alt=\"Side by side graphs. Graph on the left is a 90 degree angle and graph on the right is a 360 degree angle. Terminal side and initial side are labeled for both graphs.\" width=\"731\" height=\"365\" \/> <b>Figure 6<\/b>[\/caption]\r\n\r\nDrawing an angle in standard position always starts the same way\u2014draw the initial side along the positive <em>x<\/em>-axis. To place the terminal side of the angle, we must calculate the fraction of a full rotation the angle represents. We do that by dividing the angle measure in degrees by 360\u00b0. For example, to draw a 90\u00b0 angle, we calculate that [latex]\\frac{90^\\circ }{360^\\circ }=\\frac{1}{4}[\/latex]. So, the terminal side will be one-fourth of the way around the circle, moving counterclockwise from the positive <em>x<\/em>-axis. To draw a 360\u00b0 angle, we calculate that [latex]\\frac{360^\\circ }{360^\\circ }=1[\/latex]. So the terminal side will be 1 complete rotation around the circle, moving counterclockwise from the positive <em>x<\/em>-axis. In this case, the initial side and the terminal side overlap.<span id=\"fs-id1165134042853\">\r\n<\/span>\r\n\r\nSince we define an angle in <strong>standard position<\/strong> by its terminal side, we have a special type of angle whose terminal side lies on an axis, a <strong>quadrantal angle<\/strong>. This type of angle can have a measure of 0\u00b0, 90\u00b0, 180\u00b0, 270\u00b0 or 360\u00b0.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/923\/2015\/04\/25180209\/CNX_Precalc_Figure_05_01_0182.jpg\" alt=\"Four side by side graphs. First graph shows angle of 0 degrees. Second graph shows an angle of 90 degrees. Third graph shows an angle of 180 degrees. Fourth graph shows an angle of 270 degrees.\" width=\"975\" height=\"237\" \/> <b>Figure 7.<\/b> Quadrantal angles have a terminal side that lies along an axis. Examples are shown.[\/caption]\r\n\r\n<div class=\"textbox\">\r\n<h3>A GENERAL NOTE: QUADRANTAL ANGLES<\/h3>\r\nQuadrantal angles are angles whose terminal side lies on an axis, including 0\u00b0, 90\u00b0, 180\u00b0, 270\u00b0, or 360\u00b0.\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>HOW TO: GIVEN AN ANGLE MEASURE IN DEGREES, DRAW THE ANGLE IN STANDARD POSITION.<strong>\r\n<\/strong><\/h3>\r\n<ol>\r\n \t<li>Express the angle measure as a fraction of 360\u00b0.<\/li>\r\n \t<li>Reduce the fraction to simplest form.<\/li>\r\n \t<li>Draw an angle that contains that same fraction of the circle, beginning on the positive\u00a0<em>x<\/em>-axis and moving counterclockwise for positive angles and clockwise for negative angles.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>EXAMPLE 1: DRAWING AN ANGLE IN STANDARD POSITION MEASURED IN DEGREES<\/h3>\r\n<ol>\r\n \t<li>Sketch an angle of 30\u00b0 in standard position.<\/li>\r\n \t<li>Sketch an angle of \u2212135\u00b0 in standard position.<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"899324\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"899324\"]\r\n<ol>\r\n \t<li>Divide the angle measure by 360\u00b0.\r\n<div style=\"text-align: center;\">[latex]\\frac{30^\\circ }{360^\\circ }=\\frac{1}{12}[\/latex]<\/div>\r\nTo rewrite the fraction in a more familiar fraction, we can recognize that\r\n<div style=\"text-align: center;\">[latex]\\frac{1}{12}=\\frac{1}{3}\\left(\\frac{1}{4}\\right)[\/latex]<\/div>\r\nOne-twelfth equals one-third of a quarter, so by dividing a quarter rotation into thirds, we can sketch a line at 30\u00b0 as in Figure 8.<span id=\"fs-id1165137784208\">\r\n<\/span>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/923\/2015\/04\/25180211\/CNX_Precalc_Figure_05_01_0072.jpg\" alt=\"Graph of a 30 degree angle.\" width=\"487\" height=\"383\" \/> <b>Figure 8<\/b>[\/caption]\r\n\r\n&nbsp;<\/li>\r\n \t<li>Divide the angle measure by 360\u00b0.\r\n<div style=\"text-align: center;\">[latex]\\frac{-135^\\circ }{360^\\circ }=-\\frac{3}{8}[\/latex]<\/div>\r\nIn this case, we can recognize that\r\n<div style=\"text-align: center;\">[latex]-\\frac{3}{8}=-\\frac{3}{2}\\left(\\frac{1}{4}\\right)[\/latex]<\/div><\/li>\r\n \t<li>Negative three-eighths is one and one-half times a quarter, so we place a line by moving clockwise one full quarter and one-half of another quarter, as in Figure 9.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/923\/2015\/04\/25180213\/CNX_Precalc_Figure_05_01_0082.jpg\" alt=\"Graph of a negative 135 degree angle.\" width=\"487\" height=\"383\" \/> <b>Figure 9<\/b>[\/caption]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]<b><\/b>\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>TRY IT 1<\/h3>\r\nShow an angle of 240\u00b0 on a circle in standard position.\r\n\r\n[reveal-answer q=\"862928\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"862928\"]\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003504\/CNX_Precalc_Figure_05_01_0092.jpg\" alt=\"Graph of a 240 degree angle.\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div>Watch this video for more examples of determining angles of rotation.<\/div>\r\n[embed]https:\/\/www.youtube.com\/embed\/0yHDfG2m-44[\/embed]\r\n<h2 class=\"\">Converting Between Degrees and Radians<\/h2>\r\nDividing a circle into 360 parts is an arbitrary choice, although it creates the familiar degree measurement. We may choose other ways to divide a circle. To find another unit, think of the process of drawing a circle. Imagine that you stop before the circle is completed. The portion that you drew is referred to as an arc. An\u00a0<strong>arc<\/strong>\u00a0may be a portion of a full circle, a full circle, or more than a full circle, represented by more than one full rotation. The length of the arc around an entire circle is called the\u00a0<strong>circumference<\/strong>\u00a0of that circle.\r\n\r\nThe circumference of a circle is [latex]C=2\\pi r[\/latex]. If we divide both sides of this equation by [latex]r[\/latex], we create the ratio of the circumference to the radius, which is always [latex]2\\pi[\/latex] regardless of the length of the radius. So the circumference of any circle is [latex]2\\pi \\approx 6.28[\/latex] times the length of the radius. That means that if we took a string as long as the radius and used it to measure consecutive lengths around the circumference, there would be room for six full string-lengths and a little more than a quarter of a seventh, as shown in Figure 10.\r\n\r\n<span id=\"fs-id1165137769898\">\u00a0<\/span>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/923\/2015\/04\/25180217\/CNX_Precalc_Figure_05_01_0102.jpg\" alt=\"Illustration of a circle showing the number of radians in a circle.\" width=\"487\" height=\"340\" \/> <b>Figure 10<\/b>[\/caption]\r\n<p id=\"fs-id1165137667767\">This brings us to our new angle measure. One <span class=\"no-emphasis\">radian<\/span> is the measure of a central angle of a circle that intercepts an arc equal in length to the radius of that circle. A central angle is an angle formed at the center of a circle by two radii. Because the total circumference equals [latex]2\\pi [\/latex] times the radius, a full circular rotation is [latex]2\\pi [\/latex] radians. So<\/p>\r\n\r\n<div id=\"eip-246\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{gathered} 2\\pi \\text{ radians}={360}^{\\circ } \\\\ \\pi \\text{ radians}=\\frac{{360}^{\\circ }}{2}={180}^{\\circ } \\\\ 1\\text{ radian}=\\frac{{180}^{\\circ }}{\\pi }\\approx {57.3}^{\\circ } \\end{gathered}[\/latex]<\/div>\r\nNote that when an angle is described without a specific unit, it refers to radian measure. For example, an angle measure of 3 indicates 3 radians. In fact, radian measure is dimensionless, since it is the quotient of a length (circumference) divided by a length (radius) and the length units cancel out.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/923\/2015\/04\/25180219\/CNX_Precalc_Figure_05_01_0112.jpg\" alt=\"Illustration of a circle with angle t, radius r, and an arc of r.\" width=\"487\" height=\"323\" \/> <b>Figure 11.<\/b> The angle <i>t<\/i> sweeps out a measure of one radian. Note that the length of the intercepted arc is the same as the length of the radius of the circle.[\/caption]\r\n<h2>Converting Between Degrees and Degrees Minutes Seconds (DMS)<\/h2>\r\nUsing our definition of degree measure, we have that 1\u00b0 represents the measure of an angle which constitutes \u00a0\u00a0of a revolution. Even though it may be hard to draw, it is nonetheless not difficult to imagine an angle with measure smaller than 1\u00b0. There are two ways to subdivide degrees. The first, and most familiar, is decimal degrees. For example, an angle with a measure of 30.5\u00b0 would represent a rotation halfway between 30\u00b0 and 31\u00b0 , or equivalently, of a full rotation. This can be taken to the limit using Calculus so that measures like \u221a2\u00b0 make sense. The second way to divide degrees is the <strong>Degree - Minute - Second (DMS)<\/strong> system. In this system, one degree is divided equally into sixty minutes, and in turn, each minute is divided equally into sixty seconds. In symbols, we write 1\u00b0 = 60\u2032 and 1\u2032 = 60\u2032\u2032, from which it follows that 1\u00b0 = 3600\u2032\u2032.\r\n\r\n<\/div>\r\n<div class=\"entry-content\">\r\n<div class=\"textbox examples\">\r\n<h3>Examples(converting from Decimal degrees to dms)<\/h3>\r\nTo convert a measure of 42.125\u00b0 to the DMS system, we start by noting that\r\n\r\n42.125\u00b0= 42\u00b0 + 0.125\u00b0\r\n<ul>\r\n \t<li>Converting the partial amount of degrees to minutes, we find 0.125\u00b0 (60\u2019\/1\u00b0) = 7.5\u2032 = 7\u2032 + 0.5\u2032<\/li>\r\n \t<li>Converting the partial amount of minutes to seconds gives 0.5\u2032 (60\u201d\/1\u2019)= 30\u2032\u2032<\/li>\r\n \t<li>Putting it all together yields\r\n42.125\u00b0 = 42\u00b0 \u00a0+ 0.125\u00b0\r\n= 42\u00b0 + 7.5\u2032\r\n= 42\u00b0+ 7\u2032 + 0.5\u2032\r\n= 42\u00b0 + 7\u2032 + 30\u2032\u2032\r\n= 42\u00b07\u203230\u2032\u2032<\/li>\r\n<\/ul>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox examples\">\r\n<h3>Examples (DMS to decimal degrees)<\/h3>\r\nconvert 117\u00b015\u203245\u2032\u2032 to decimal degrees\r\n<ul>\r\n \t<li>compute 15\u2032 ( 1\u00b0\/60\u2032) = (1\/4)\u00b0<\/li>\r\n \t<li>Second, 45\u2032\u2032 ( 1\u00b0\/3600\u2032\u2032) = (1\/80)\u00b0<\/li>\r\n \t<li>Then we find117\u00b015\u201945\u201d=117\u00b0+15\u2019+45\u201d\r\n\r\n= 117\u00b0+ (1\/4)\u00b0 + (1\/80)\u00b0\r\n\r\n= (9381\/80) \u00b0=117.2625\u00b0\r\n\r\n&nbsp;<\/li>\r\n<\/ul>\r\n<\/div>\r\n&nbsp;\r\n\r\nhttps:\/\/youtu.be\/m86G_02jdPA\r\n<h2>Relating Arc Lengths to Radius<\/h2>\r\nAn <strong>arc length<\/strong> [latex]s[\/latex] is the length of the curve along the arc. Just as the full circumference of a circle always has a constant ratio to the radius, the arc length produced by any given angle also has a constant relation to the radius, regardless of the length of the radius.\r\n\r\nThis ratio, called the <strong>radian measure<\/strong>, is the same regardless of the radius of the circle\u2014it depends only on the angle. This property allows us to define a measure of any angle as the ratio of the arc length [latex]s[\/latex] to the radius [latex]r[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{gathered}s=r\\theta \\\\ \\theta =\\frac{s}{r}\\end{gathered}[\/latex]<\/div>\r\nIf [latex]s=r[\/latex], then [latex]\\theta =\\frac{r}{r}=\\text{ 1 radian}\\text{.}[\/latex]\r\n\r\n<span id=\"fs-id1165135347607\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003429\/CNX_Precalc_Figure_05_01_013abc2.jpg\" alt=\"Three side by side graphs of circles. First graph has a circle with radius r and arc s, with an equivalence between r and s. The second graph shows a circle with radius r and an arc of length 2r. The third graph shows a circle with a full revolution, showing 6.28 radians.\" \/><\/span>\r\n<p style=\"text-align: center;\"><strong>Figure 12.<\/strong>\u00a0(a) In an angle of 1 radian, the arc length [latex]s[\/latex] equals the radius [latex]r[\/latex]. (b) An angle of 2 radians has an arc length [latex]s=2r[\/latex]. (c) A full revolution is [latex]2\\pi [\/latex] or about 6.28 radians.<\/p>\r\nTo elaborate on this idea, consider two circles, one with radius 2 and the other with radius 3. Recall the circumference of a circle is [latex]C=2\\pi r[\/latex], where [latex]r[\/latex] is the radius. The smaller circle then has circumference [latex]2\\pi \\left(2\\right)=4\\pi [\/latex] and the larger has circumference [latex]2\\pi \\left(3\\right)=6\\pi [\/latex].\u00a0Now we draw a 45\u00b0 angle on the two circles, as in\u00a0Figure 13.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/923\/2015\/04\/25180222\/CNX_Precalc_Figure_05_01_0122.jpg\" alt=\"Graph of a circle with a 45 degree angle and a label for pi\/4 radians.\" width=\"487\" height=\"369\" \/> <b>Figure 13.<\/b> A 45\u00b0 angle contains one-eighth of the circumference of a circle, regardless of the radius.[\/caption]\r\n\r\nNotice what happens if we find the ratio of the arc length divided by the radius of the circle.\r\n<div style=\"text-align: center;\">[latex]\\begin{gathered}\\text{Smaller circle: }\\frac{\\frac{1}{2}\\pi }{2}=\\frac{1}{4}\\pi \\\\ \\text{ Larger circle: }\\frac{\\frac{3}{4}\\pi }{3}=\\frac{1}{4}\\pi \\end{gathered}[\/latex]<\/div>\r\nSince both ratios are [latex]\\frac{1}{4}\\pi [\/latex], the angle measures of both circles are the same, even though the arc length and radius differ.\r\n<div class=\"textbox\">\r\n<h3>A GENERAL NOTE: RADIANS<\/h3>\r\nOne <strong>radian<\/strong> is the measure of the central angle of a circle such that the length of the arc between the initial side and the terminal side is equal to the radius of the circle. A full revolution (360\u00b0) equals [latex]2\\pi [\/latex] radians. A half revolution (180\u00b0) is equivalent to [latex]\\pi [\/latex] radians.\r\n\r\nThe <strong>radian measure<\/strong> of an angle is the ratio of the length of the arc subtended by the angle to the radius of the circle. In other words, if [latex]s[\/latex] is the length of an arc of a circle, and [latex]r[\/latex] is the radius of the circle, then the central angle containing that arc measures [latex]\\frac{s}{r}[\/latex] radians. In a circle of radius 1, the radian measure corresponds to the length of the arc.\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n\r\n<strong>Q &amp; A<\/strong>\r\n<h3>A MEASURE OF 1 RADIAN LOOKS TO BE ABOUT 60\u00b0. IS THAT CORRECT?<\/h3>\r\n<em>Yes. It is approximately 57.3\u00b0. Because [latex]2\\pi [\/latex] radians equals 360\u00b0, [latex]1[\/latex] radian equals [latex]\\frac{360^\\circ }{2\\pi }\\approx 57.3^\\circ [\/latex].<\/em>\r\n\r\n<\/div>\r\n<h2>Using Radians<\/h2>\r\nBecause <strong>radian<\/strong> measure is the ratio of two lengths, it is a unitless measure. For example, in Figure 12, suppose the radius were 2 inches and the distance along the arc were also 2 inches. When we calculate the radian measure of the angle, the \"inches\" cancel, and we have a result without units. Therefore, it is not necessary to write the label \"radians\" after a radian measure, and if we see an angle that is not labeled with \"degrees\" or the degree symbol, we can assume that it is a radian measure.\r\n\r\nConsidering the most basic case, the <strong>unit circle<\/strong> (a circle with radius 1), we know that 1 rotation equals 360 degrees, 360\u00b0. We can also track one rotation around a circle by finding the circumference, [latex]C=2\\pi r[\/latex], and for the unit circle [latex]C=2\\pi [\/latex]. These two different ways to rotate around a circle give us a way to convert from degrees to radians.