{"id":1147,"date":"2021-11-11T17:37:23","date_gmt":"2021-11-11T17:37:23","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/skills-review-for-polar-coordinates\/"},"modified":"2022-11-09T16:26:40","modified_gmt":"2022-11-09T16:26:40","slug":"skills-review-for-polar-coordinates","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/skills-review-for-polar-coordinates\/","title":{"raw":"Skills Review for Polar Coordinates","rendered":"Skills Review for Polar Coordinates"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Identify reference angles for angles measured in both radians and degrees<\/li>\r\n \t<li>Evaluate trigonometric functions using the unit circle<\/li>\r\n \t<li>Evaluate inverse trigonometric functions<\/li>\r\n \t<li>Draw an Angle in Standard Position<\/li>\r\n \t<li>Write the equation of a circle in standard form<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn the Polar Coordinates section, we will be introduced to polar coordinates and equations. Here we will review how to evaluate sine and cosine functions at specific angle measures, evaluate inverse trigonometric functions, draw angles, and write the equation of a circle in standard form.\r\n<h2>Find Reference Angles<\/h2>\r\nIt is easiest to evaluate trigonometric functions when an angle is in the first quadrant. When the original angle is given in quadrant two, three, or four, a reference angle should be found.\r\n\r\nAn angle\u2019s <strong>reference angle<\/strong> is the acute angle, [latex]t[\/latex], formed by the terminal side of the angle [latex]t[\/latex] and the horizontal axis. A reference angle is always an angle between [latex]0[\/latex] and [latex]90^\\circ [\/latex], or [latex]0[\/latex] and [latex]\\frac{\\pi }{2}[\/latex] radians. As we can see in the figure below, for any angle in quadrants II, III, or IV, there is a reference angle in quadrant I.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003604\/CNX_Precalc_Figure_05_01_0195.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>A visual of the corresponding reference angles for each of the quadrants.<\/b>[\/caption]\r\n\r\n<div class=\"textbox\">\r\n<h3>How To: Given an angle between [latex]0[\/latex] and [latex]2\\pi [\/latex], find its reference angle<\/h3>\r\n<ol>\r\n \t<li>An angle in the first quadrant is its own reference angle.<\/li>\r\n \t<li>For an angle in the second or third quadrant, the reference angle is [latex]|\\pi -t|[\/latex] or [latex]|180^\\circ \\mathrm{-t}|[\/latex].<\/li>\r\n \t<li>For an angle in the fourth quadrant, the reference angle is [latex]2\\pi -t[\/latex] or [latex]360^\\circ \\mathrm{-t}[\/latex].<\/li>\r\n \t<li>If an angle is less than [latex]0[\/latex] or greater than [latex]2\\pi [\/latex], add or subtract [latex]2\\pi [\/latex] as many times as needed to find an equivalent angle between [latex]0[\/latex] and [latex]2\\pi [\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding a Reference Angle<\/h3>\r\nFind the reference angle of [latex]225^\\circ [\/latex] as shown in below.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003606\/CNX_Precalc_Figure_05_02_0162.jpg\" alt=\"Graph of circle with 225 degree angle inscribed.\" width=\"487\" height=\"383\" \/>\r\n\r\n[reveal-answer q=\"770468\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"770468\"]\r\n\r\nBecause [latex]225^\\circ [\/latex] is in the third quadrant, the reference angle is\r\n<p style=\"text-align: center;\">[latex]|\\left(180^\\circ -225^\\circ \\right)|=|-45^\\circ |=45^\\circ [\/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\nFind the reference angle of [latex]\\frac{5\\pi }{3}[\/latex].\r\n\r\n[reveal-answer q=\"227547\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"227547\"]\r\n\r\n[latex]\\frac{\\pi }{3}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWe can evaluate trigonometric functions of angles outside the first quadrant using reference angles. The quadrant of the original angle determines whether the answer is positive or negative. To help us remember which of the six trigonometric functions are positive in each quadrant, we can use the mnemonic phrase \"A Smart Trig Class.\" Each of the four words in the phrase corresponds to one of the four quadrants, starting with quadrant I and rotating counterclockwise. In quadrant I, which is \"<strong>A<\/strong>,\" <strong><u>a<\/u><\/strong>ll of the six trigonometric functions are positive. In quadrant II, \"<strong>S<\/strong>mart,\" only <strong><u>s<\/u><\/strong>ine and its reciprocal function, cosecant, are positive. In quadrant III, \"<strong>T<\/strong>rig,\" only <strong><u>t<\/u><\/strong>angent and its reciprocal function, cotangent, are positive. Finally, in quadrant IV, \"<strong>C<\/strong>lass,\" only <strong><u>c<\/u><\/strong>osine and its reciprocal function, secant, are positive.\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\/27003705\/CNX_Precalc_Figure_05_03_0042.jpg\" alt=\"Graph of circle with each quadrant labeled. Under quadrant 1, labels fro sin t, cos t, tan t, sec t, csc t, and cot t. Under quadrant 2, labels for sin t and csc t. Under quadrant 3, labels for tan t and cot t. Under quadrant 4, labels for cos t, sec t.\" width=\"487\" height=\"363\" \/> <b>An illustration of which trigonometric functions are positive in each of the quadrants.<\/b>[\/caption]\r\n<h2>Evaluate Trigonometric Functions Using the Unit Circle<\/h2>\r\nThe unit circle tells us the value of cosine and sine at any of the given angle measures seen below. The first coordinate in each ordered pair is the value of cosine at the given angle measure, while the second coordinate in each ordered pair is the value of sine at the given angle measure. You will learn in Section 1.3 that all trigonometric functions can be written in terms of sine and cosine. Thus, if you can evaluate sine and cosine at various angle values, you can also evaluate the other trigonometric functions at various angle values. Take time to learn the [latex]\\left(x,y\\right)[\/latex] coordinates of all of the major angles in the first quadrant of the unit circle.\r\n\r\nRemember, every angle in quadrant two, three, or four has a reference angle that lies in quadrant one. The quadrant of the original angle only affects the sign (positive or negative) of a trigonometric function's value at a given angle.\r\n\r\n<a href=\"https:\/\/courses.lumenlearning.com\/precalctwoxmaster\/wp-content\/uploads\/sites\/145\/2015\/11\/f-d-43392176e093fa07f39e1f3687226d4d751b809be928c16abac5dcb3-IMAGE-IMAGE.png\"><img class=\"aligncenter wp-image-12625 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003609\/f-d-43392176e093fa07f39e1f3687226d4d751b809be928c16abac5dcb3-IMAGE-IMAGE.png\" alt=\"f-d-43392176e093fa07f39e1f3687226d4d751b809be928c16abac5dcb3+IMAGE+IMAGE\" width=\"800\" height=\"728\" \/><\/a>\r\n<div class=\"textbox\">\r\n<h3>How To: Given the angle of a point on The Unit circle, find the Value of Cosine (Or Sine) using quadrant one.<\/h3>\r\n<ol>\r\n \t<li>Find the reference angle using the appropriate reference angle formula from the first portion of this review section.<\/li>\r\n \t<li>Find the value of cosine (or sine) at the reference angle by looking at quadrant one of the unit circle.<\/li>\r\n \t<li>Determine the appropriate sign of your found value for cosine (or sine) based on the quadrant of the original angle.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Using the Unit Circle to Find the Value of cosine<\/h3>\r\nUse quadrant one of the unit circle to find the value of cosine at an angle of [latex]\\frac{7\\pi }{6}[\/latex].\r\n\r\n[reveal-answer q=\"865133\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"865133\"]\r\n\r\nWe know that the angle [latex]\\frac{7\\pi }{6}[\/latex] is in the third quadrant.\r\n\r\nFirst, let\u2019s find the reference angle. The reference angle is:\r\n<p style=\"text-align: center;\">[latex]\\frac{7\\pi }{6}-\\pi =\\frac{\\pi }{6}[\/latex]<\/p>\r\nNext, we find the value of cosine at the reference angle which is represented by the first coordinate of the ordered pair at\u00a0[latex]\\frac{\\pi }{6}[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\cos \\left(\\frac{\\pi }{6}\\right)=\\frac{\\sqrt{3}}{2}[\/latex]<\/p>\r\nBecause our original angle is in the third quadrant, where cosine is always negative, we have:\r\n<p style=\"text-align: center;\">[latex]\\cos \\left(\\frac{7\\pi }{6}\\right)=-\\frac{\\sqrt{3}}{2}[\/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\nUse quadrant one of the unit circle to find the value of sine at an angle of [latex]\\frac{5\\pi }{3}[\/latex].\r\n\r\n[reveal-answer q=\"913342\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"913342\"]\r\n\r\n[latex]-\\frac{\\sqrt{3}}{2}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Evaluate Inverse Trigonometric Functions<\/h2>\r\nIn order to use inverse trigonometric functions, we need to understand that an inverse trigonometric function \u201cundoes\u201d what the original trigonometric function \u201cdoes,\u201d as is the case with any other function and its inverse. In other words, the domain of the inverse function is the range of the original function, and vice versa, as summarized below.