\r\n<div style=\"text-align: center;\">[latex]\\begin{gathered}1\\text{ rotation }=360^\\circ =2\\pi \\text{radians} \\\\ \\frac{1}{2}\\text{ rotation}=180^\\circ =\\pi \\text{radians} \\\\ \\frac{1}{4}\\text{ rotation}=90^\\circ =\\frac{\\pi }{2} \\text{radians} \\end{gathered}[\/latex]<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]148258[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Identifying Special Angles Measured in Radians<\/h2>\r\nIn addition to knowing the measurements in degrees and radians of a quarter revolution, a half revolution, and a full revolution, there are other frequently encountered angles in one revolution of a circle with which we should be familiar. It is common to encounter multiples of 30, 45, 60, and 90 degrees. These values are shown in Figure 14. Memorizing these angles will be very useful as we study the properties associated with angles.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/923\/2015\/04\/25180223\/CNX_Precalc_Figure_05_01_0162.jpg\" alt=\"A graph of a circle with angles of 0, 30, 45, 60, 90, 120, 135, 150, 180, 210, 225, 240, 270, 300, 315, and 330 degrees.\" width=\"487\" height=\"406\" \/> <b>Figure 14.<\/b> Commonly encountered angles measured in degrees[\/caption]\r\n\r\nNow, we can list the corresponding radian values for the common measures of a circle corresponding to those listed in Figure 14, which are shown in Figure 15. Be sure you can verify each of these measures.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/923\/2015\/04\/25180224\/CNX_Precalc_Figure_05_01_0172.jpg\" alt=\"A graph of a circle with angles of 0, 30, 45, 60, 90, 120, 135, 150, 180, 210, 225, 240, 270, 300, 315, and 330 degrees. The graph also shows the equivalent amount of radians for each angle of degrees. For example, 30 degrees is equal to pi\/6 radians.\" width=\"487\" height=\"485\" \/> <b>Figure 15.<\/b> Commonly encountered angles measured in radians[\/caption]\r\n\r\n<div class=\"textbox shaded\">\r\n<h3>EXAMPLE 2: FINDING A RADIAN MEASURE<\/h3>\r\nFind the radian measure of one-third of a full rotation.\r\n\r\n[reveal-answer q=\"791790\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"791790\"]\r\n\r\nFor any circle, the arc length along such a rotation would be one-third of the circumference. We know that\r\n<p style=\"text-align: center;\">[latex]1\\text{ rotation}=2\\pi r[\/latex]<\/p>\r\nSo,\r\n<p style=\"text-align: center;\">[latex]\\begin{align} s&amp;=\\frac{1}{3}\\left(2\\pi r\\right) \\\\ &amp;=\\frac{2\\pi r}{3} \\end{align}[\/latex]<\/p>\r\nThe radian measure would be the arc length divided by the radius.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\text{radian measure}&amp;=\\frac{\\frac{2\\pi r}{3}}{r} \\\\ &amp;=\\frac{2\\pi r}{3r} \\\\ &amp;=\\frac{2\\pi }{3} \\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>TRY IT 3<\/h3>\r\nFind the radian measure of three-fourths of a full rotation.\r\n\r\n[reveal-answer q=\"558820\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"558820\"]\r\n\r\n[latex]\\frac{3\\pi }{2}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Converting between Radians and Degrees<\/h2>\r\nBecause degrees and radians both measure angles, we need to be able to convert between them. We can easily do so using a proportion.\r\n<div style=\"text-align: center;\">[latex]\\frac{\\theta }{180}=\\frac{{\\theta }^{R}}{\\pi }[\/latex]<\/div>\r\nThis proportion shows that the measure of angle [latex]\\theta [\/latex] in degrees divided by 180 equals the measure of angle [latex]\\theta [\/latex] in radians divided by [latex]\\pi . [\/latex] Or, phrased another way, degrees is to 180 as radians is to [latex]\\pi [\/latex].\r\n<div style=\"text-align: center;\">[latex]\\frac{\\text{Degrees}}{180}=\\frac{\\text{Radians}}{\\pi }[\/latex]<span class=\"fontsize-ensurer reset-size5 size5\" style=\"font-size: 1rem; text-align: initial;\"><span class=\"\">\u200b<\/span><\/span><span style=\"font-size: 1rem; text-align: initial;\">\u200b<\/span><\/div>\r\n<div>\r\n<h2>Converting between Radians and Degrees<\/h2>\r\nTo convert between degrees and radians, use the proportion\r\n<div style=\"text-align: center;\">[latex]\\frac{\\theta }{180}=\\frac{{\\theta }^{R}}{\\pi }[\/latex]<\/div>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>EXAMPLE 3: CONVERTING RADIANS TO DEGREES<\/h3>\r\nConvert each radian measure to degrees.\r\n<p style=\"padding-left: 60px;\">a. [latex]\\frac{\\pi }{6}[\/latex]<\/p>\r\n<p style=\"padding-left: 60px;\">b. 3<\/p>\r\n[reveal-answer q=\"620590\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"620590\"]\r\n\r\nBecause we are given radians and we want degrees, we should set up a proportion and solve it.\r\n<p style=\"padding-left: 60px;\">a. We use the proportion, substituting the given information.<\/p>\r\n<p style=\"padding-left: 60px; text-align: center;\">[latex]\\begin{gathered} \\frac{\\theta }{180}=\\frac{{\\theta }^{R}}{\\pi } \\\\ \\frac{\\theta }{180}=\\frac{\\frac{\\pi }{6}}{\\pi } \\\\ \\theta =\\frac{180}{6} \\\\ \\theta ={30}^{\\circ } \\end{gathered}[\/latex]<\/p>\r\n<p style=\"padding-left: 60px;\">b. We use the proportion, substituting the given information.<\/p>\r\n<p style=\"padding-left: 60px; text-align: center;\">[latex]\\begin{gathered} \\frac{\\theta }{180}=\\frac{{\\theta }^{R}}{\\pi } \\\\ \\frac{\\theta }{180}=\\frac{3}{\\pi } \\\\ \\theta =\\frac{3\\left(180\\right)}{\\pi } \\\\ \\theta \\approx {172}^{\\circ } \\end{gathered}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>TRY IT 4<\/h3>\r\nConvert [latex]-\\frac{3\\pi }{4}[\/latex] radians to degrees.\r\n\r\n[reveal-answer q=\"362624\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"362624\"]\r\n\r\n\u2212135\u00b0\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]99889[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>EXAMPLE 4: CONVERTING DEGREES TO RADIANS<\/h3>\r\nConvert [latex]15[\/latex] degrees to radians.\r\n\r\n[reveal-answer q=\"867109\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"867109\"]\r\n\r\nIn this example, we start with degrees and want radians, so we again set up a proportion and solve it, but we substitute the given information into a different part of the proportion.\r\n<p style=\"text-align: center;\">[latex]\\begin{gathered} \\frac{\\theta }{180}=\\frac{{\\theta }^{R}}{\\pi } \\\\ \\frac{15}{180}=\\frac{{\\theta }^{R}}{\\pi }\\\\ \\frac{15\\pi }{180}={\\theta }^{R}\\\\ \\frac{\\pi }{12}={\\theta }^{R} \\end{gathered}[\/latex]<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\nAnother way to think about this problem is by remembering that [latex]{30}^{\\circ }=\\frac{\\pi }{6}[\/latex].\r\nBecause [latex]{15}^{\\circ }=\\frac{1}{2}\\left({30}^{\\circ }\\right)[\/latex], we can find that [latex]\\frac{1}{2}\\left(\\frac{\\pi }{6}\\right)[\/latex] is [latex]\\frac{\\pi }{12}[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>TRY IT 6<\/h3>\r\nConvert 126\u00b0 to radians.\r\n\r\n[reveal-answer q=\"164537\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"164537\"]\r\n\r\n[latex]\\frac{7\\pi }{10}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]99887[\/ohm_question]\r\n\r\n<\/div>\r\n<div>\r\n<div>Watch the following video for an explanation of radian measure and examples of converting between radians and degrees.<\/div>\r\nhttps:\/\/youtu.be\/nAJqXtzwpXQ\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<section class=\"citations-section focusable\" role=\"contentinfo\">\r\n<div id=\"post-10774\" class=\"standard post-10774 chapter type-chapter status-publish hentry\">\r\n<div class=\"entry-content\">\r\n<h2>Finding Coterminal Angles<\/h2>\r\nConverting between degrees and radians can make working with angles easier in some applications. For other applications, we may need another type of conversion. Negative angles and angles greater than a full revolution are more awkward to work with than those in the range of 0\u00b0 to 360\u00b0, or 0 to [latex]2\\pi [\/latex]. It would be convenient to replace those out-of-range angles with a corresponding angle within the range of a single revolution.\r\n\r\nIt is possible for more than one angle to have the same terminal side. Look at Figure 16. The angle of 140\u00b0 is a\u00a0<strong>positive angle<\/strong>, measured counterclockwise. The angle of \u2013220\u00b0 is a\u00a0<strong>negative angle<\/strong>, measured clockwise. But both angles have the same terminal side. If two angles in standard position have the same terminal side, they are\u00a0<strong>coterminal angles<\/strong>. Every angle greater than 360\u00b0 or less than 0\u00b0 is coterminal with an angle between 0\u00b0 and 360\u00b0, and it is often more convenient to find the coterminal angle within the range of 0\u00b0 to 360\u00b0 than to work with an angle that is outside that range.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/923\/2015\/04\/25180226\/CNX_Precalc_Figure_05_01_0202.jpg\" alt=\"A graph showing the equivalence between a 140 degree angle and a negative 220 degree angle.\" width=\"487\" height=\"383\" \/> <b>Figure 16.<\/b> An angle of 140\u00b0 and an angle of \u2013220\u00b0 are coterminal angles.[\/caption]\r\n\r\nThis video shows examples of how to determine if two angles are coterminal.\r\n\r\nhttps:\/\/www.youtube.com\/watch?v=TuyF8fFg3B0\r\n\r\nAny angle has infinitely many\u00a0<strong>coterminal angles<\/strong>\u00a0because each time we add 360\u00b0 to that angle\u2014or subtract 360\u00b0 from it\u2014the resulting value has a terminal side in the same location. For example, 100\u00b0 and 460\u00b0 are coterminal for this reason, as is \u2212260\u00b0. Recognizing that any angle has infinitely many coterminal angles explains the repetitive shape in the graphs of trigonometric functions.\r\n\r\nAn angle\u2019s reference angle is the measure of the smallest, positive, acute angle [latex]t[\/latex] formed by the terminal side of the angle [latex]t[\/latex] and the horizontal axis. Thus positive reference angles have terminal sides that lie in the first quadrant and can be used as models for angles in other quadrants. See Figure 17\u00a0for examples of reference angles for angles in different quadrants.<span id=\"fs-id1165137542464\">\u00a0<\/span>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/923\/2015\/04\/25180227\/CNX_Precalc_Figure_05_01_0194.jpg\" alt=\"Four side by side graphs. First graph shows an angle of t in quadrant 1 in it's normal position. Second graph shows an angle of t in quadrant 2 due to a rotation of pi minus t. Third graph shows an angle of t in quadrant 3 due to a rotation of t minus pi. Fourth graph shows an angle of t in quadrant 4 due to a rotation of two pi minus t.\" width=\"975\" height=\"331\" \/> <b>Figure 17<\/b>[\/caption]\r\n\r\n&nbsp;\r\n<div class=\"textbox\">\r\n<h3>A GENERAL NOTE: COTERMINAL AND REFERENCE ANGLES<\/h3>\r\nCoterminal angles are two angles in standard position that have the same terminal side.\r\n\r\nAn angle\u2019s <strong>reference angle<\/strong> is the size of the smallest acute angle, [latex]{t}^{\\prime }[\/latex], formed by the terminal side of the angle [latex]t[\/latex]\u00a0and the horizontal axis.\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>HOW TO: GIVEN AN ANGLE GREATER THAN 360\u00b0, FIND A COTERMINAL ANGLE BETWEEN 0\u00b0 AND 360\u00b0.<\/h3>\r\n<ol>\r\n \t<li>Subtract 360\u00b0 from the given angle.<\/li>\r\n \t<li>If the result is still greater than 360\u00b0, subtract 360\u00b0 again till the result is between 0\u00b0 and 360\u00b0.<\/li>\r\n \t<li>The resulting angle is coterminal with the original angle.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>EXAMPLE 5: FINDING AN ANGLE COTERMINAL WITH AN ANGLE OF MEASURE GREATER THAN 360\u00b0<\/h3>\r\nFind the least positive angle [latex]\\theta [\/latex] that is coterminal with an angle measuring 800\u00b0, where [latex]0^\\circ \\le \\theta &lt;360^\\circ [\/latex].\r\n\r\n[reveal-answer q=\"964169\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"964169\"]\r\n\r\nAn angle with measure 800\u00b0 is coterminal with an angle with measure 800 \u2212 360 = 440\u00b0, but 440\u00b0 is still greater than 360\u00b0, so we subtract 360\u00b0 again to find another coterminal angle: 440 \u2212 360 = 80\u00b0.\r\n\r\nThe angle [latex]\\theta =80^\\circ [\/latex] is coterminal with 800\u00b0. To put it another way, 800\u00b0 equals 80\u00b0 plus two full rotations, as shown in Figure 18.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/923\/2015\/04\/25180229\/CNX_Precalc_Figure_05_01_0212.jpg\" alt=\"A graph showing the equivalence between an 80 degree angle and an 800 degree angle.\" width=\"487\" height=\"383\" \/> <b>Figure 18<\/b>[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>TRY IT 8<\/h3>\r\nFind an angle [latex]\\alpha [\/latex] that is coterminal with an angle measuring 870\u00b0, where [latex]0^\\circ \\le \\alpha &lt;360^\\circ [\/latex].\r\n\r\n[reveal-answer q=\"363809\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"363809\"]\r\n\r\n[latex]\\alpha =150^\\circ [\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>HOW TO: GIVEN AN ANGLE WITH MEASURE LESS THAN 0\u00b0, FIND A COTERMINAL ANGLE HAVING A MEASURE BETWEEN 0\u00b0 AND 360\u00b0.<\/h3>\r\n<ol>\r\n \t<li>Add 360\u00b0 to the given angle.<\/li>\r\n \t<li>If the result is still less than 0\u00b0, add 360\u00b0 again until the result is between 0\u00b0 and 360\u00b0.<\/li>\r\n \t<li>The resulting angle is coterminal with the original angle.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>EXAMPLE 6: FINDING AN ANGLE COTERMINAL WITH AN ANGLE MEASURING LESS THAN 0\u00b0<\/h3>\r\nShow the angle with measure \u221245\u00b0 on a circle and find a positive coterminal angle [latex]\\alpha [\/latex] such that 0\u00b0 \u2264 <em>\u03b1<\/em> &lt; 360\u00b0.\r\n\r\n[reveal-answer q=\"713003\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"713003\"]\r\n\r\nSince 45\u00b0 is half of 90\u00b0, we can start at the positive horizontal axis and measure clockwise half of a 90\u00b0 angle.\r\n\r\nBecause we can find coterminal angles by adding or subtracting a full rotation of 360\u00b0, we can find a positive coterminal angle here by adding 360\u00b0:\r\n<p style=\"text-align: center;\">[latex]-45^\\circ +360^\\circ =315^\\circ [\/latex]<\/p>\r\nWe can then show the angle on a circle, as in Figure 19.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/923\/2015\/04\/25180230\/CNX_Precalc_Figure_05_01_0222.jpg\" alt=\"A graph showing the equivalence of a 315 degree angle and a negative 45 degree angle.\" width=\"487\" height=\"383\" \/> <strong>Figure 19<\/strong>[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div>\r\n\r\nWatch this video for another example of how to determine positive and negative coterminal angles.\r\n\r\nhttps:\/\/www.youtube.com\/watch?v=m7jTGVVzb0s\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>TRY IT 9<\/h3>\r\nFind an angle [latex]\\beta [\/latex] that is coterminal with an angle measuring \u2212300\u00b0 such that [latex]0^\\circ \\le \\beta &lt;360^\\circ [\/latex].\r\n\r\n[reveal-answer q=\"113901\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"113901\"]\r\n\r\n[latex]\\beta =60^\\circ [\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]147466[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Finding Coterminal Angles Measured in Radians<\/h2>\r\nWe can find\u00a0<strong>coterminal angles<\/strong>\u00a0measured in radians in much the same way as we have found them using degrees. In both cases, we find coterminal angles by adding or subtracting one or more full rotations.\r\n<div>\r\n\r\n<strong>Given an angle greater than<\/strong> [latex]2\\pi [\/latex], <strong>find a coterminal angle between 0 and<\/strong> [latex]2\\pi [\/latex].\r\n<ol>\r\n \t<li>Subtract [latex]2\\pi [\/latex] from the given angle.<\/li>\r\n \t<li>If the result is still greater than [latex]2\\pi [\/latex], subtract [latex]2\\pi [\/latex] again until the result is between [latex]0[\/latex] and [latex]2\\pi [\/latex].<\/li>\r\n \t<li>The resulting angle is coterminal with the original angle.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>EXAMPLE 7: FINDING COTERMINAL ANGLES USING RADIANS<\/h3>\r\nFind an angle [latex]\\beta [\/latex] that is coterminal with [latex]\\frac{19\\pi }{4}[\/latex], where [latex]0\\le \\beta &lt;2\\pi [\/latex].\r\n\r\n[reveal-answer q=\"742225\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"742225\"]\r\n\r\nWhen working in degrees, we found coterminal angles by adding or subtracting 360 degrees, a full rotation. Likewise, in radians, we can find coterminal angles by adding or subtracting full rotations of [latex]2\\pi [\/latex] radians:\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\frac{19\\pi }{4}-2\\pi =\\frac{19\\pi }{4}-\\frac{8\\pi }{4} =\\frac{11\\pi }{4} \\end{align}[\/latex]<\/p>\r\nThe angle [latex]\\frac{11\\pi }{4}[\/latex] is coterminal, but not less than [latex]2\\pi [\/latex], so we subtract another rotation:\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\frac{11\\pi }{4}-2\\pi =\\frac{11\\pi }{4}-\\frac{8\\pi }{4} =\\frac{3\\pi }{4} \\end{align}[\/latex]<\/p>\r\nThe angle [latex]\\frac{3\\pi }{4}[\/latex] is coterminal with [latex]\\frac{19\\pi }{4}[\/latex], as shown in Figure 20.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/923\/2015\/04\/25180232\/CNX_Precalc_Figure_05_01_0232.jpg\" alt=\"A graph showing a circle and the equivalence between angles of 3pi\/4 radians and 19pi\/4 radians.\" width=\"487\" height=\"383\" \/> <b>Figure 20<\/b>[\/caption]\r\n\r\n[\/hidden-answer]<b><\/b>\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>TRY IT 11<\/h3>\r\nFind an angle of measure [latex]\\theta [\/latex] that is coterminal with an angle of measure [latex]-\\frac{17\\pi }{6}[\/latex] where [latex]0\\le \\theta &lt;2\\pi [\/latex].\r\n\r\n[reveal-answer q=\"436085\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"436085\"]\r\n\r\n[latex]\\frac{7\\pi }{6}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]99894[\/ohm_question]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<h2>Determining the Length of an Arc<\/h2>\r\n<\/section>Recall that the <strong>radian measure<\/strong> [latex]\\theta [\/latex] of an angle was defined as the ratio of the <strong>arc length<\/strong> [latex]s[\/latex] of a circular arc to the radius [latex]r[\/latex] of the circle, [latex]\\theta =\\frac{s}{r}[\/latex]. From this relationship, we can find arc length along a circle, given an angle.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Arc Length on a Circle<\/h3>\r\nIn a circle of radius <em>r<\/em>, the length of an arc [latex]s[\/latex] subtended by an angle with measure [latex]\\theta [\/latex] in radians, shown in Figure 20, is\r\n<div style=\"text-align: center;\">\r\n\r\n[latex]s=r\\theta [\/latex]\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"349\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003450\/CNX_Precalc_Figure_05_01_024F2.jpg\" alt=\"Illustration of circle with angle theta, radius r, and arc with length s.\" width=\"349\" height=\"348\" \/> <b>Figure 20<\/b>[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a circle of radius [latex]r[\/latex], calculate the length [latex]s[\/latex] of the arc subtended by a given angle of measure [latex]\\theta [\/latex].<\/h3>\r\n<ol>\r\n \t<li>If necessary, convert [latex]\\theta [\/latex] to radians.<\/li>\r\n \t<li>Multiply the radius [latex]r[\/latex] by the radian measure of [latex]\\theta :s=r\\theta [\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section><section>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 8: Finding the Length of an Arc<\/h3>\r\nAssume the orbit of Mercury around the sun is a perfect circle. Mercury is approximately 36 million miles from the sun.\r\n<ol>\r\n \t<li>In one Earth day, Mercury completes 0.0114 of its total revolution. How many miles does it travel in one day?<\/li>\r\n \t<li>Use your answer from part (a) to determine the radian measure for Mercury\u2019s movement in one Earth day.<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"107850\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"107850\"]\r\n<ol>\r\n \t<li>Let\u2019s begin by finding the circumference of Mercury\u2019s orbit.\r\n<div style=\"text-align: center;\">[latex]\\begin{align}C&amp;=2\\pi r \\\\ &amp;=2\\pi \\left(36\\text{ million miles}\\right) \\\\ &amp;\\approx 226\\text{ million miles} \\end{align}[\/latex]<\/div>\r\nSince Mercury completes 0.0114 of its total revolution in one Earth day, we can now find the distance traveled:\r\n<div style=\"text-align: center;\">[latex]\\left(0.0114\\right)226\\text{ million miles = 2}\\text{.58 million miles}[\/latex]<\/div><\/li>\r\n \t<li>Now, we convert to radians:\r\n<div style=\"text-align: center;\">[latex]\\begin{align}\\text{radian}&amp;=\\frac{\\text{arclength}}{\\text{radius}} \\\\ &amp;=\\frac{2.\\text{58 million miles}}{36\\text{ million miles}} \\\\ &amp;=0.0717 \\end{align}[\/latex]<\/div><\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nFind the arc length along a circle of radius 10 units subtended by an angle of 215\u00b0.\r\n\r\n[reveal-answer q=\"188778\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"188778\"]\r\n\r\n[latex]\\frac{215\\pi }{18}=37.525\\text{ units}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/section><\/section>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]172921[\/ohm_question]\r\n\r\n<\/div>\r\n<span style=\"color: #077fab; font-size: 1.15em; font-weight: 600;\">Finding the Area of a Sector of a Circle<\/span>\r\n\r\n<section><section>In addition to arc length, we can also use angles to find the area of a <strong>sector of a circle<\/strong>. A sector is a region of a circle bounded by two radii and the intercepted arc, like a slice of pizza or pie. Recall that the area of a circle with radius [latex]r[\/latex] can be found using the formula [latex]A=\\pi {r}^{2}[\/latex]. If the two radii form an angle of [latex]\\theta [\/latex], measured in radians, then [latex]\\frac{\\theta }{2\\pi }[\/latex] is the ratio of the angle measure to the measure of a full rotation and is also, therefore, the ratio of the area of the sector to the area of the circle. Thus, the <strong>area of a sector<\/strong> is the fraction [latex]\\frac{\\theta }{2\\pi }[\/latex]\u00a0multiplied by the entire area. (Always remember that this formula only applies if [latex]\\theta [\/latex] is in radians.)\r\n<div style=\"text-align: center;\">[latex]\\begin{align}\\text{Area of sector}&amp;=\\left(\\frac{\\theta }{2\\pi }\\right)\\pi {r}^{2} \\\\ &amp;=\\frac{\\theta \\pi {r}^{2}}{2\\pi } \\\\ &amp;=\\frac{1}{2}\\theta {r}^{2} \\end{align}[\/latex]<\/div>\r\n<div class=\"textbox\">\r\n<h3>A General Note: Area of a Sector<\/h3>\r\nThe <strong>area of a sector<\/strong> of a circle with radius [latex]r[\/latex] subtended by an angle [latex]\\theta [\/latex], measured in radians, is\r\n<div>\r\n<p style=\"text-align: center;\">[latex]A=\\frac{1}{2}\\theta {r}^{2}[\/latex]<\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"442\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003452\/CNX_Precalc_Figure_05_01_026F2.jpg\" alt=\"Graph showing a circle with angle theta and radius r, and the area of the slice of circle created by the initial side and terminal side of the angle.\" width=\"442\" height=\"394\" \/> <b>Figure 21.<\/b> The area of the sector equals half the square of the radius times the central angle measured in radians.[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a circle of radius [latex]r[\/latex], find the area of a sector defined by a given angle [latex]\\theta [\/latex].<\/h3>\r\n<ol>\r\n \t<li>If necessary, convert [latex]\\theta [\/latex] to radians.<\/li>\r\n \t<li>Multiply half the radian measure of [latex]\\theta [\/latex] by the square of the radius [latex]r:\\text{ } A=\\frac{1}{2}\\theta {r}^{2}[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 9: Finding the Area of a Sector<\/h3>\r\nAn automatic lawn sprinkler sprays a distance of 20 feet while rotating 30 degrees, as shown in Figure 22. What is the area of the sector of grass the sprinkler waters?\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003454\/CNX_Precalc_Figure_05_01_0272.jpg\" alt=\"Illustration of a 30 degree ange with a terminal and initial side with length of 20 feet.\" width=\"487\" height=\"169\" \/> <b>Figure 22.<\/b> The sprinkler sprays 20 ft within an arc of 30\u00b0.[\/caption]\r\n\r\n[reveal-answer q=\"785904\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"785904\"]\r\n\r\nFirst, we need to convert the angle measure into radians. Because 30 degrees is one of our special angles, we already know the equivalent radian measure, but we can also convert:\r\n<p style=\"text-align: center;\">[latex]\\begin{align}30\\text{ degrees}&amp;=30\\cdot \\frac{\\pi }{180} \\\\ &amp;=\\frac{\\pi }{6}\\text{ radians} \\end{align}[\/latex]<\/p>\r\nThe area of the sector is then\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\text{Area} &amp;= \\frac{1}{2}\\left(\\frac{\\pi }{6}\\right){\\left(20\\right)}^{2} \\\\ &amp;\\approx 104.72 \\end{align}[\/latex]<\/p>\r\nSo the area is about [latex]104.72{\\text{ ft}}^{2}[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nIn central pivot irrigation, a large irrigation pipe on wheels rotates around a center point. A farmer has a central pivot system with a radius of 400 meters. If water restrictions only allow her to water 150 thousand square meters a day, what angle should she set the system to cover? Write the answer in radian measure to two decimal places.\r\n\r\n[reveal-answer q=\"495928\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"495928\"]\r\n\r\n1.88\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video you will see how to calculate arc length and area of a sector of a circle.\r\n\r\nhttps:\/\/youtu.be\/zD4CsKIYEHo\r\n<h2>Use Linear and Angular Speed to Describe Motion on a Circular Path<\/h2>\r\n<\/section><\/section>In addition to finding the area of a sector, we can use angles to describe the speed of a moving object. An object traveling in a circular path has two types of speed.<strong> Linear speed<\/strong> is speed along a straight path and can be determined by the distance it moves along (its <strong>displacement<\/strong>) in a given time interval. For instance, if a wheel with radius 5 inches rotates once a second, a point on the edge of the wheel moves a distance equal to the circumference, or [latex]10\\pi [\/latex] inches, every second. So the linear speed of the point is [latex]10\\pi [\/latex] in.\/s. The equation for linear speed is as follows where [latex]v[\/latex] is linear speed, [latex]s[\/latex] is displacement, and [latex]t[\/latex]\r\nis time.\r\n<div style=\"text-align: center;\">[latex]v=\\frac{s}{t}[\/latex]<\/div>\r\n<strong>Angular speed<\/strong> results from circular motion and can be determined by the angle through which a point rotates in a given time interval. In other words, angular speed is angular rotation per unit time. So, for instance, if a gear makes a full rotation every 4 seconds, we can calculate its angular speed as [latex]\\frac{360\\text{ degrees}}{4\\text{ seconds}}=[\/latex] 90 degrees per second. Angular speed can be given in radians per second, rotations per minute, or degrees per hour for example. The equation for angular speed is as follows, where [latex]\\omega [\/latex] (read as omega) is angular speed, [latex]\\theta [\/latex] is the angle traversed, and [latex]t[\/latex] is time.\r\n<div style=\"text-align: center;\">[latex]\\omega =\\frac{\\theta }{t}[\/latex]<\/div>\r\nCombining the definition of angular speed with the arc length equation, [latex]s=r\\theta [\/latex], we can find a relationship between angular and linear speeds. The angular speed equation can be solved for [latex]\\theta [\/latex], giving [latex]\\theta =\\omega t[\/latex]. Substituting this into the arc length equation gives:\r\n<div style=\"text-align: center;\">[latex]\\begin{align}s&amp;=r\\theta \\\\ &amp;=r\\omega t \\end{align}[\/latex]<\/div>\r\nSubstituting this into the linear speed equation gives:\r\n<div style=\"text-align: center;\">[latex]\\begin{align} v&amp;=\\frac{s}{t} \\\\ &amp;=\\frac{r\\omega t}{t} \\\\ &amp;=r\\omega \\end{align}[\/latex]<\/div>\r\n<div class=\"textbox\">\r\n<h3>A General Note: Angular and Linear Speed<\/h3>\r\nAs a point moves along a circle of radius [latex]r[\/latex], its <strong>angular speed<\/strong>, [latex]\\omega [\/latex], is the angular rotation [latex]\\theta [\/latex] per unit time, [latex]t[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\omega =\\frac{\\theta }{t}[\/latex]<\/p>\r\nThe <strong>linear speed<\/strong>. [latex]v[\/latex], of the point can be found as the distance traveled, arc length [latex]s[\/latex], per unit time, [latex]t[\/latex].\r\n<p style=\"text-align: center;\">[latex]v=\\frac{s}{t}[\/latex]<\/p>\r\nWhen the angular speed is measured in radians per unit time, linear speed and angular speed are related by the equation\r\n<p style=\"text-align: center;\">[latex]v=r\\omega[\/latex]<\/p>\r\nThis equation states that the angular speed in radians, [latex]\\omega [\/latex], representing the amount of rotation occurring in a unit of time, can be multiplied by the radius [latex]r[\/latex] to calculate the total arc length traveled in a unit of time, which is the definition of linear speed.\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given the amount of angle rotation and the time elapsed, calculate the angular speed.<strong>\r\n<\/strong><\/h3>\r\n<ol>\r\n \t<li>If necessary, convert the angle measure to radians.<\/li>\r\n \t<li>Divide the angle in radians by the number of time units elapsed: [latex]\\omega =\\frac{\\theta }{t}[\/latex].<\/li>\r\n \t<li>The resulting speed will be in radians per time unit.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 10: Finding Angular Speed<\/h3>\r\nA water wheel, shown in Figure 23, completes 1 rotation every 5 seconds. Find the angular speed in radians per second.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003457\/CNX_Precalc_Figure_05_01_028n.jpg\" alt=\"Illustration of a water wheel.\" width=\"487\" height=\"359\" \/> <b>Figure 23<\/b>[\/caption]\r\n\r\n[reveal-answer q=\"602649\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"602649\"]\r\n\r\nThe wheel completes 1 rotation, or passes through an angle of [latex]2\\pi [\/latex]\u00a0radians in 5 seconds, so the angular speed would be [latex]\\omega =\\frac{2\\pi }{5}\\approx 1.257[\/latex] radians per second.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nAn old vinyl record is played on a turntable rotating clockwise at a rate of 45 rotations per minute. Find the angular speed in radians per second.\r\n\r\n[reveal-answer q=\"737350\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"737350\"]\r\n\r\n[latex]\\frac{-3\\pi }{2}[\/latex]\u00a0rad\/s\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]155235[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given the radius of a circle, an angle of rotation, and a length of elapsed time, determine the linear speed.<strong>\r\n<\/strong><\/h3>\r\n<ol>\r\n \t<li>Convert the total rotation to radians if necessary.<\/li>\r\n \t<li>Divide the total rotation in radians by the elapsed time to find the angular speed: apply [latex]\\omega =\\frac{\\theta }{t}[\/latex].<\/li>\r\n \t<li>Multiply the angular speed by the length of the radius to find the linear speed, expressed in terms of the length unit used for the radius and the time unit used for the elapsed time: apply [latex]v=r\\mathrm{\\omega}[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\nhttps:\/\/youtu.be\/bfWkgA5GSE0\r\n<div class=\"textbox shaded\">\r\n<h3>Example 11: Finding a Linear Speed<\/h3>\r\nA bicycle has wheels 28 inches in diameter. A tachometer determines the wheels are rotating at 180 RPM (revolutions per minute). Find the speed the bicycle is traveling down the road.\r\n\r\n[reveal-answer q=\"461618\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"461618\"]\r\n\r\nHere, we have an angular speed and need to find the corresponding linear speed, since the linear speed of the outside of the tires is the speed at which the bicycle travels down the road.