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163959\/CNX_Precalc_Figure_06_03_013.jpg\" alt=\"A chart that says \u201cTrig Functinos\u201d, \u201cInverse Trig Functions\u201d, \u201cDomain: Measure of an angle\u201d, \u201cDomain: Ratio\u201d, \u201cRange: Ratio\u201d, and \u201cRange: Measure of an angle\u201d.\" width=\"731\" height=\"78\" \/>\r\n\r\nFor example, if [latex]f(x)=\\sin x[\/latex], then we would write\u00a0[latex]f^{-1}(x)={\\sin}^{-1}{x}[\/latex]. Be aware that [latex]{\\sin}^{-1}x[\/latex] does not mean [latex]\\frac{1}{\\sin{x}}[\/latex]. The following examples illustrate the inverse trigonometric functions:\r\n<ul>\r\n \t<li>Since [latex]\\sin\\left(\\frac{\\pi}{6}\\right)=\\frac{1}{2}[\/latex], then [latex]\\frac{\\pi}{6}=\\sin^{\u22121}(\\frac{1}{2})[\/latex].<\/li>\r\n \t<li>Since [latex]\\cos(\\pi)=\u22121[\/latex], then [latex]\\pi=\\cos^{\u22121}(\u22121)[\/latex].<\/li>\r\n \t<li>Since [latex]\\tan\\left(\\frac{\\pi}{4}\\right)=1[\/latex], then [latex]\\frac{\\pi}{4}=\\tan^{\u22121}(1)[\/latex].<\/li>\r\n<\/ul>\r\n<div>\r\n<div style=\"text-align: center;\"><\/div>\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>A General Note: Relations for Inverse Sine, Cosine, and Tangent Functions<\/h3>\r\nFor angles in the interval [latex]\\left[\u2212\\frac{\\pi}{2}\\text{, }\\frac{\\pi}{2}\\right][\/latex], if [latex]\\sin y=x[\/latex], then [latex]\\sin^{\u22121}x=y[\/latex].\r\n\r\nFor angles in the interval [0, \u03c0], if [latex]\\cos y=x[\/latex], then [latex]\\cos^{\u22121}x=y[\/latex].\r\n\r\nFor angles in the interval [latex]\\left(\u2212\\frac{\\pi}{2}\\text{, }\\frac{\\pi}{2}\\right)[\/latex], if [latex]\\tan y=x[\/latex], then [latex]\\tan^{\u22121}x=y[\/latex].\r\n\r\n<\/div>\r\nJust as we did with the original trigonometric functions, we can give exact values for the inverse functions when we are using the special angles, specifically [latex]\\frac{\\pi}{ 6} (30^\\circ)\\text{, }\\frac{\\pi}{ 4} (45^\\circ),\\text{ and } \\frac{\\pi}{ 3} (60^\\circ)[\/latex], and their reflections into other quadrants.\r\n<div class=\"textbox\">\r\n<h3>How To: Given a \u201cspecial\u201d input value, evaluate an inverse trigonometric function.<\/h3>\r\n<ol>\r\n \t<li>Find angle\u00a0<em>x<\/em>\u00a0for which the original trigonometric function has an output equal to the given input for the inverse trigonometric function.<\/li>\r\n \t<li>If\u00a0<em>x<\/em>\u00a0is not in the defined range of the inverse, find another angle\u00a0<em>y<\/em>\u00a0that is in the defined range and has the same sine, cosine, or tangent as\u00a0<em>x<\/em>, depending on which corresponds to the given inverse function.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Evaluating Inverse Trigonometric Functions for Special Input Values<\/h3>\r\nEvaluate each of the following.\r\n<p style=\"padding-left: 60px;\">a. [latex]\\sin\u22121\\left(\\frac{1}{2}\\right)[\/latex]<\/p>\r\n<p style=\"padding-left: 60px;\">b. [latex]\\sin\u22121\\left(\u2212\\frac{2}{\\sqrt{2}}\\right)[\/latex]<\/p>\r\n<p style=\"padding-left: 60px;\">c. [latex]\\cos\u22121\\left(\u2212\\frac{3}{\\sqrt{2}}\\right)[\/latex]<\/p>\r\n<p style=\"padding-left: 60px;\">d. [latex]\\tan^{\u2212 1}(1)[\/latex]<\/p>\r\n[reveal-answer q=\"666370\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"666370\"]\r\n<p style=\"padding-left: 60px;\">a. Evaluating [latex]\\sin^{\u22121}(\\frac{1}{2})[\/latex] is the same as determining the angle that would have a sine value of [latex]\\frac{1}{2}[\/latex]. In other words, what angle <em>x<\/em> would satisfy [latex]\\sin(x)=\\frac{1}{2}[\/latex]? There are multiple values that would satisfy this relationship, such as [latex]\\frac{\\pi}{6}[\/latex] and [latex]\\frac{5\\pi}{6}[\/latex], but we know we need the angle in the interval [latex]\\left[\u2212\\frac{\\pi}{2}\\text{, }\\frac{\\pi}{2}\\right][\/latex], so the answer will be [latex]\\sin^{\u22121}(\\frac{1}{2})=\\frac{\\pi}{6}[\/latex]. Remember that the inverse is a function, so for each input, we will get exactly one output.<\/p>\r\n<p style=\"padding-left: 60px;\">b. To evaluate [latex]\\sin^{\u22121}\\left(\u2212\\frac{\\sqrt{2}}{2}\\right)[\/latex], we know that [latex]\\frac{5\\pi}{4}[\/latex] and [latex]\\frac{7\\pi}{4}[\/latex] both have a sine value of [latex]\u2212\\frac{\\sqrt{2}}{2}[\/latex], but neither is in the interval [latex]\\left[\u2212\\frac{\\pi}{2}\\text{, }\\frac{\\pi}{2}\\right][\/latex]. For that, we need the negative angle coterminal with [latex]\\frac{7\\pi}{4}:\\sin^{\u22121}\\left(\u2212\\frac{\\sqrt{2}}{2}\\right)=\u2212\\frac{\\pi}{4}[\/latex].<\/p>\r\n<p style=\"padding-left: 60px;\">c. To evaluate [latex]\\cos^{\u22121}\\left(\u2212\\frac{\\sqrt{3}}{2}\\right)[\/latex], we are looking for an angle in the interval [0,\u03c0] with a cosine value of [latex]\u2212\\frac{\\sqrt{3}}{2}[\/latex]. The angle that satisfies this is [latex]\\cos^{\u22121}\\left(\u2212\\frac{\\sqrt{3}}{2}\\right)=\\frac{5\\pi}{6}[\/latex].<\/p>\r\n<p style=\"padding-left: 60px;\">d. Evaluating [latex]\\tan^{\u22121}(1)[\/latex], we are looking for an angle in the interval [latex](\u2212\\frac{\\pi}{2}\\text{, }\\frac{\\pi}{2})[\/latex] with a tangent value of 1. The correct angle is [latex]\\tan^{\u22121}(1)=\\frac{\\pi}{4}[\/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\nEvaluate each of the following.\r\n<ol>\r\n \t<li>[latex]\\sin^{\u22121}(\u22121)[\/latex]<\/li>\r\n \t<li>[latex]\\tan^{\u22121}(\u22121)[\/latex]<\/li>\r\n \t<li>[latex]\\cos^{\u22121}(\u22121)[\/latex]<\/li>\r\n \t<li>[latex]\\cos^{\u22121}(\\frac{1}{2})[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"333778\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"333778\"]\r\n\r\n1. [latex]\u2212\\frac{\\pi}{2}[\/latex];\r\n\r\n2. [latex]\u2212\\frac{\\pi}{4}[\/latex]\r\n\r\n3. [latex]\\pi[\/latex]\r\n\r\n4. [latex]\\frac{\\pi}{3}[\/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]173433[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Draw an Angle in Standard Position<\/h2>\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>.\u00a0The\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>An angle has both an initial side and a terminal side.<\/b>[\/caption]\r\n\r\nWhen drawing an angle on an <em>x<\/em>-<em>y<\/em> coordinate plane, 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<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 exercises\">\r\n<h3>Example: Drawing an Angle<\/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<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\" \/>\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.<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\" \/><\/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\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<h2>Write the Equation of a Circle in Standard Form<\/h2>\r\nA circle is all points in a plane that are a fixed distance from a given point in the plane. The given point is called the <strong>center<\/strong>, (h,k), and the fixed distance is called the <strong>radius<\/strong>, r, of the circle.\r\n\r\nGiven a circle with center (h,k) and radius r, the equation of the circle in standard form is:\r\n<p style=\"text-align: center;\">[latex](x-h)^2+(y-k)^2=r^2[\/latex]<\/p>\r\n\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Writing the Equation of a Circle in Standard Form<\/h3>\r\nWrite the equation of a circle in standard form with a center of (5,7) and a radius of 4.\r\n\r\n[reveal-answer q=\"18285444\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"18285444\"]\r\n<p style=\"text-align: left;\">Standard form of a circle is\u00a0[latex](x-h)^2+(y-k)^2=r^2[\/latex]. In this case, [latex]h=5[\/latex], [latex]k=7[\/latex], and [latex]r=4[\/latex].<\/p>\r\nTherefore, the equation of the circle in standard form is:\r\n<p style=\"text-align: center;\">[latex](x-5)^2+(y-7)^2=(4)^2[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex](x-5)^2+(y-7)^2=16[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nWrite the equation of a circle in standard form with a center of (4,-2) and a radius of 3.\r\n\r\n[reveal-answer q=\"182854444\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"182854444\"]\r\n<p style=\"text-align: left;\">[latex](x-4)^2+(y+2)^2=9[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Identify reference angles for angles measured in both radians and degrees<\/li>\n<li>Evaluate trigonometric functions using the unit circle<\/li>\n<li>Evaluate inverse trigonometric functions<\/li>\n<li>Draw an Angle in Standard Position<\/li>\n<li>Write the equation of a circle in standard form<\/li>\n<\/ul>\n<\/div>\n<p>In the Polar Coordinates section, we will be introduced to polar coordinates and equations. Here we will review how to evaluate sine and cosine functions at specific angle measures, evaluate inverse trigonometric functions, draw angles, and write the equation of a circle in standard form.<\/p>\n<h2>Find Reference Angles<\/h2>\n<p>It is easiest to evaluate trigonometric functions when an angle is in the first quadrant. When the original angle is given in quadrant two, three, or four, a reference angle should be found.