\r\n\r\nWe begin by converting from rotations per minute to radians per minute. It can be helpful to utilize the units to make this conversion:\r\n<p style=\"text-align: center;\">[latex]180\\frac{\\cancel{\\text{rotations}}}{\\text{minute}}\\cdot \\frac{2\\pi \\text{radians}}{\\cancel{\\text{rotation}}}=360\\pi \\frac{\\text{radians}}{\\text{minute}}[\/latex]<\/p>\r\nUsing the formula from above along with the radius of the wheels, we can find the linear speed:\r\n<p style=\"text-align: center;\">[latex]\\begin{align} v&amp;=\\left(14\\text{inches}\\right)\\left(360\\pi \\frac{\\text{radians}}{\\text{minute}}\\right) \\\\ &amp;=5040\\pi \\frac{\\text{inches}}{\\text{minute}} \\end{align}[\/latex]<\/p>\r\nRemember that radians are a unitless measure, so it is not necessary to include them.\r\n\r\nFinally, we may wish to convert this linear speed into a more familiar measurement, like miles per hour.\r\n<p style=\"text-align: center;\">[latex]5040\\pi \\frac{\\cancel{\\text{inches}}}{\\cancel{\\text{minute}}}\\cdot \\frac{\\text{1}\\cancel{\\text{feet}}}{\\text{12}\\cancel{\\text{inches}}}\\cdot \\frac{\\text{1 mile}}{\\text{5280}\\cancel{\\text{feet}}}\\cdot \\frac{\\text{60}\\cancel{\\text{minutes}}}{\\text{1 hour}}\\approx 14.99\\text{miles per hour (mph)}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nA satellite is rotating around Earth at 0.25 radians per hour at an altitude of 242 km above Earth. If the radius of Earth is 6378 kilometers, find the linear speed of the satellite in kilometers per hour.\r\n\r\n[reveal-answer q=\"851398\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"851398\"]\r\n\r\n1655 kilometers per hour\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<section id=\"fs-id1165135369540\" class=\"key-equations\">\r\n<h2>Key Equations<\/h2>\r\n<table id=\"eip-id1165134342478\" summary=\"..\">\r\n<tbody>\r\n<tr>\r\n<td>arc length<\/td>\r\n<td>[latex]s=r\\theta [\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>area of a sector<\/td>\r\n<td>[latex]A=\\frac{1}{2}\\theta {r}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>angular speed<\/td>\r\n<td>[latex]\\omega =\\frac{\\theta }{t}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>linear speed<\/td>\r\n<td>[latex]v=\\frac{s}{t}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>linear speed related to angular speed<\/td>\r\n<td>[latex]v=r\\omega [\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/section><section id=\"fs-id1165135431038\" class=\"key-concepts\">\r\n<h2>Key Concepts<\/h2>\r\n<ul id=\"fs-id1165135431044\">\r\n \t<li>An angle is formed from the union of two rays, by keeping the initial side fixed and rotating the terminal side. The amount of rotation determines the measure of the angle.<\/li>\r\n \t<li>An angle is in standard position if its vertex is at the origin and its initial side lies along the positive <em>x<\/em>-axis. A positive angle is measured counterclockwise from the initial side and a negative angle is measured clockwise.<\/li>\r\n \t<li>To draw an angle in standard position, draw the initial side along the positive <em>x<\/em>-axis and then place the terminal side according to the fraction of a full rotation the angle represents.<\/li>\r\n \t<li>In addition to degrees, the measure of an angle can be described in radians.<\/li>\r\n \t<li>To convert between degrees and radians, use the proportion [latex]\\frac{\\theta }{180}=\\frac{{\\theta }^{R}}{\\pi }[\/latex].<\/li>\r\n \t<li>Two angles that have the same terminal side are called coterminal angles.<\/li>\r\n \t<li>We can find coterminal angles by adding or subtracting 360\u00b0 or [latex]2\\pi [\/latex].<\/li>\r\n \t<li>Coterminal angles can be found using radians just as they are for degrees.<\/li>\r\n \t<li>The length of a circular arc is a fraction of the circumference of the entire circle.<\/li>\r\n \t<li>The area of sector is a fraction of the area of the entire circle.<\/li>\r\n \t<li>An object moving in a circular path has both linear and angular speed.<\/li>\r\n \t<li>The angular speed of an object traveling in a circular path is the measure of the angle through which it turns in a unit of time.<\/li>\r\n \t<li>The linear speed of an object traveling along a circular path is the distance it travels in a unit of time.<\/li>\r\n<\/ul>\r\n<div>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1165135487231\" class=\"definition\">\r\n \t<dt>angle<\/dt>\r\n \t<dd id=\"fs-id1165135487236\">the union of two rays having a common endpoint<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135487241\" class=\"definition\">\r\n \t<dt>angular speed<\/dt>\r\n \t<dd id=\"fs-id1165135487246\">the angle through which a rotating object travels in a unit of time<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135487250\" class=\"definition\">\r\n \t<dt>arc length<\/dt>\r\n \t<dd id=\"fs-id1165135170996\">the length of the curve formed by an arc<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135171000\" class=\"definition\">\r\n \t<dt>area of a sector<\/dt>\r\n \t<dd id=\"fs-id1165135171006\">area of a portion of a circle bordered by two radii and the intercepted arc; the fraction [latex]\\frac{\\theta }{2\\pi }[\/latex] multiplied by the area of the entire circle<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165134223298\" class=\"definition\">\r\n \t<dt>coterminal angles<\/dt>\r\n \t<dd id=\"fs-id1165134223303\">description of positive and negative angles in standard position sharing the same terminal side<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165134223307\" class=\"definition\">\r\n \t<dt>degree<\/dt>\r\n \t<dd id=\"fs-id1165134084943\">a unit of measure describing the size of an angle as one-360th of a full revolution of a circle<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165134084947\" class=\"definition\">\r\n \t<dt>initial side<\/dt>\r\n \t<dd id=\"fs-id1165134084952\">the side of an angle from which rotation begins<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165134084957\" class=\"definition\">\r\n \t<dt>linear speed<\/dt>\r\n \t<dd id=\"fs-id1165134084962\">the distance along a straight path a rotating object travels in a unit of time; determined by the arc length<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135580273\" class=\"definition\">\r\n \t<dt>measure of an angle<\/dt>\r\n \t<dd id=\"fs-id1165135580278\">the amount of rotation from the initial side to the terminal side<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135580283\" class=\"definition\">\r\n \t<dt>negative angle<\/dt>\r\n \t<dd id=\"fs-id1165135580288\">description of an angle measured clockwise from the positive <em>x<\/em>-axis<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135580297\" class=\"definition\">\r\n \t<dt>positive angle<\/dt>\r\n \t<dd id=\"fs-id1165135519264\">description of an angle measured counterclockwise from the positive <em>x<\/em>-axis<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135519274\" class=\"definition\">\r\n \t<dt>quadrantal angle<\/dt>\r\n \t<dd id=\"fs-id1165135519280\">an angle whose terminal side lies on an axis<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135519284\" class=\"definition\">\r\n \t<dt>radian measure<\/dt>\r\n \t<dd id=\"fs-id1165135519289\">the ratio of the arc length formed by an angle divided by the radius of the circle<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135193270\" class=\"definition\">\r\n \t<dt>radian<\/dt>\r\n \t<dd id=\"fs-id1165135193275\">the measure of a central angle of a circle that intercepts an arc equal in length to the radius of that circle<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135193281\" class=\"definition\">\r\n \t<dt>ray<\/dt>\r\n \t<dd id=\"fs-id1165135193286\">one point on a line and all points extending in one direction from that point; one side of an angle<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135193291\" class=\"definition\">\r\n \t<dt>reference angle<\/dt>\r\n \t<dd id=\"fs-id1165134298995\">the measure of the acute angle formed by the terminal side of the angle and the horizontal axis<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165134299000\" class=\"definition\">\r\n \t<dt>standard position<\/dt>\r\n \t<dd id=\"fs-id1165134299005\">the position of an angle having the vertex at the origin and the initial side along the positive <em>x<\/em>-axis<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165134299015\" class=\"definition\">\r\n \t<dt>terminal side<\/dt>\r\n \t<dd id=\"fs-id1165134299020\">the side of an angle at which rotation ends<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165133309879\" class=\"definition\">\r\n \t<dt>vertex<\/dt>\r\n \t<dd id=\"fs-id1165133309884\">the common endpoint of two rays that form an angle<\/dd>\r\n<\/dl>\r\n<\/div>\r\n<\/section>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Draw angles in standard position.<\/li>\n<li>Convert between degrees and radians.<\/li>\n<li>Convert between degrees\/radians and Degrees Minutes Seconds (DMS)<\/li>\n<li>Find coterminal angles.<\/li>\n<li>Find the length of a circular arc.<\/li>\n<li>Find the area of a sector of a circle.<\/li>\n<li>Use linear and angular speed to describe motion on a circular path.<\/li>\n<\/ul>\n<\/div>\n<p>A golfer swings to hit a ball over a sand trap and onto the green. An airline pilot maneuvers a plane toward a narrow runway. A dress designer creates the latest fashion. What do they all have in common? They all work with angles, and so do all of us at one time or another. Sometimes we need to measure angles exactly with instruments. Other times we estimate them or judge them by eye. Either way, the proper angle can make the difference between success and failure in many undertakings. In this section, we will examine properties of angles.<\/p>\n<h2>Draw angles in standard position<\/h2>\n<div id=\"post-10770\" class=\"standard post-10770 chapter type-chapter status-publish hentry\">\n<div class=\"entry-content\">\n<p>Properly defining an angle first requires that we define a ray. A\u00a0<strong>ray<\/strong>\u00a0consists of one point on a line and all points extending in one direction from that point. The first point is called the\u00a0<strong>endpoint<\/strong>\u00a0of the ray. We can refer to a specific ray by stating its endpoint and any other point on it. The ray in Figure 1\u00a0can be named as ray EF, or in symbolic form [latex]\\overrightarrow{EF}[\/latex].<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/923\/2015\/04\/25180159\/CNX_Precalc_Figure_05_01_0012.jpg\" alt=\"Illustration of Ray EF, with point F and endpoint E.\" width=\"487\" height=\"173\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 1<\/b><\/p>\n<\/div>\n<p>An\u00a0<strong>angle<\/strong>\u00a0is the union of two rays having a common endpoint. The endpoint is called the\u00a0<strong>vertex<\/strong>\u00a0of the angle, and the two rays are the sides of the angle. The angle in Figure 2\u00a0is formed from\u00a0[latex]\\overrightarrow{ED}[\/latex] and\u00a0[latex]\\overrightarrow{EF}[\/latex]. Angles can be named using a point on each ray and the vertex, such as angle [latex]{DEF}[\/latex], or in symbol form [latex]\\angle{DEF}[\/latex].<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/923\/2015\/04\/25180201\/CNX_Precalc_Figure_05_01_0022.jpg\" alt=\"Illustration of Angle DEF, with vertex E and points D and F.\" width=\"487\" height=\"246\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 2<\/b><\/p>\n<\/div>\n<p>Greek letters are often used as variables for the measure of an angle. The table below\u00a0is a list of Greek letters commonly used to represent angles, and a sample angle is shown in Figure 2.<\/p>\n<table id=\"Table_05_01_01\" style=\"height: 66px;\" summary=\"Two rows and five columns. First row shows symbols for theta, phi, alpha, beta, and gamma. Second row spells out name for each symbol.\">\n<tbody>\n<tr style=\"height: 53px;\">\n<td style=\"height: 53px; width: 130.313px;\"><span class=\"katex-html\"><span class=\"base textstyle uncramped\"><span class=\"reset-textstyle displaystyle textstyle uncramped\"><span class=\"mord mathit\">[latex]\\theta[\/latex]<\/span><\/span><\/span><\/span><\/td>\n<td style=\"height: 53px; width: 180.313px;\"><span class=\"katex-html\"><span class=\"base textstyle uncramped\"><span class=\"reset-textstyle displaystyle textstyle uncramped\"><span class=\"mord mathit\">[latex]\\phi \\text{ or }\\varphi[\/latex]<\/span><\/span><\/span><\/span><\/td>\n<td style=\"height: 53px; width: 106.676px;\"><span class=\"katex-html\"><span class=\"base textstyle uncramped\"><span class=\"reset-textstyle displaystyle textstyle uncramped\"><span class=\"mord mathit\">[latex]\\alpha[\/latex]<\/span><\/span><\/span><\/span><\/td>\n<td style=\"height: 53px; width: 102.131px;\"><span class=\"katex-html\"><span class=\"base textstyle uncramped\"><span class=\"reset-textstyle displaystyle textstyle uncramped\"><span class=\"mord mathit\">[latex]\\beta[\/latex]<\/span><\/span><\/span><\/span><\/td>\n<td style=\"height: 53px; width: 122.131px;\"><span class=\"katex-html\"><span class=\"base textstyle uncramped\"><span class=\"reset-textstyle displaystyle textstyle uncramped\"><span class=\"mord mathit\">[latex]\\gamma[\/latex]<\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<tr style=\"height: 13px;\">\n<td style=\"height: 13px; width: 130.313px;\">theta<\/td>\n<td style=\"height: 13px; width: 180.313px;\">phi<\/td>\n<td style=\"height: 13px; width: 106.676px;\">alpha<\/td>\n<td style=\"height: 13px; width: 102.131px;\">beta<\/td>\n<td style=\"height: 13px; width: 122.131px;\">gamma<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<figure id=\"Figure_05_01_003\" class=\"small\">\n<figure id=\"Figure_05_01_003\" class=\"small\"><span id=\"fs-id1165135192939\"> <img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/923\/2015\/04\/25180202\/CNX_Precalc_Figure_05_01_0032.jpg\" alt=\"Illustration of angle theta.\" \/><\/span><\/figure>\n<p style=\"text-align: center;\"><strong>Figure 3.\u00a0<\/strong>Angle theta, shown as [latex]\\angle \\theta[\/latex]<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/923\/2015\/04\/25180203\/CNX_Precalc_Figure_05_01_0042.jpg\" alt=\"Illustration of an angle with labels for initial side, terminal side, and vertex.\" width=\"487\" height=\"247\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 4<\/b><\/p>\n<\/div>\n<\/figure>\n<p>Angle creation is a dynamic process. We start with two rays lying on top of one another. We leave one fixed in place, and rotate the other. The fixed ray is the\u00a0<strong>initial side<\/strong>, and the rotated ray is the\u00a0<strong>terminal side<\/strong>. In order to identify the different sides, we indicate the rotation with a small arc and arrow close to the vertex as in Figure 4.<span id=\"fs-id1165137737991\"><br \/>\n<\/span><\/p>\n<p>The following video provides an illustration of angles in standard position.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Animation:  Angles in Standard Position\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/hpIjaKLOo6o?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>As we discussed at the beginning of the section, there are many applications for angles, but in order to use them correctly, we must be able to measure them. The\u00a0<strong>measure of an angle<\/strong>\u00a0is the amount of rotation from the initial side to the terminal side. Probably the most familiar unit of angle measurement is the degree. One\u00a0<strong>degree<\/strong>\u00a0is [latex]\\frac{1}{360}[\/latex] of a circular rotation, so a complete circular rotation contains 360 degrees. An angle measured in degrees should always include the unit \u201cdegrees\u201d after the number, or include the degree symbol \u00b0. For example, 90 degrees = 90\u00b0.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/923\/2015\/04\/25180205\/CNX_Precalc_Figure_05_01_0052.jpg\" alt=\"Graph of an angle in standard position with labels for the initial side and terminal side.\" width=\"487\" height=\"417\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 5<\/b><\/p>\n<\/div>\n<p>To formalize our work, we will begin by drawing angles on an <em>x<\/em>&#8211;<em>y<\/em> coordinate plane. Angles can occur in any position on the coordinate plane, but for the purpose of comparison, the convention is to illustrate them in the same position whenever possible. An angle is in <strong>standard position<\/strong> if its vertex is located at the origin, and its initial side extends along the positive <em>x<\/em>-axis.\u00a0<span id=\"fs-id1165137804556\"><br \/>\n<\/span><\/p>\n<p>If the angle is measured in a counterclockwise direction from the initial side to the terminal side, the angle is said to be a <strong>positive angle<\/strong>. If the angle is measured in a clockwise direction, the angle is said to be a <strong>negative angl<\/strong><strong style=\"font-size: 1em;\">e<\/strong><span style=\"font-size: 1em;\">.<\/span><\/p>\n<div style=\"width: 741px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/923\/2015\/04\/25180207\/CNX_Precalc_Figure_05_01_0062.jpg\" alt=\"Side by side graphs. Graph on the left is a 90 degree angle and graph on the right is a 360 degree angle. Terminal side and initial side are labeled for both graphs.\" width=\"731\" height=\"365\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 6<\/b><\/p>\n<\/div>\n<p>Drawing an angle in standard position always starts the same way\u2014draw the initial side along the positive <em>x<\/em>-axis. To place the terminal side of the angle, we must calculate the fraction of a full rotation the angle represents. We do that by dividing the angle measure in degrees by 360\u00b0. For example, to draw a 90\u00b0 angle, we calculate that [latex]\\frac{90^\\circ }{360^\\circ }=\\frac{1}{4}[\/latex]. So, the terminal side will be one-fourth of the way around the circle, moving counterclockwise from the positive <em>x<\/em>-axis. To draw a 360\u00b0 angle, we calculate that [latex]\\frac{360^\\circ }{360^\\circ }=1[\/latex]. So the terminal side will be 1 complete rotation around the circle, moving counterclockwise from the positive <em>x<\/em>-axis. In this case, the initial side and the terminal side overlap.<span id=\"fs-id1165134042853\"><br \/>\n<\/span><\/p>\n<p>Since we define an angle in <strong>standard position<\/strong> by its terminal side, we have a special type of angle whose terminal side lies on an axis, a <strong>quadrantal angle<\/strong>. This type of angle can have a measure of 0\u00b0, 90\u00b0, 180\u00b0, 270\u00b0 or 360\u00b0.<\/p>\n<div style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/923\/2015\/04\/25180209\/CNX_Precalc_Figure_05_01_0182.jpg\" alt=\"Four side by side graphs. First graph shows angle of 0 degrees. Second graph shows an angle of 90 degrees. Third graph shows an angle of 180 degrees. Fourth graph shows an angle of 270 degrees.\" width=\"975\" height=\"237\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 7.<\/b> Quadrantal angles have a terminal side that lies along an axis. Examples are shown.<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>A GENERAL NOTE: QUADRANTAL ANGLES<\/h3>\n<p>Quadrantal angles are angles whose terminal side lies on an axis, including 0\u00b0, 90\u00b0, 180\u00b0, 270\u00b0, or 360\u00b0.<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>HOW TO: GIVEN AN ANGLE MEASURE IN DEGREES, DRAW THE ANGLE IN STANDARD POSITION.<strong><br \/>\n<\/strong><\/h3>\n<ol>\n<li>Express the angle measure as a fraction of 360\u00b0.<\/li>\n<li>Reduce the fraction to simplest form.<\/li>\n<li>Draw an angle that contains that same fraction of the circle, beginning on the positive\u00a0<em>x<\/em>-axis and moving counterclockwise for positive angles and clockwise for negative angles.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>EXAMPLE 1: DRAWING AN ANGLE IN STANDARD POSITION MEASURED IN DEGREES<\/h3>\n<ol>\n<li>Sketch an angle of 30\u00b0 in standard position.<\/li>\n<li>Sketch an angle of \u2212135\u00b0 in standard position.<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q899324\">Show Solution<\/span><\/p>\n<div id=\"q899324\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>Divide the angle measure by 360\u00b0.\n<div style=\"text-align: center;\">[latex]\\frac{30^\\circ }{360^\\circ }=\\frac{1}{12}[\/latex]<\/div>\n<p>To rewrite the fraction in a more familiar fraction, we can recognize that<\/p>\n<div style=\"text-align: center;\">[latex]\\frac{1}{12}=\\frac{1}{3}\\left(\\frac{1}{4}\\right)[\/latex]<\/div>\n<p>One-twelfth equals one-third of a quarter, so by dividing a quarter rotation into thirds, we can sketch a line at 30\u00b0 as in Figure 8.<span id=\"fs-id1165137784208\"><br \/>\n<\/span><\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/923\/2015\/04\/25180211\/CNX_Precalc_Figure_05_01_0072.jpg\" alt=\"Graph of a 30 degree angle.\" width=\"487\" height=\"383\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 8<\/b><\/p>\n<\/div>\n<p>&nbsp;<\/li>\n<li>Divide the angle measure by 360\u00b0.\n<div style=\"text-align: center;\">[latex]\\frac{-135^\\circ }{360^\\circ }=-\\frac{3}{8}[\/latex]<\/div>\n<p>In this case, we can recognize that<\/p>\n<div style=\"text-align: center;\">[latex]-\\frac{3}{8}=-\\frac{3}{2}\\left(\\frac{1}{4}\\right)[\/latex]<\/div>\n<\/li>\n<li>Negative three-eighths is one and one-half times a quarter, so we place a line by moving clockwise one full quarter and one-half of another quarter, as in Figure 9.\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/923\/2015\/04\/25180213\/CNX_Precalc_Figure_05_01_0082.jpg\" alt=\"Graph of a negative 135 degree angle.\" width=\"487\" height=\"383\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 9<\/b><\/p>\n<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p><b><\/b><\/p>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>TRY IT 1<\/h3>\n<p>Show an angle of 240\u00b0 on a circle in standard position.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q862928\">Show Solution<\/span><\/p>\n<div id=\"q862928\" class=\"hidden-answer\" style=\"display: none\">\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003504\/CNX_Precalc_Figure_05_01_0092.jpg\" alt=\"Graph of a 240 degree angle.\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div>Watch this video for more examples of determining angles of rotation.<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Examples:  Determine Angles of Rotation\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/0yHDfG2m-44?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2 class=\"\">Converting Between Degrees and Radians<\/h2>\n<p>Dividing a circle into 360 parts is an arbitrary choice, although it creates the familiar degree measurement. We may choose other ways to divide a circle. To find another unit, think of the process of drawing a circle. Imagine that you stop before the circle is completed. The portion that you drew is referred to as an arc. An\u00a0<strong>arc<\/strong>\u00a0may be a portion of a full circle, a full circle, or more than a full circle, represented by more than one full rotation. The length of the arc around an entire circle is called the\u00a0<strong>circumference<\/strong>\u00a0of that circle.<\/p>\n<p>The circumference of a circle is [latex]C=2\\pi r[\/latex]. If we divide both sides of this equation by [latex]r[\/latex], we create the ratio of the circumference to the radius, which is always [latex]2\\pi[\/latex] regardless of the length of the radius. So the circumference of any circle is [latex]2\\pi \\approx 6.28[\/latex] times the length of the radius. That means that if we took a string as long as the radius and used it to measure consecutive lengths around the circumference, there would be room for six full string-lengths and a little more than a quarter of a seventh, as shown in Figure 10.<\/p>\n<p><span id=\"fs-id1165137769898\">\u00a0<\/span><\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/923\/2015\/04\/25180217\/CNX_Precalc_Figure_05_01_0102.jpg\" alt=\"Illustration of a circle showing the number of radians in a circle.\" width=\"487\" height=\"340\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 10<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165137667767\">This brings us to our new angle measure. One <span class=\"no-emphasis\">radian<\/span> is the measure of a central angle of a circle that intercepts an arc equal in length to the radius of that circle. A central angle is an angle formed at the center of a circle by two radii. Because the total circumference equals [latex]2\\pi[\/latex] times the radius, a full circular rotation is [latex]2\\pi[\/latex] radians. So<\/p>\n<div id=\"eip-246\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{gathered} 2\\pi \\text{ radians}={360}^{\\circ } \\\\ \\pi \\text{ radians}=\\frac{{360}^{\\circ }}{2}={180}^{\\circ } \\\\ 1\\text{ radian}=\\frac{{180}^{\\circ }}{\\pi }\\approx {57.3}^{\\circ } \\end{gathered}[\/latex]<\/div>\n<p>Note that when an angle is described without a specific unit, it refers to radian measure. For example, an angle measure of 3 indicates 3 radians. In fact, radian measure is dimensionless, since it is the quotient of a length (circumference) divided by a length (radius) and the length units cancel out.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/923\/2015\/04\/25180219\/CNX_Precalc_Figure_05_01_0112.jpg\" alt=\"Illustration of a circle with angle t, radius r, and an arc of r.\" width=\"487\" height=\"323\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 11.<\/b> The angle <i>t<\/i> sweeps out a measure of one radian. Note that the length of the intercepted arc is the same as the length of the radius of the circle.<\/p>\n<\/div>\n<h2>Converting Between Degrees and Degrees Minutes Seconds (DMS)<\/h2>\n<p>Using our definition of degree measure, we have that 1\u00b0 represents the measure of an angle which constitutes \u00a0\u00a0of a revolution. Even though it may be hard to draw, it is nonetheless not difficult to imagine an angle with measure smaller than 1\u00b0. There are two ways to subdivide degrees. The first, and most familiar, is decimal degrees. For example, an angle with a measure of 30.5\u00b0 would represent a rotation halfway between 30\u00b0 and 31\u00b0 , or equivalently, of a full rotation. This can be taken to the limit using Calculus so that measures like \u221a2\u00b0 make sense. The second way to divide degrees is the <strong>Degree &#8211; Minute &#8211; Second (DMS)<\/strong> system. In this system, one degree is divided equally into sixty minutes, and in turn, each minute is divided equally into sixty seconds. In symbols, we write 1\u00b0 = 60\u2032 and 1\u2032 = 60\u2032\u2032, from which it follows that 1\u00b0 = 3600\u2032\u2032.<\/p>\n<\/div>\n<div class=\"entry-content\">\n<div class=\"textbox examples\">\n<h3>Examples(converting from Decimal degrees to dms)<\/h3>\n<p>To convert a measure of 42.125\u00b0 to the DMS system, we start by noting that<\/p>\n<p>42.125\u00b0= 42\u00b0 + 0.125\u00b0<\/p>\n<ul>\n<li>Converting the partial amount of degrees to minutes, we find 0.125\u00b0 (60\u2019\/1\u00b0) = 7.5\u2032 = 7\u2032 + 0.5\u2032<\/li>\n<li>Converting the partial amount of minutes to seconds gives 0.5\u2032 (60\u201d\/1\u2019)= 30\u2032\u2032<\/li>\n<li>Putting it all together yields<br \/>\n42.125\u00b0 = 42\u00b0 \u00a0+ 0.125\u00b0<br \/>\n= 42\u00b0 + 7.5\u2032<br \/>\n= 42\u00b0+ 7\u2032 + 0.5\u2032<br \/>\n= 42\u00b0 + 7\u2032 + 30\u2032\u2032<br \/>\n= 42\u00b07\u203230\u2032\u2032<\/li>\n<\/ul>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox examples\">\n<h3>Examples (DMS to decimal degrees)<\/h3>\n<p>convert 117\u00b015\u203245\u2032\u2032 to decimal degrees<\/p>\n<ul>\n<li>compute 15\u2032 ( 1\u00b0\/60\u2032) = (1\/4)\u00b0<\/li>\n<li>Second, 45\u2032\u2032 ( 1\u00b0\/3600\u2032\u2032) = (1\/80)\u00b0<\/li>\n<li>Then we find117\u00b015\u201945\u201d=117\u00b0+15\u2019+45\u201d\n<p>= 117\u00b0+ (1\/4)\u00b0 + (1\/80)\u00b0<\/p>\n<p>= (9381\/80) \u00b0=117.2625\u00b0<\/p>\n<p>&nbsp;<\/li>\n<\/ul>\n<\/div>\n<p>&nbsp;<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Using the TI-84 to Convert Degrees to Degree, Minutes, and Seconds and to Radians\" width=\"500\" height=\"375\" src=\"https:\/\/www.youtube.com\/embed\/m86G_02jdPA?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Relating Arc Lengths to Radius<\/h2>\n<p>An <strong>arc length<\/strong> [latex]s[\/latex] is the length of the curve along the arc. Just as the full circumference of a circle always has a constant ratio to the radius, the arc length produced by any given angle also has a constant relation to the radius, regardless of the length of the radius.<\/p>\n<p>This ratio, called the <strong>radian measure<\/strong>, is the same regardless of the radius of the circle\u2014it depends only on the angle. This property allows us to define a measure of any angle as the ratio of the arc length [latex]s[\/latex] to the radius [latex]r[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{gathered}s=r\\theta \\\\ \\theta =\\frac{s}{r}\\end{gathered}[\/latex]<\/div>\n<p>If [latex]s=r[\/latex], then [latex]\\theta =\\frac{r}{r}=\\text{ 1 radian}\\text{.}[\/latex]<\/p>\n<p><span id=\"fs-id1165135347607\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003429\/CNX_Precalc_Figure_05_01_013abc2.jpg\" alt=\"Three side by side graphs of circles. First graph has a circle with radius r and arc s, with an equivalence between r and s. The second graph shows a circle with radius r and an arc of length 2r. The third graph shows a circle with a full revolution, showing 6.28 radians.\" \/><\/span><\/p>\n<p style=\"text-align: center;\"><strong>Figure 12.<\/strong>\u00a0(a) In an angle of 1 radian, the arc length [latex]s[\/latex] equals the radius [latex]r[\/latex]. (b) An angle of 2 radians has an arc length [latex]s=2r[\/latex]. (c) A full revolution is [latex]2\\pi[\/latex] or about 6.28 radians.<\/p>\n<p>To elaborate on this idea, consider two circles, one with radius 2 and the other with radius 3. Recall the circumference of a circle is [latex]C=2\\pi r[\/latex], where [latex]r[\/latex] is the radius. The smaller circle then has circumference [latex]2\\pi \\left(2\\right)=4\\pi[\/latex] and the larger has circumference [latex]2\\pi \\left(3\\right)=6\\pi[\/latex].\u00a0Now we draw a 45\u00b0 angle on the two circles, as in\u00a0Figure 13.<\/p>\n<p>&nbsp;<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/923\/2015\/04\/25180222\/CNX_Precalc_Figure_05_01_0122.jpg\" alt=\"Graph of a circle with a 45 degree angle and a label for pi\/4 radians.\" width=\"487\" height=\"369\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 13.<\/b> A 45\u00b0 angle contains one-eighth of the circumference of a circle, regardless of the radius.<\/p>\n<\/div>\n<p>Notice what happens if we find the ratio of the arc length divided by the radius of the circle.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{gathered}\\text{Smaller circle: }\\frac{\\frac{1}{2}\\pi }{2}=\\frac{1}{4}\\pi \\\\ \\text{ Larger circle: }\\frac{\\frac{3}{4}\\pi }{3}=\\frac{1}{4}\\pi \\end{gathered}[\/latex]<\/div>\n<p>Since both ratios are [latex]\\frac{1}{4}\\pi[\/latex], the angle measures of both circles are the same, even though the arc length and radius differ.<\/p>\n<div class=\"textbox\">\n<h3>A GENERAL NOTE: RADIANS<\/h3>\n<p>One <strong>radian<\/strong> is the measure of the central angle of a circle such that the length of the arc between the initial side and the terminal side is equal to the radius of the circle. A full revolution (360\u00b0) equals [latex]2\\pi[\/latex] radians. A half revolution (180\u00b0) is equivalent to [latex]\\pi[\/latex] radians.