<\/p>\n<p>An angle\u2019s <strong>reference angle<\/strong> is the acute angle, [latex]t[\/latex], formed by the terminal side of the angle [latex]t[\/latex] and the horizontal axis. A reference angle is always an angle between [latex]0[\/latex] and [latex]90^\\circ[\/latex], or [latex]0[\/latex] and [latex]\\frac{\\pi }{2}[\/latex] radians. As we can see in the figure below, for any angle in quadrants II, III, or IV, there is a reference angle in quadrant I.<\/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\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003604\/CNX_Precalc_Figure_05_01_0195.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>A visual of the corresponding reference angles for each of the quadrants.<\/b><\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given an angle between [latex]0[\/latex] and [latex]2\\pi[\/latex], find its reference angle<\/h3>\n<ol>\n<li>An angle in the first quadrant is its own reference angle.<\/li>\n<li>For an angle in the second or third quadrant, the reference angle is [latex]|\\pi -t|[\/latex] or [latex]|180^\\circ \\mathrm{-t}|[\/latex].<\/li>\n<li>For an angle in the fourth quadrant, the reference angle is [latex]2\\pi -t[\/latex] or [latex]360^\\circ \\mathrm{-t}[\/latex].<\/li>\n<li>If an angle is less than [latex]0[\/latex] or greater than [latex]2\\pi[\/latex], add or subtract [latex]2\\pi[\/latex] as many times as needed to find an equivalent angle between [latex]0[\/latex] and [latex]2\\pi[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding a Reference Angle<\/h3>\n<p>Find the reference angle of [latex]225^\\circ[\/latex] as shown in below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003606\/CNX_Precalc_Figure_05_02_0162.jpg\" alt=\"Graph of circle with 225 degree angle inscribed.\" width=\"487\" height=\"383\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q770468\">Show Solution<\/span><\/p>\n<div id=\"q770468\" class=\"hidden-answer\" style=\"display: none\">\n<p>Because [latex]225^\\circ[\/latex] is in the third quadrant, the reference angle is<\/p>\n<p style=\"text-align: center;\">[latex]|\\left(180^\\circ -225^\\circ \\right)|=|-45^\\circ |=45^\\circ[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Find the reference angle of [latex]\\frac{5\\pi }{3}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q227547\">Show Solution<\/span><\/p>\n<div id=\"q227547\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\frac{\\pi }{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>We can evaluate trigonometric functions of angles outside the first quadrant using reference angles. The quadrant of the original angle determines whether the answer is positive or negative. To help us remember which of the six trigonometric functions are positive in each quadrant, we can use the mnemonic phrase &#8220;A Smart Trig Class.&#8221; Each of the four words in the phrase corresponds to one of the four quadrants, starting with quadrant I and rotating counterclockwise. In quadrant I, which is &#8220;<strong>A<\/strong>,&#8221; <strong><u>a<\/u><\/strong>ll of the six trigonometric functions are positive. In quadrant II, &#8220;<strong>S<\/strong>mart,&#8221; only <strong><u>s<\/u><\/strong>ine and its reciprocal function, cosecant, are positive. In quadrant III, &#8220;<strong>T<\/strong>rig,&#8221; only <strong><u>t<\/u><\/strong>angent and its reciprocal function, cotangent, are positive. Finally, in quadrant IV, &#8220;<strong>C<\/strong>lass,&#8221; only <strong><u>c<\/u><\/strong>osine and its reciprocal function, secant, are positive.<\/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\/27003705\/CNX_Precalc_Figure_05_03_0042.jpg\" alt=\"Graph of circle with each quadrant labeled. Under quadrant 1, labels fro sin t, cos t, tan t, sec t, csc t, and cot t. Under quadrant 2, labels for sin t and csc t. Under quadrant 3, labels for tan t and cot t. Under quadrant 4, labels for cos t, sec t.\" width=\"487\" height=\"363\" \/><\/p>\n<p class=\"wp-caption-text\"><b>An illustration of which trigonometric functions are positive in each of the quadrants.<\/b><\/p>\n<\/div>\n<h2>Evaluate Trigonometric Functions Using the Unit Circle<\/h2>\n<p>The unit circle tells us the value of cosine and sine at any of the given angle measures seen below. The first coordinate in each ordered pair is the value of cosine at the given angle measure, while the second coordinate in each ordered pair is the value of sine at the given angle measure. You will learn in Section 1.3 that all trigonometric functions can be written in terms of sine and cosine. Thus, if you can evaluate sine and cosine at various angle values, you can also evaluate the other trigonometric functions at various angle values. Take time to learn the [latex]\\left(x,y\\right)[\/latex] coordinates of all of the major angles in the first quadrant of the unit circle.<\/p>\n<p>Remember, every angle in quadrant two, three, or four has a reference angle that lies in quadrant one. The quadrant of the original angle only affects the sign (positive or negative) of a trigonometric function&#8217;s value at a given angle.<\/p>\n<p><a href=\"https:\/\/courses.lumenlearning.com\/precalctwoxmaster\/wp-content\/uploads\/sites\/145\/2015\/11\/f-d-43392176e093fa07f39e1f3687226d4d751b809be928c16abac5dcb3-IMAGE-IMAGE.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-12625 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003609\/f-d-43392176e093fa07f39e1f3687226d4d751b809be928c16abac5dcb3-IMAGE-IMAGE.png\" alt=\"f-d-43392176e093fa07f39e1f3687226d4d751b809be928c16abac5dcb3+IMAGE+IMAGE\" width=\"800\" height=\"728\" \/><\/a><\/p>\n<div class=\"textbox\">\n<h3>How To: Given the angle of a point on The Unit circle, find the Value of Cosine (Or Sine) using quadrant one.<\/h3>\n<ol>\n<li>Find the reference angle using the appropriate reference angle formula from the first portion of this review section.<\/li>\n<li>Find the value of cosine (or sine) at the reference angle by looking at quadrant one of the unit circle.<\/li>\n<li>Determine the appropriate sign of your found value for cosine (or sine) based on the quadrant of the original angle.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using the Unit Circle to Find the Value of cosine<\/h3>\n<p>Use quadrant one of the unit circle to find the value of cosine at an angle of [latex]\\frac{7\\pi }{6}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q865133\">Show Solution<\/span><\/p>\n<div id=\"q865133\" class=\"hidden-answer\" style=\"display: none\">\n<p>We know that the angle [latex]\\frac{7\\pi }{6}[\/latex] is in the third quadrant.<\/p>\n<p>First, let\u2019s find the reference angle. The reference angle is:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{7\\pi }{6}-\\pi =\\frac{\\pi }{6}[\/latex]<\/p>\n<p>Next, we find the value of cosine at the reference angle which is represented by the first coordinate of the ordered pair at\u00a0[latex]\\frac{\\pi }{6}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\cos \\left(\\frac{\\pi }{6}\\right)=\\frac{\\sqrt{3}}{2}[\/latex]<\/p>\n<p>Because our original angle is in the third quadrant, where cosine is always negative, we have:<\/p>\n<p style=\"text-align: center;\">[latex]\\cos \\left(\\frac{7\\pi }{6}\\right)=-\\frac{\\sqrt{3}}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Use quadrant one of the unit circle to find the value of sine at an angle of [latex]\\frac{5\\pi }{3}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q913342\">Show Solution<\/span><\/p>\n<div id=\"q913342\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]-\\frac{\\sqrt{3}}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Evaluate Inverse Trigonometric Functions<\/h2>\n<p>In order to use inverse trigonometric functions, we need to understand that an inverse trigonometric function \u201cundoes\u201d what the original trigonometric function \u201cdoes,\u201d as is the case with any other function and its inverse. In other words, the domain of the inverse function is the range of the original function, and vice versa, as summarized below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163959\/CNX_Precalc_Figure_06_03_013.jpg\" alt=\"A chart that says \u201cTrig Functinos\u201d, \u201cInverse Trig Functions\u201d, \u201cDomain: Measure of an angle\u201d, \u201cDomain: Ratio\u201d, \u201cRange: Ratio\u201d, and \u201cRange: Measure of an angle\u201d.\" width=\"731\" height=\"78\" \/><\/p>\n<p>For example, if [latex]f(x)=\\sin x[\/latex], then we would write\u00a0[latex]f^{-1}(x)={\\sin}^{-1}{x}[\/latex]. Be aware that [latex]{\\sin}^{-1}x[\/latex] does not mean [latex]\\frac{1}{\\sin{x}}[\/latex]. The following examples illustrate the inverse trigonometric functions:<\/p>\n<ul>\n<li>Since [latex]\\sin\\left(\\frac{\\pi}{6}\\right)=\\frac{1}{2}[\/latex], then [latex]\\frac{\\pi}{6}=\\sin^{\u22121}(\\frac{1}{2})[\/latex].<\/li>\n<li>Since [latex]\\cos(\\pi)=\u22121[\/latex], then [latex]\\pi=\\cos^{\u22121}(\u22121)[\/latex].<\/li>\n<li>Since [latex]\\tan\\left(\\frac{\\pi}{4}\\right)=1[\/latex], then [latex]\\frac{\\pi}{4}=\\tan^{\u22121}(1)[\/latex].