<\/p>\n<p>The <strong>radian measure<\/strong> of an angle is the ratio of the length of the arc subtended by the angle to the radius of the circle. In other words, if [latex]s[\/latex] is the length of an arc of a circle, and [latex]r[\/latex] is the radius of the circle, then the central angle containing that arc measures [latex]\\frac{s}{r}[\/latex] radians. In a circle of radius 1, the radian measure corresponds to the length of the arc.<\/p>\n<\/div>\n<div class=\"textbox\">\n<p><strong>Q &amp; A<\/strong><\/p>\n<h3>A MEASURE OF 1 RADIAN LOOKS TO BE ABOUT 60\u00b0. IS THAT CORRECT?<\/h3>\n<p><em>Yes. It is approximately 57.3\u00b0. Because [latex]2\\pi[\/latex] radians equals 360\u00b0, [latex]1[\/latex] radian equals [latex]\\frac{360^\\circ }{2\\pi }\\approx 57.3^\\circ[\/latex].<\/em><\/p>\n<\/div>\n<h2>Using Radians<\/h2>\n<p>Because <strong>radian<\/strong> measure is the ratio of two lengths, it is a unitless measure. For example, in Figure 12, suppose the radius were 2 inches and the distance along the arc were also 2 inches. When we calculate the radian measure of the angle, the &#8220;inches&#8221; cancel, and we have a result without units. Therefore, it is not necessary to write the label &#8220;radians&#8221; after a radian measure, and if we see an angle that is not labeled with &#8220;degrees&#8221; or the degree symbol, we can assume that it is a radian measure.<\/p>\n<p>Considering the most basic case, the <strong>unit circle<\/strong> (a circle with radius 1), we know that 1 rotation equals 360 degrees, 360\u00b0. We can also track one rotation around a circle by finding the circumference, [latex]C=2\\pi r[\/latex], and for the unit circle [latex]C=2\\pi[\/latex]. These two different ways to rotate around a circle give us a way to convert from degrees to radians.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{gathered}1\\text{ rotation }=360^\\circ =2\\pi \\text{radians} \\\\ \\frac{1}{2}\\text{ rotation}=180^\\circ =\\pi \\text{radians} \\\\ \\frac{1}{4}\\text{ rotation}=90^\\circ =\\frac{\\pi }{2} \\text{radians} \\end{gathered}[\/latex]<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm148258\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=148258&theme=oea&iframe_resize_id=ohm148258\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Identifying Special Angles Measured in Radians<\/h2>\n<p>In addition to knowing the measurements in degrees and radians of a quarter revolution, a half revolution, and a full revolution, there are other frequently encountered angles in one revolution of a circle with which we should be familiar. It is common to encounter multiples of 30, 45, 60, and 90 degrees. These values are shown in Figure 14. Memorizing these angles will be very useful as we study the properties associated with angles.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/923\/2015\/04\/25180223\/CNX_Precalc_Figure_05_01_0162.jpg\" alt=\"A graph of a circle with angles of 0, 30, 45, 60, 90, 120, 135, 150, 180, 210, 225, 240, 270, 300, 315, and 330 degrees.\" width=\"487\" height=\"406\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 14.<\/b> Commonly encountered angles measured in degrees<\/p>\n<\/div>\n<p>Now, we can list the corresponding radian values for the common measures of a circle corresponding to those listed in Figure 14, which are shown in Figure 15. Be sure you can verify each of these measures.<\/p>\n<p>&nbsp;<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/923\/2015\/04\/25180224\/CNX_Precalc_Figure_05_01_0172.jpg\" alt=\"A graph of a circle with angles of 0, 30, 45, 60, 90, 120, 135, 150, 180, 210, 225, 240, 270, 300, 315, and 330 degrees. The graph also shows the equivalent amount of radians for each angle of degrees. For example, 30 degrees is equal to pi\/6 radians.\" width=\"487\" height=\"485\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 15.<\/b> Commonly encountered angles measured in radians<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>EXAMPLE 2: FINDING A RADIAN MEASURE<\/h3>\n<p>Find the radian measure of one-third of a full rotation.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q791790\">Show Solution<\/span><\/p>\n<div id=\"q791790\" class=\"hidden-answer\" style=\"display: none\">\n<p>For any circle, the arc length along such a rotation would be one-third of the circumference. We know that<\/p>\n<p style=\"text-align: center;\">[latex]1\\text{ rotation}=2\\pi r[\/latex]<\/p>\n<p>So,<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align} s&=\\frac{1}{3}\\left(2\\pi r\\right) \\\\ &=\\frac{2\\pi r}{3} \\end{align}[\/latex]<\/p>\n<p>The radian measure would be the arc length divided by the radius.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\text{radian measure}&=\\frac{\\frac{2\\pi r}{3}}{r} \\\\ &=\\frac{2\\pi r}{3r} \\\\ &=\\frac{2\\pi }{3} \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>TRY IT 3<\/h3>\n<p>Find the radian measure of three-fourths of a full rotation.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q558820\">Show Solution<\/span><\/p>\n<div id=\"q558820\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\frac{3\\pi }{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Converting between Radians and Degrees<\/h2>\n<p>Because degrees and radians both measure angles, we need to be able to convert between them. We can easily do so using a proportion.<\/p>\n<div style=\"text-align: center;\">[latex]\\frac{\\theta }{180}=\\frac{{\\theta }^{R}}{\\pi }[\/latex]<\/div>\n<p>This proportion shows that the measure of angle [latex]\\theta[\/latex] in degrees divided by 180 equals the measure of angle [latex]\\theta[\/latex] in radians divided by [latex]\\pi .[\/latex] Or, phrased another way, degrees is to 180 as radians is to [latex]\\pi[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]\\frac{\\text{Degrees}}{180}=\\frac{\\text{Radians}}{\\pi }[\/latex]<span class=\"fontsize-ensurer reset-size5 size5\" style=\"font-size: 1rem; text-align: initial;\"><span class=\"\">\u200b<\/span><\/span><span style=\"font-size: 1rem; text-align: initial;\">\u200b<\/span><\/div>\n<div>\n<h2>Converting between Radians and Degrees<\/h2>\n<p>To convert between degrees and radians, use the proportion<\/p>\n<div style=\"text-align: center;\">[latex]\\frac{\\theta }{180}=\\frac{{\\theta }^{R}}{\\pi }[\/latex]<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>EXAMPLE 3: CONVERTING RADIANS TO DEGREES<\/h3>\n<p>Convert each radian measure to degrees.<\/p>\n<p style=\"padding-left: 60px;\">a. [latex]\\frac{\\pi }{6}[\/latex]<\/p>\n<p style=\"padding-left: 60px;\">b. 3<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q620590\">Show Solution<\/span><\/p>\n<div id=\"q620590\" class=\"hidden-answer\" style=\"display: none\">\n<p>Because we are given radians and we want degrees, we should set up a proportion and solve it.<\/p>\n<p style=\"padding-left: 60px;\">a. We use the proportion, substituting the given information.<\/p>\n<p style=\"padding-left: 60px; text-align: center;\">[latex]\\begin{gathered} \\frac{\\theta }{180}=\\frac{{\\theta }^{R}}{\\pi } \\\\ \\frac{\\theta }{180}=\\frac{\\frac{\\pi }{6}}{\\pi } \\\\ \\theta =\\frac{180}{6} \\\\ \\theta ={30}^{\\circ } \\end{gathered}[\/latex]<\/p>\n<p style=\"padding-left: 60px;\">b. We use the proportion, substituting the given information.<\/p>\n<p style=\"padding-left: 60px; text-align: center;\">[latex]\\begin{gathered} \\frac{\\theta }{180}=\\frac{{\\theta }^{R}}{\\pi } \\\\ \\frac{\\theta }{180}=\\frac{3}{\\pi } \\\\ \\theta =\\frac{3\\left(180\\right)}{\\pi } \\\\ \\theta \\approx {172}^{\\circ } \\end{gathered}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>TRY IT 4<\/h3>\n<p>Convert [latex]-\\frac{3\\pi }{4}[\/latex] radians to degrees.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q362624\">Show Solution<\/span><\/p>\n<div id=\"q362624\" class=\"hidden-answer\" style=\"display: none\">\n<p>\u2212135\u00b0<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm99889\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=99889&theme=oea&iframe_resize_id=ohm99889\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>EXAMPLE 4: CONVERTING DEGREES TO RADIANS<\/h3>\n<p>Convert [latex]15[\/latex] degrees to radians.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q867109\">Show Solution<\/span><\/p>\n<div id=\"q867109\" class=\"hidden-answer\" style=\"display: none\">\n<p>In this example, we start with degrees and want radians, so we again set up a proportion and solve it, but we substitute the given information into a different part of the proportion.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{gathered} \\frac{\\theta }{180}=\\frac{{\\theta }^{R}}{\\pi } \\\\ \\frac{15}{180}=\\frac{{\\theta }^{R}}{\\pi }\\\\ \\frac{15\\pi }{180}={\\theta }^{R}\\\\ \\frac{\\pi }{12}={\\theta }^{R} \\end{gathered}[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>Another way to think about this problem is by remembering that [latex]{30}^{\\circ }=\\frac{\\pi }{6}[\/latex].<br \/>\nBecause [latex]{15}^{\\circ }=\\frac{1}{2}\\left({30}^{\\circ }\\right)[\/latex], we can find that [latex]\\frac{1}{2}\\left(\\frac{\\pi }{6}\\right)[\/latex] is [latex]\\frac{\\pi }{12}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>TRY IT 6<\/h3>\n<p>Convert 126\u00b0 to radians.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q164537\">Show Solution<\/span><\/p>\n<div id=\"q164537\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\frac{7\\pi }{10}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm99887\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=99887&theme=oea&iframe_resize_id=ohm99887\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div>\n<div>Watch the following video for an explanation of radian measure and examples of converting between radians and degrees.<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"Radian Measure\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/nAJqXtzwpXQ?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<\/div>\n<\/div>\n<\/div>\n<section class=\"citations-section focusable\" role=\"contentinfo\">\n<div id=\"post-10774\" class=\"standard post-10774 chapter type-chapter status-publish hentry\">\n<div class=\"entry-content\">\n<h2>Finding Coterminal Angles<\/h2>\n<p>Converting between degrees and radians can make working with angles easier in some applications. For other applications, we may need another type of conversion. Negative angles and angles greater than a full revolution are more awkward to work with than those in the range of 0\u00b0 to 360\u00b0, or 0 to [latex]2\\pi[\/latex]. It would be convenient to replace those out-of-range angles with a corresponding angle within the range of a single revolution.<\/p>\n<p>It is possible for more than one angle to have the same terminal side. Look at Figure 16. The angle of 140\u00b0 is a\u00a0<strong>positive angle<\/strong>, measured counterclockwise. The angle of \u2013220\u00b0 is a\u00a0<strong>negative angle<\/strong>, measured clockwise. But both angles have the same terminal side. If two angles in standard position have the same terminal side, they are\u00a0<strong>coterminal angles<\/strong>. Every angle greater than 360\u00b0 or less than 0\u00b0 is coterminal with an angle between 0\u00b0 and 360\u00b0, and it is often more convenient to find the coterminal angle within the range of 0\u00b0 to 360\u00b0 than to work with an angle that is outside that range.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/923\/2015\/04\/25180226\/CNX_Precalc_Figure_05_01_0202.jpg\" alt=\"A graph showing the equivalence between a 140 degree angle and a negative 220 degree angle.\" width=\"487\" height=\"383\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 16.<\/b> An angle of 140\u00b0 and an angle of \u2013220\u00b0 are coterminal angles.<\/p>\n<\/div>\n<p>This video shows examples of how to determine if two angles are coterminal.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-5\" title=\"Example:  Determine if Two Angles Are Coterminal\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/TuyF8fFg3B0?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>Any angle has infinitely many\u00a0<strong>coterminal angles<\/strong>\u00a0because each time we add 360\u00b0 to that angle\u2014or subtract 360\u00b0 from it\u2014the resulting value has a terminal side in the same location. For example, 100\u00b0 and 460\u00b0 are coterminal for this reason, as is \u2212260\u00b0. Recognizing that any angle has infinitely many coterminal angles explains the repetitive shape in the graphs of trigonometric functions.<\/p>\n<p>An angle\u2019s reference angle is the measure of the smallest, positive, acute angle [latex]t[\/latex] formed by the terminal side of the angle [latex]t[\/latex] and the horizontal axis. Thus positive reference angles have terminal sides that lie in the first quadrant and can be used as models for angles in other quadrants. See Figure 17\u00a0for examples of reference angles for angles in different quadrants.<span id=\"fs-id1165137542464\">\u00a0<\/span><\/p>\n<div style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/923\/2015\/04\/25180227\/CNX_Precalc_Figure_05_01_0194.jpg\" alt=\"Four side by side graphs. First graph shows an angle of t in quadrant 1 in it's normal position. Second graph shows an angle of t in quadrant 2 due to a rotation of pi minus t. Third graph shows an angle of t in quadrant 3 due to a rotation of t minus pi. Fourth graph shows an angle of t in quadrant 4 due to a rotation of two pi minus t.\" width=\"975\" height=\"331\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 17<\/b><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox\">\n<h3>A GENERAL NOTE: COTERMINAL AND REFERENCE ANGLES<\/h3>\n<p>Coterminal angles are two angles in standard position that have the same terminal side.<\/p>\n<p>An angle\u2019s <strong>reference angle<\/strong> is the size of the smallest acute angle, [latex]{t}^{\\prime }[\/latex], formed by the terminal side of the angle [latex]t[\/latex]\u00a0and the horizontal axis.<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>HOW TO: GIVEN AN ANGLE GREATER THAN 360\u00b0, FIND A COTERMINAL ANGLE BETWEEN 0\u00b0 AND 360\u00b0.<\/h3>\n<ol>\n<li>Subtract 360\u00b0 from the given angle.<\/li>\n<li>If the result is still greater than 360\u00b0, subtract 360\u00b0 again till the result is between 0\u00b0 and 360\u00b0.<\/li>\n<li>The resulting angle is coterminal with the original angle.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>EXAMPLE 5: FINDING AN ANGLE COTERMINAL WITH AN ANGLE OF MEASURE GREATER THAN 360\u00b0<\/h3>\n<p>Find the least positive angle [latex]\\theta[\/latex] that is coterminal with an angle measuring 800\u00b0, where [latex]0^\\circ \\le \\theta <360^\\circ[\/latex].\n\n\n\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q964169\">Show Solution<\/span><\/p>\n<div id=\"q964169\" class=\"hidden-answer\" style=\"display: none\">\n<p>An angle with measure 800\u00b0 is coterminal with an angle with measure 800 \u2212 360 = 440\u00b0, but 440\u00b0 is still greater than 360\u00b0, so we subtract 360\u00b0 again to find another coterminal angle: 440 \u2212 360 = 80\u00b0.<\/p>\n<p>The angle [latex]\\theta =80^\\circ[\/latex] is coterminal with 800\u00b0. To put it another way, 800\u00b0 equals 80\u00b0 plus two full rotations, as shown in Figure 18.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/923\/2015\/04\/25180229\/CNX_Precalc_Figure_05_01_0212.jpg\" alt=\"A graph showing the equivalence between an 80 degree angle and an 800 degree angle.\" width=\"487\" height=\"383\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 18<\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>TRY IT 8<\/h3>\n<p>Find an angle [latex]\\alpha[\/latex] that is coterminal with an angle measuring 870\u00b0, where [latex]0^\\circ \\le \\alpha <360^\\circ[\/latex].