<\/li>\n<\/ul>\n<div>\n<div style=\"text-align: center;\"><\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>A General Note: Relations for Inverse Sine, Cosine, and Tangent Functions<\/h3>\n<p>For angles in the interval [latex]\\left[\u2212\\frac{\\pi}{2}\\text{, }\\frac{\\pi}{2}\\right][\/latex], if [latex]\\sin y=x[\/latex], then [latex]\\sin^{\u22121}x=y[\/latex].<\/p>\n<p>For angles in the interval [0, \u03c0], if [latex]\\cos y=x[\/latex], then [latex]\\cos^{\u22121}x=y[\/latex].<\/p>\n<p>For angles in the interval [latex]\\left(\u2212\\frac{\\pi}{2}\\text{, }\\frac{\\pi}{2}\\right)[\/latex], if [latex]\\tan y=x[\/latex], then [latex]\\tan^{\u22121}x=y[\/latex].<\/p>\n<\/div>\n<p>Just as we did with the original trigonometric functions, we can give exact values for the inverse functions when we are using the special angles, specifically [latex]\\frac{\\pi}{ 6} (30^\\circ)\\text{, }\\frac{\\pi}{ 4} (45^\\circ),\\text{ and } \\frac{\\pi}{ 3} (60^\\circ)[\/latex], and their reflections into other quadrants.<\/p>\n<div class=\"textbox\">\n<h3>How To: Given a \u201cspecial\u201d input value, evaluate an inverse trigonometric function.<\/h3>\n<ol>\n<li>Find angle\u00a0<em>x<\/em>\u00a0for which the original trigonometric function has an output equal to the given input for the inverse trigonometric function.<\/li>\n<li>If\u00a0<em>x<\/em>\u00a0is not in the defined range of the inverse, find another angle\u00a0<em>y<\/em>\u00a0that is in the defined range and has the same sine, cosine, or tangent as\u00a0<em>x<\/em>, depending on which corresponds to the given inverse function.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Evaluating Inverse Trigonometric Functions for Special Input Values<\/h3>\n<p>Evaluate each of the following.<\/p>\n<p style=\"padding-left: 60px;\">a. [latex]\\sin\u22121\\left(\\frac{1}{2}\\right)[\/latex]<\/p>\n<p style=\"padding-left: 60px;\">b. [latex]\\sin\u22121\\left(\u2212\\frac{2}{\\sqrt{2}}\\right)[\/latex]<\/p>\n<p style=\"padding-left: 60px;\">c. [latex]\\cos\u22121\\left(\u2212\\frac{3}{\\sqrt{2}}\\right)[\/latex]<\/p>\n<p style=\"padding-left: 60px;\">d. [latex]\\tan^{\u2212 1}(1)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q666370\">Show Solution<\/span><\/p>\n<div id=\"q666370\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"padding-left: 60px;\">a. Evaluating [latex]\\sin^{\u22121}(\\frac{1}{2})[\/latex] is the same as determining the angle that would have a sine value of [latex]\\frac{1}{2}[\/latex]. In other words, what angle <em>x<\/em> would satisfy [latex]\\sin(x)=\\frac{1}{2}[\/latex]? There are multiple values that would satisfy this relationship, such as [latex]\\frac{\\pi}{6}[\/latex] and [latex]\\frac{5\\pi}{6}[\/latex], but we know we need the angle in the interval [latex]\\left[\u2212\\frac{\\pi}{2}\\text{, }\\frac{\\pi}{2}\\right][\/latex], so the answer will be [latex]\\sin^{\u22121}(\\frac{1}{2})=\\frac{\\pi}{6}[\/latex]. Remember that the inverse is a function, so for each input, we will get exactly one output.<\/p>\n<p style=\"padding-left: 60px;\">b. To evaluate [latex]\\sin^{\u22121}\\left(\u2212\\frac{\\sqrt{2}}{2}\\right)[\/latex], we know that [latex]\\frac{5\\pi}{4}[\/latex] and [latex]\\frac{7\\pi}{4}[\/latex] both have a sine value of [latex]\u2212\\frac{\\sqrt{2}}{2}[\/latex], but neither is in the interval [latex]\\left[\u2212\\frac{\\pi}{2}\\text{, }\\frac{\\pi}{2}\\right][\/latex]. For that, we need the negative angle coterminal with [latex]\\frac{7\\pi}{4}:\\sin^{\u22121}\\left(\u2212\\frac{\\sqrt{2}}{2}\\right)=\u2212\\frac{\\pi}{4}[\/latex].<\/p>\n<p style=\"padding-left: 60px;\">c. To evaluate [latex]\\cos^{\u22121}\\left(\u2212\\frac{\\sqrt{3}}{2}\\right)[\/latex], we are looking for an angle in the interval [0,\u03c0] with a cosine value of [latex]\u2212\\frac{\\sqrt{3}}{2}[\/latex]. The angle that satisfies this is [latex]\\cos^{\u22121}\\left(\u2212\\frac{\\sqrt{3}}{2}\\right)=\\frac{5\\pi}{6}[\/latex].<\/p>\n<p style=\"padding-left: 60px;\">d. Evaluating [latex]\\tan^{\u22121}(1)[\/latex], we are looking for an angle in the interval [latex](\u2212\\frac{\\pi}{2}\\text{, }\\frac{\\pi}{2})[\/latex] with a tangent value of 1. The correct angle is [latex]\\tan^{\u22121}(1)=\\frac{\\pi}{4}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Evaluate each of the following.