\n\n\n\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q363809\">Show Solution<\/span><\/p>\n<div id=\"q363809\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\alpha =150^\\circ[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>HOW TO: GIVEN AN ANGLE WITH MEASURE LESS THAN 0\u00b0, FIND A COTERMINAL ANGLE HAVING A MEASURE BETWEEN 0\u00b0 AND 360\u00b0.<\/h3>\n<ol>\n<li>Add 360\u00b0 to the given angle.<\/li>\n<li>If the result is still less than 0\u00b0, add 360\u00b0 again until the result is between 0\u00b0 and 360\u00b0.<\/li>\n<li>The resulting angle is coterminal with the original angle.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>EXAMPLE 6: FINDING AN ANGLE COTERMINAL WITH AN ANGLE MEASURING LESS THAN 0\u00b0<\/h3>\n<p>Show the angle with measure \u221245\u00b0 on a circle and find a positive coterminal angle [latex]\\alpha[\/latex] such that 0\u00b0 \u2264 <em>\u03b1<\/em> &lt; 360\u00b0.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q713003\">Show Solution<\/span><\/p>\n<div id=\"q713003\" class=\"hidden-answer\" style=\"display: none\">\n<p>Since 45\u00b0 is half of 90\u00b0, we can start at the positive horizontal axis and measure clockwise half of a 90\u00b0 angle.<\/p>\n<p>Because we can find coterminal angles by adding or subtracting a full rotation of 360\u00b0, we can find a positive coterminal angle here by adding 360\u00b0:<\/p>\n<p style=\"text-align: center;\">[latex]-45^\\circ +360^\\circ =315^\\circ[\/latex]<\/p>\n<p>We can then show the angle on a circle, as in Figure 19.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/923\/2015\/04\/25180230\/CNX_Precalc_Figure_05_01_0222.jpg\" alt=\"A graph showing the equivalence of a 315 degree angle and a negative 45 degree angle.\" width=\"487\" height=\"383\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 19<\/strong><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div>\n<p>Watch this video for another example of how to determine positive and negative coterminal angles.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-6\" title=\"Example:  Determine Positive and Negative Coterminal Angles\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/m7jTGVVzb0s?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>TRY IT 9<\/h3>\n<p>Find an angle [latex]\\beta[\/latex] that is coterminal with an angle measuring \u2212300\u00b0 such that [latex]0^\\circ \\le \\beta <360^\\circ[\/latex].\n\n\n\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q113901\">Show Solution<\/span><\/p>\n<div id=\"q113901\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\beta =60^\\circ[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm147466\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=147466&theme=oea&iframe_resize_id=ohm147466\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Finding Coterminal Angles Measured in Radians<\/h2>\n<p>We can find\u00a0<strong>coterminal angles<\/strong>\u00a0measured in radians in much the same way as we have found them using degrees. In both cases, we find coterminal angles by adding or subtracting one or more full rotations.<\/p>\n<div>\n<p><strong>Given an angle greater than<\/strong> [latex]2\\pi[\/latex], <strong>find a coterminal angle between 0 and<\/strong> [latex]2\\pi[\/latex].<\/p>\n<ol>\n<li>Subtract [latex]2\\pi[\/latex] from the given angle.<\/li>\n<li>If the result is still greater than [latex]2\\pi[\/latex], subtract [latex]2\\pi[\/latex] again until the result is between [latex]0[\/latex] and [latex]2\\pi[\/latex].<\/li>\n<li>The resulting angle is coterminal with the original angle.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>EXAMPLE 7: FINDING COTERMINAL ANGLES USING RADIANS<\/h3>\n<p>Find an angle [latex]\\beta[\/latex] that is coterminal with [latex]\\frac{19\\pi }{4}[\/latex], where [latex]0\\le \\beta <2\\pi[\/latex].\n\n\n\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q742225\">Show Solution<\/span><\/p>\n<div id=\"q742225\" class=\"hidden-answer\" style=\"display: none\">\n<p>When working in degrees, we found coterminal angles by adding or subtracting 360 degrees, a full rotation. Likewise, in radians, we can find coterminal angles by adding or subtracting full rotations of [latex]2\\pi[\/latex] radians:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\frac{19\\pi }{4}-2\\pi =\\frac{19\\pi }{4}-\\frac{8\\pi }{4} =\\frac{11\\pi }{4} \\end{align}[\/latex]<\/p>\n<p>The angle [latex]\\frac{11\\pi }{4}[\/latex] is coterminal, but not less than [latex]2\\pi[\/latex], so we subtract another rotation:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\frac{11\\pi }{4}-2\\pi =\\frac{11\\pi }{4}-\\frac{8\\pi }{4} =\\frac{3\\pi }{4} \\end{align}[\/latex]<\/p>\n<p>The angle [latex]\\frac{3\\pi }{4}[\/latex] is coterminal with [latex]\\frac{19\\pi }{4}[\/latex], as shown in Figure 20.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/923\/2015\/04\/25180232\/CNX_Precalc_Figure_05_01_0232.jpg\" alt=\"A graph showing a circle and the equivalence between angles of 3pi\/4 radians and 19pi\/4 radians.\" width=\"487\" height=\"383\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 20<\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><b><\/b><\/p>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>TRY IT 11<\/h3>\n<p>Find an angle of measure [latex]\\theta[\/latex] that is coterminal with an angle of measure [latex]-\\frac{17\\pi }{6}[\/latex] where [latex]0\\le \\theta <2\\pi[\/latex].\n\n\n\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q436085\">Show Solution<\/span><\/p>\n<div id=\"q436085\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\frac{7\\pi }{6}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm99894\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=99894&theme=oea&iframe_resize_id=ohm99894\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Determining the Length of an Arc<\/h2>\n<\/section>\n<p>Recall that the <strong>radian measure<\/strong> [latex]\\theta[\/latex] of an angle was defined as the ratio of the <strong>arc length<\/strong> [latex]s[\/latex] of a circular arc to the radius [latex]r[\/latex] of the circle, [latex]\\theta =\\frac{s}{r}[\/latex]. From this relationship, we can find arc length along a circle, given an angle.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Arc Length on a Circle<\/h3>\n<p>In a circle of radius <em>r<\/em>, the length of an arc [latex]s[\/latex] subtended by an angle with measure [latex]\\theta[\/latex] in radians, shown in Figure 20, is<\/p>\n<div style=\"text-align: center;\">\n<p>[latex]s=r\\theta[\/latex]<\/p>\n<div style=\"width: 359px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003450\/CNX_Precalc_Figure_05_01_024F2.jpg\" alt=\"Illustration of circle with angle theta, radius r, and arc with length s.\" width=\"349\" height=\"348\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 20<\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a circle of radius [latex]r[\/latex], calculate the length [latex]s[\/latex] of the arc subtended by a given angle of measure [latex]\\theta[\/latex].<\/h3>\n<ol>\n<li>If necessary, convert [latex]\\theta[\/latex] to radians.<\/li>\n<li>Multiply the radius [latex]r[\/latex] by the radian measure of [latex]\\theta :s=r\\theta[\/latex].<\/li>\n<\/ol>\n<\/div>\n<section>\n<section>\n<div class=\"textbox shaded\">\n<h3>Example 8: Finding the Length of an Arc<\/h3>\n<p>Assume the orbit of Mercury around the sun is a perfect circle. Mercury is approximately 36 million miles from the sun.<\/p>\n<ol>\n<li>In one Earth day, Mercury completes 0.0114 of its total revolution. How many miles does it travel in one day?<\/li>\n<li>Use your answer from part (a) to determine the radian measure for Mercury\u2019s movement in one Earth day.<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q107850\">Show Solution<\/span><\/p>\n<div id=\"q107850\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>Let\u2019s begin by finding the circumference of Mercury\u2019s orbit.\n<div style=\"text-align: center;\">[latex]\\begin{align}C&=2\\pi r \\\\ &=2\\pi \\left(36\\text{ million miles}\\right) \\\\ &\\approx 226\\text{ million miles} \\end{align}[\/latex]<\/div>\n<p>Since Mercury completes 0.0114 of its total revolution in one Earth day, we can now find the distance traveled:<\/p>\n<div style=\"text-align: center;\">[latex]\\left(0.0114\\right)226\\text{ million miles = 2}\\text{.58 million miles}[\/latex]<\/div>\n<\/li>\n<li>Now, we convert to radians:\n<div style=\"text-align: center;\">[latex]\\begin{align}\\text{radian}&=\\frac{\\text{arclength}}{\\text{radius}} \\\\ &=\\frac{2.\\text{58 million miles}}{36\\text{ million miles}} \\\\ &=0.0717 \\end{align}[\/latex]<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Find the arc length along a circle of radius 10 units subtended by an angle of 215\u00b0.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q188778\">Show Solution<\/span><\/p>\n<div id=\"q188778\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\frac{215\\pi }{18}=37.525\\text{ units}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/section>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm172921\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=172921&theme=oea&iframe_resize_id=ohm172921\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p><span style=\"color: #077fab; font-size: 1.15em; font-weight: 600;\">Finding the Area of a Sector of a Circle<\/span><\/p>\n<section>\n<section>In addition to arc length, we can also use angles to find the area of a <strong>sector of a circle<\/strong>. A sector is a region of a circle bounded by two radii and the intercepted arc, like a slice of pizza or pie. Recall that the area of a circle with radius [latex]r[\/latex] can be found using the formula [latex]A=\\pi {r}^{2}[\/latex]. If the two radii form an angle of [latex]\\theta[\/latex], measured in radians, then [latex]\\frac{\\theta }{2\\pi }[\/latex] is the ratio of the angle measure to the measure of a full rotation and is also, therefore, the ratio of the area of the sector to the area of the circle. Thus, the <strong>area of a sector<\/strong> is the fraction [latex]\\frac{\\theta }{2\\pi }[\/latex]\u00a0multiplied by the entire area. (Always remember that this formula only applies if [latex]\\theta[\/latex] is in radians.)<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align}\\text{Area of sector}&=\\left(\\frac{\\theta }{2\\pi }\\right)\\pi {r}^{2} \\\\ &=\\frac{\\theta \\pi {r}^{2}}{2\\pi } \\\\ &=\\frac{1}{2}\\theta {r}^{2} \\end{align}[\/latex]<\/div>\n<div class=\"textbox\">\n<h3>A General Note: Area of a Sector<\/h3>\n<p>The <strong>area of a sector<\/strong> of a circle with radius [latex]r[\/latex] subtended by an angle [latex]\\theta[\/latex], measured in radians, is<\/p>\n<div>\n<p style=\"text-align: center;\">[latex]A=\\frac{1}{2}\\theta {r}^{2}[\/latex]<\/p>\n<div style=\"width: 452px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003452\/CNX_Precalc_Figure_05_01_026F2.jpg\" alt=\"Graph showing a circle with angle theta and radius r, and the area of the slice of circle created by the initial side and terminal side of the angle.\" width=\"442\" height=\"394\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 21.<\/b> The area of the sector equals half the square of the radius times the central angle measured in radians.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a circle of radius [latex]r[\/latex], find the area of a sector defined by a given angle [latex]\\theta[\/latex].<\/h3>\n<ol>\n<li>If necessary, convert [latex]\\theta[\/latex] to radians.<\/li>\n<li>Multiply half the radian measure of [latex]\\theta[\/latex] by the square of the radius [latex]r:\\text{ } A=\\frac{1}{2}\\theta {r}^{2}[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 9: Finding the Area of a Sector<\/h3>\n<p>An automatic lawn sprinkler sprays a distance of 20 feet while rotating 30 degrees, as shown in Figure 22. What is the area of the sector of grass the sprinkler waters?<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003454\/CNX_Precalc_Figure_05_01_0272.jpg\" alt=\"Illustration of a 30 degree ange with a terminal and initial side with length of 20 feet.\" width=\"487\" height=\"169\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 22.<\/b> The sprinkler sprays 20 ft within an arc of 30\u00b0.<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q785904\">Show Solution<\/span><\/p>\n<div id=\"q785904\" class=\"hidden-answer\" style=\"display: none\">\n<p>First, we need to convert the angle measure into radians. Because 30 degrees is one of our special angles, we already know the equivalent radian measure, but we can also convert:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}30\\text{ degrees}&=30\\cdot \\frac{\\pi }{180} \\\\ &=\\frac{\\pi }{6}\\text{ radians} \\end{align}[\/latex]<\/p>\n<p>The area of the sector is then<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\text{Area} &= \\frac{1}{2}\\left(\\frac{\\pi }{6}\\right){\\left(20\\right)}^{2} \\\\ &\\approx 104.72 \\end{align}[\/latex]<\/p>\n<p>So the area is about [latex]104.72{\\text{ ft}}^{2}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>In central pivot irrigation, a large irrigation pipe on wheels rotates around a center point. A farmer has a central pivot system with a radius of 400 meters. If water restrictions only allow her to water 150 thousand square meters a day, what angle should she set the system to cover? Write the answer in radian measure to two decimal places.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q495928\">Show Solution<\/span><\/p>\n<div id=\"q495928\" class=\"hidden-answer\" style=\"display: none\">\n<p>1.88<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video you will see how to calculate arc length and area of a sector of a circle.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-7\" title=\"Arc Length and Area of a Sector\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/zD4CsKIYEHo?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Use Linear and Angular Speed to Describe Motion on a Circular Path<\/h2>\n<\/section>\n<\/section>\n<p>In addition to finding the area of a sector, we can use angles to describe the speed of a moving object. An object traveling in a circular path has two types of speed.<strong> Linear speed<\/strong> is speed along a straight path and can be determined by the distance it moves along (its <strong>displacement<\/strong>) in a given time interval. For instance, if a wheel with radius 5 inches rotates once a second, a point on the edge of the wheel moves a distance equal to the circumference, or [latex]10\\pi[\/latex] inches, every second. So the linear speed of the point is [latex]10\\pi[\/latex] in.\/s. The equation for linear speed is as follows where [latex]v[\/latex] is linear speed, [latex]s[\/latex] is displacement, and [latex]t[\/latex]<br \/>\nis time.<\/p>\n<div style=\"text-align: center;\">[latex]v=\\frac{s}{t}[\/latex]<\/div>\n<p><strong>Angular speed<\/strong> results from circular motion and can be determined by the angle through which a point rotates in a given time interval. In other words, angular speed is angular rotation per unit time. So, for instance, if a gear makes a full rotation every 4 seconds, we can calculate its angular speed as [latex]\\frac{360\\text{ degrees}}{4\\text{ seconds}}=[\/latex] 90 degrees per second. Angular speed can be given in radians per second, rotations per minute, or degrees per hour for example. The equation for angular speed is as follows, where [latex]\\omega[\/latex] (read as omega) is angular speed, [latex]\\theta[\/latex] is the angle traversed, and [latex]t[\/latex] is time.<\/p>\n<div style=\"text-align: center;\">[latex]\\omega =\\frac{\\theta }{t}[\/latex]<\/div>\n<p>Combining the definition of angular speed with the arc length equation, [latex]s=r\\theta[\/latex], we can find a relationship between angular and linear speeds. The angular speed equation can be solved for [latex]\\theta[\/latex], giving [latex]\\theta =\\omega t[\/latex]. Substituting this into the arc length equation gives:<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align}s&=r\\theta \\\\ &=r\\omega t \\end{align}[\/latex]<\/div>\n<p>Substituting this into the linear speed equation gives:<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align} v&=\\frac{s}{t} \\\\ &=\\frac{r\\omega t}{t} \\\\ &=r\\omega \\end{align}[\/latex]<\/div>\n<div class=\"textbox\">\n<h3>A General Note: Angular and Linear Speed<\/h3>\n<p>As a point moves along a circle of radius [latex]r[\/latex], its <strong>angular speed<\/strong>, [latex]\\omega[\/latex], is the angular rotation [latex]\\theta[\/latex] per unit time, [latex]t[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\omega =\\frac{\\theta }{t}[\/latex]<\/p>\n<p>The <strong>linear speed<\/strong>. [latex]v[\/latex], of the point can be found as the distance traveled, arc length [latex]s[\/latex], per unit time, [latex]t[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]v=\\frac{s}{t}[\/latex]<\/p>\n<p>When the angular speed is measured in radians per unit time, linear speed and angular speed are related by the equation<\/p>\n<p style=\"text-align: center;\">[latex]v=r\\omega[\/latex]<\/p>\n<p>This equation states that the angular speed in radians, [latex]\\omega[\/latex], representing the amount of rotation occurring in a unit of time, can be multiplied by the radius [latex]r[\/latex] to calculate the total arc length traveled in a unit of time, which is the definition of linear speed.<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given the amount of angle rotation and the time elapsed, calculate the angular speed.<strong><br \/>\n<\/strong><\/h3>\n<ol>\n<li>If necessary, convert the angle measure to radians.<\/li>\n<li>Divide the angle in radians by the number of time units elapsed: [latex]\\omega =\\frac{\\theta }{t}[\/latex].<\/li>\n<li>The resulting speed will be in radians per time unit.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 10: Finding Angular Speed<\/h3>\n<p>A water wheel, shown in Figure 23, completes 1 rotation every 5 seconds. Find the angular speed in radians per second.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003457\/CNX_Precalc_Figure_05_01_028n.jpg\" alt=\"Illustration of a water wheel.\" width=\"487\" height=\"359\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 23<\/b><\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q602649\">Show Solution<\/span><\/p>\n<div id=\"q602649\" class=\"hidden-answer\" style=\"display: none\">\n<p>The wheel completes 1 rotation, or passes through an angle of [latex]2\\pi[\/latex]\u00a0radians in 5 seconds, so the angular speed would be [latex]\\omega =\\frac{2\\pi }{5}\\approx 1.257[\/latex] radians per second.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>An old vinyl record is played on a turntable rotating clockwise at a rate of 45 rotations per minute. Find the angular speed in radians per second.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q737350\">Show Solution<\/span><\/p>\n<div id=\"q737350\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\frac{-3\\pi }{2}[\/latex]\u00a0rad\/s<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm155235\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=155235&theme=oea&iframe_resize_id=ohm155235\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given the radius of a circle, an angle of rotation, and a length of elapsed time, determine the linear speed.<strong><br \/>\n<\/strong><\/h3>\n<ol>\n<li>Convert the total rotation to radians if necessary.<\/li>\n<li>Divide the total rotation in radians by the elapsed time to find the angular speed: apply [latex]\\omega =\\frac{\\theta }{t}[\/latex].<\/li>\n<li>Multiply the angular speed by the length of the radius to find the linear speed, expressed in terms of the length unit used for the radius and the time unit used for the elapsed time: apply [latex]v=r\\mathrm{\\omega}[\/latex].<\/li>\n<\/ol>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-8\" title=\"Example:  Determine Angular and Linear Velocity\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/bfWkgA5GSE0?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox shaded\">\n<h3>Example 11: Finding a Linear Speed<\/h3>\n<p>A bicycle has wheels 28 inches in diameter. A tachometer determines the wheels are rotating at 180 RPM (revolutions per minute). Find the speed the bicycle is traveling down the road.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q461618\">Show Solution<\/span><\/p>\n<div id=\"q461618\" class=\"hidden-answer\" style=\"display: none\">\n<p>Here, we have an angular speed and need to find the corresponding linear speed, since the linear speed of the outside of the tires is the speed at which the bicycle travels down the road.<\/p>\n<p>We begin by converting from rotations per minute to radians per minute. It can be helpful to utilize the units to make this conversion:<\/p>\n<p style=\"text-align: center;\">[latex]180\\frac{\\cancel{\\text{rotations}}}{\\text{minute}}\\cdot \\frac{2\\pi \\text{radians}}{\\cancel{\\text{rotation}}}=360\\pi \\frac{\\text{radians}}{\\text{minute}}[\/latex]<\/p>\n<p>Using the formula from above along with the radius of the wheels, we can find the linear speed:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align} v&=\\left(14\\text{inches}\\right)\\left(360\\pi \\frac{\\text{radians}}{\\text{minute}}\\right) \\\\ &=5040\\pi \\frac{\\text{inches}}{\\text{minute}} \\end{align}[\/latex]<\/p>\n<p>Remember that radians are a unitless measure, so it is not necessary to include them.<\/p>\n<p>Finally, we may wish to convert this linear speed into a more familiar measurement, like miles per hour.<\/p>\n<p style=\"text-align: center;\">[latex]5040\\pi \\frac{\\cancel{\\text{inches}}}{\\cancel{\\text{minute}}}\\cdot \\frac{\\text{1}\\cancel{\\text{feet}}}{\\text{12}\\cancel{\\text{inches}}}\\cdot \\frac{\\text{1 mile}}{\\text{5280}\\cancel{\\text{feet}}}\\cdot \\frac{\\text{60}\\cancel{\\text{minutes}}}{\\text{1 hour}}\\approx 14.99\\text{miles per hour (mph)}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>A satellite is rotating around Earth at 0.25 radians per hour at an altitude of 242 km above Earth. If the radius of Earth is 6378 kilometers, find the linear speed of the satellite in kilometers per hour.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q851398\">Show Solution<\/span><\/p>\n<div id=\"q851398\" class=\"hidden-answer\" style=\"display: none\">\n<p>1655 kilometers per hour<\/p>\n<\/div>\n<\/div>\n<\/div>\n<section id=\"fs-id1165135369540\" class=\"key-equations\">\n<h2>Key Equations<\/h2>\n<table id=\"eip-id1165134342478\" summary=\"..\">\n<tbody>\n<tr>\n<td>arc length<\/td>\n<td>[latex]s=r\\theta[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>area of a sector<\/td>\n<td>[latex]A=\\frac{1}{2}\\theta {r}^{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>angular speed<\/td>\n<td>[latex]\\omega =\\frac{\\theta }{t}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>linear speed<\/td>\n<td>[latex]v=\\frac{s}{t}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>linear speed related to angular speed<\/td>\n<td>[latex]v=r\\omega[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/section>\n<section id=\"fs-id1165135431038\" class=\"key-concepts\">\n<h2>Key Concepts<\/h2>\n<ul id=\"fs-id1165135431044\">\n<li>An angle is formed from the union of two rays, by keeping the initial side fixed and rotating the terminal side. The amount of rotation determines the measure of the angle.<\/li>\n<li>An angle is in standard position if its vertex is at the origin and its initial side lies along the positive <em>x<\/em>-axis. A positive angle is measured counterclockwise from the initial side and a negative angle is measured clockwise.<\/li>\n<li>To draw an angle in standard position, draw the initial side along the positive <em>x<\/em>-axis and then place the terminal side according to the fraction of a full rotation the angle represents.<\/li>\n<li>In addition to degrees, the measure of an angle can be described in radians.<\/li>\n<li>To convert between degrees and radians, use the proportion [latex]\\frac{\\theta }{180}=\\frac{{\\theta }^{R}}{\\pi }[\/latex].<\/li>\n<li>Two angles that have the same terminal side are called coterminal angles.<\/li>\n<li>We can find coterminal angles by adding or subtracting 360\u00b0 or [latex]2\\pi[\/latex].<\/li>\n<li>Coterminal angles can be found using radians just as they are for degrees.<\/li>\n<li>The length of a circular arc is a fraction of the circumference of the entire circle.<\/li>\n<li>The area of sector is a fraction of the area of the entire circle.<\/li>\n<li>An object moving in a circular path has both linear and angular speed.<\/li>\n<li>The angular speed of an object traveling in a circular path is the measure of the angle through which it turns in a unit of time.<\/li>\n<li>The linear speed of an object traveling along a circular path is the distance it travels in a unit of time.<\/li>\n<\/ul>\n<div>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165135487231\" class=\"definition\">\n<dt>angle<\/dt>\n<dd id=\"fs-id1165135487236\">the union of two rays having a common endpoint<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135487241\" class=\"definition\">\n<dt>angular speed<\/dt>\n<dd id=\"fs-id1165135487246\">the angle through which a rotating object travels in a unit of time<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135487250\" class=\"definition\">\n<dt>arc length<\/dt>\n<dd id=\"fs-id1165135170996\">the length of the curve formed by an arc<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135171000\" class=\"definition\">\n<dt>area of a sector<\/dt>\n<dd id=\"fs-id1165135171006\">area of a portion of a circle bordered by two radii and the intercepted arc; the fraction [latex]\\frac{\\theta }{2\\pi }[\/latex] multiplied by the area of the entire circle<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134223298\" class=\"definition\">\n<dt>coterminal angles<\/dt>\n<dd id=\"fs-id1165134223303\">description of positive and negative angles in standard position sharing the same terminal side<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134223307\" class=\"definition\">\n<dt>degree<\/dt>\n<dd id=\"fs-id1165134084943\">a unit of measure describing the size of an angle as one-360th of a full revolution of a circle<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134084947\" class=\"definition\">\n<dt>initial side<\/dt>\n<dd id=\"fs-id1165134084952\">the side of an angle from which rotation begins<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134084957\" class=\"definition\">\n<dt>linear speed<\/dt>\n<dd id=\"fs-id1165134084962\">the distance along a straight path a rotating object travels in a unit of time; determined by the arc length<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135580273\" class=\"definition\">\n<dt>measure of an angle<\/dt>\n<dd id=\"fs-id1165135580278\">the amount of rotation from the initial side to the terminal side<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135580283\" class=\"definition\">\n<dt>negative angle<\/dt>\n<dd id=\"fs-id1165135580288\">description of an angle measured clockwise from the positive <em>x<\/em>-axis<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135580297\" class=\"definition\">\n<dt>positive angle<\/dt>\n<dd id=\"fs-id1165135519264\">description of an angle measured counterclockwise from the positive <em>x<\/em>-axis<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135519274\" class=\"definition\">\n<dt>quadrantal angle<\/dt>\n<dd id=\"fs-id1165135519280\">an angle whose terminal side lies on an axis<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135519284\" class=\"definition\">\n<dt>radian measure<\/dt>\n<dd id=\"fs-id1165135519289\">the ratio of the arc length formed by an angle divided by the radius of the circle<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135193270\" class=\"definition\">\n<dt>radian<\/dt>\n<dd id=\"fs-id1165135193275\">the measure of a central angle of a circle that intercepts an arc equal in length to the radius of that circle<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135193281\" class=\"definition\">\n<dt>ray<\/dt>\n<dd id=\"fs-id1165135193286\">one point on a line and all points extending in one direction from that point; one side of an angle<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135193291\" class=\"definition\">\n<dt>reference angle<\/dt>\n<dd id=\"fs-id1165134298995\">the measure of the acute angle formed by the terminal side of the angle and the horizontal axis<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134299000\" class=\"definition\">\n<dt>standard position<\/dt>\n<dd id=\"fs-id1165134299005\">the position of an angle having the vertex at the origin and the initial side along the positive <em>x<\/em>-axis<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134299015\" class=\"definition\">\n<dt>terminal side<\/dt>\n<dd id=\"fs-id1165134299020\">the side of an angle at which rotation ends<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133309879\" class=\"definition\">\n<dt>vertex<\/dt>\n<dd id=\"fs-id1165133309884\">the common endpoint of two rays that form an angle<\/dd>\n<\/dl>\n<\/div>\n<\/section>\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-1363\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Angles. <strong>Authored by<\/strong>: OpenStax College. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">All rights reserved content<\/div><ul class=\"citation-list\"><li>Animation: Angles in Standard Position. <strong>Authored by<\/strong>: Mathispower4u. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/hpIjaKLOo6o\">https:\/\/youtu.be\/hpIjaKLOo6o<\/a>. <strong>License<\/strong>: <em>All Rights Reserved<\/em>. <strong>License Terms<\/strong>: Standard YouTube License<\/li><li>Examples: Determine Angles of Rotation. <strong>Authored by<\/strong>: Mathispower4u. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/0yHDfG2m-44.\">https:\/\/youtu.be\/0yHDfG2m-44.<\/a>. <strong>License<\/strong>: <em>All Rights Reserved<\/em>. <strong>License Terms<\/strong>: Standard YouTube License<\/li><li>Radian Measure. <strong>Authored by<\/strong>: Mathispower4u. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/nAJqXtzwpXQ.\">https:\/\/youtu.be\/nAJqXtzwpXQ.<\/a>. <strong>License<\/strong>: <em>All Rights Reserved<\/em>. <strong>License Terms<\/strong>: Standard YouTube License<\/li><li>Example: Determine if Two Angles Are Coterminal. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.youtube.com\/watch?v=TuyF8fFg3B0\">https:\/\/www.youtube.com\/watch?v=TuyF8fFg3B0<\/a>. <strong>License<\/strong>: <em>All Rights Reserved<\/em>. <strong>License Terms<\/strong>: Standard YouTube License<\/li><li>Example: Determine Positive and Negative Coterminal Angles. <strong>Authored by<\/strong>: Mathispower4u. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.youtube.com\/watch?v=m7jTGVVzb0s\">https:\/\/www.youtube.com\/watch?v=m7jTGVVzb0s<\/a>. <strong>License<\/strong>: <em>All Rights Reserved<\/em>. <strong>License Terms<\/strong>: Standard YouTube License<\/li><li>Arc Length and Area of a Sector. <strong>Authored by<\/strong>: Mathispower4u. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/zD4CsKIYEHo\">https:\/\/youtu.be\/zD4CsKIYEHo<\/a>. <strong>License<\/strong>: <em>All Rights Reserved<\/em>. <strong>License Terms<\/strong>: Standard YouTube License<\/li><li>Example: Determine Angular and Linear Velocity. <strong>Authored by<\/strong>: Mathispower4u. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/bfWkgA5GSE0\">https:\/\/youtu.be\/bfWkgA5GSE0<\/a>. <strong>License<\/strong>: <em>All Rights Reserved<\/em>. <strong>License Terms<\/strong>: Standard YouTube License<\/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":503070,"menu_order":1,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Angles\",\"author\":\"OpenStax 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