<\/p>\n<ol>\n<li>[latex]\\sin^{\u22121}(\u22121)[\/latex]<\/li>\n<li>[latex]\\tan^{\u22121}(\u22121)[\/latex]<\/li>\n<li>[latex]\\cos^{\u22121}(\u22121)[\/latex]<\/li>\n<li>[latex]\\cos^{\u22121}(\\frac{1}{2})[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q333778\">Show Solution<\/span><\/p>\n<div id=\"q333778\" class=\"hidden-answer\" style=\"display: none\">\n<p>1. [latex]\u2212\\frac{\\pi}{2}[\/latex];<\/p>\n<p>2. [latex]\u2212\\frac{\\pi}{4}[\/latex]<\/p>\n<p>3. [latex]\\pi[\/latex]<\/p>\n<p>4. [latex]\\frac{\\pi}{3}[\/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=\"ohm173433\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=173433&theme=oea&iframe_resize_id=ohm173433\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Draw an Angle in Standard Position<\/h2>\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>.\u00a0The\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>An angle has both an initial side and a terminal side.<\/b><\/p>\n<\/div>\n<p>When drawing an angle on an <em>x<\/em>&#8211;<em>y<\/em> coordinate plane, 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 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 exercises\">\n<h3>Example: Drawing an Angle<\/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<p><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>&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.<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\" \/><\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>TRY IT<\/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<h2>Write the Equation of a Circle in Standard Form<\/h2>\n<p>A circle is all points in a plane that are a fixed distance from a given point in the plane. The given point is called the <strong>center<\/strong>, (h,k), and the fixed distance is called the <strong>radius<\/strong>, r, of the circle.<\/p>\n<p>Given a circle with center (h,k) and radius r, the equation of the circle in standard form is:<\/p>\n<p style=\"text-align: center;\">[latex](x-h)^2+(y-k)^2=r^2[\/latex]<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Writing the Equation of a Circle in Standard Form<\/h3>\n<p>Write the equation of a circle in standard form with a center of (5,7) and a radius of 4.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q18285444\">Show Solution<\/span><\/p>\n<div id=\"q18285444\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: left;\">Standard form of a circle is\u00a0[latex](x-h)^2+(y-k)^2=r^2[\/latex]. In this case, [latex]h=5[\/latex], [latex]k=7[\/latex], and [latex]r=4[\/latex].<\/p>\n<p>Therefore, the equation of the circle in standard form is:<\/p>\n<p style=\"text-align: center;\">[latex](x-5)^2+(y-7)^2=(4)^2[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex](x-5)^2+(y-7)^2=16[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Write the equation of a circle in standard form with a center of (4,-2) and a radius of 3.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q182854444\">Show Solution<\/span><\/p>\n<div id=\"q182854444\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: left;\">[latex](x-4)^2+(y+2)^2=9[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1147\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 1. <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.lumenlearning.com\/calculus1\/\">https:\/\/courses.lumenlearning.com\/calculus1\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Calculus Volume 2. <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.lumenlearning.com\/calculus2\/\">https:\/\/courses.lumenlearning.com\/calculus2\/<\/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>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":349141,"menu_order":2,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 1\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"https:\/\/courses.lumenlearning.com\/calculus1\/\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Calculus Volume 2\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"https:\/\/courses.lumenlearning.com\/calculus2\/\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1147","chapter","type-chapter","status-publish","hentry"],"part":1145,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/1147","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/349141"}],"version-history":[{"count":2,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/1147\/revisions"}],"predecessor-version":[{"id":6480,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/1147\/revisions\/6480"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/1145"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/1147\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=1147"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=1147"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=1147"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=1147"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}