{"id":2338,"date":"2021-09-29T16:29:05","date_gmt":"2021-09-29T16:29:05","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/skills-review-for-parametric-equations\/"},"modified":"2021-11-19T03:22:12","modified_gmt":"2021-11-19T03:22:12","slug":"skills-review-for-parametric-equations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/skills-review-for-parametric-equations\/","title":{"raw":"Skills Review for Parametric Equations and Calculus of Parametric Equations","rendered":"Skills Review for Parametric Equations and Calculus of Parametric Equations"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Evaluate algebraic expressions<\/li>\r\n \t<li>Evaluate trigonometric functions using the unit circle<\/li>\r\n \t<li>Use substitution to rewrite a mathematical equation<\/li>\r\n \t<li>Write the equation of a line using slope and a point on the line<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn the Parametric Equations and Calculus of Parametric Equations sections, we will be working with parametric equations, graphing them, and using calculus to differentiate and integrate them. Here we will review how to evaluate both algebraic and trigonometric expressions, how to use substitution to rewrite a mathematical equation, and how to write the equation of a line.\r\n<h2>Evaluate Algebraic Expressions<\/h2>\r\nAn <strong>algebraic expression<\/strong> is a collection of constants and variables joined together by the algebraic operations of addition, subtraction, multiplication, and division.\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Evaluating Algebraic Expressions<\/h3>\r\nEvaluate each expression for the given values.\r\n<ol>\r\n \t<li>[latex]x+5[\/latex] for [latex]x=-5[\/latex]<\/li>\r\n \t<li>[latex]\\frac{t}{2t - 1}[\/latex] for [latex]t=10[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{4}{3}\\pi {r}^{3}[\/latex] for [latex]r=5[\/latex]<\/li>\r\n \t<li>[latex]a+ab+b[\/latex] for [latex]a=11,b=-8[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{2{m}^{3}{n}^{2}}[\/latex] for [latex]m=2,n=3[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"182854\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"182854\"]\r\n<ol>\r\n \t<li>Substitute [latex]-5[\/latex] for [latex]x[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{align}x+5 &amp;=\\left(-5\\right)+5 \\\\ &amp;=0\\end{align}[\/latex]<\/div><\/li>\r\n \t<li>Substitute 10 for [latex]t[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{align}\\frac{t}{2t-1} &amp; =\\frac{\\left(10\\right)}{2\\left(10\\right)-1} \\\\ &amp; =\\frac{10}{20-1} \\\\ &amp; =\\frac{10}{19}\\end{align}[\/latex]<\/div><\/li>\r\n \t<li>Substitute 5 for [latex]r[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{align}\\frac{4}{3}\\pi r^{3} &amp; =\\frac{4}{3}\\pi\\left(5\\right)^{3} \\\\ &amp; =\\frac{4}{3}\\pi\\left(125\\right) \\\\ &amp; =\\frac{500}{3}\\pi\\end{align}[\/latex]<\/div><\/li>\r\n \t<li>Substitute 11 for [latex]a[\/latex] and \u20138 for [latex]b[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{align}a+ab+b &amp; =\\left(11\\right)+\\left(11\\right)\\left(-8\\right)+\\left(-8\\right) \\\\ &amp; =11-8-8 \\\\ &amp; =-85\\end{align}[\/latex]<\/div><\/li>\r\n \t<li>Substitute 2 for [latex]m[\/latex] and 3 for [latex]n[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{align}\\sqrt{2m^{3}n^{2}} &amp; =\\sqrt{2\\left(2\\right)^{3}\\left(3\\right)^{2}} \\\\ &amp; =\\sqrt{2\\left(8\\right)\\left(9\\right)} \\\\ &amp; =\\sqrt{144} \\\\ &amp; =12\\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\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=483&amp;theme=oea&amp;iframe_resize_id=mom10[\/embed]\r\n\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92388&amp;theme=oea&amp;iframe_resize_id=mom12[\/embed]\r\n\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=109700&amp;theme=oea&amp;iframe_resize_id=mom13[\/embed]\r\n\r\n<\/div>\r\n<h2>Evaluate Trigonometric Functions Using the Unit Circle<\/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\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>Use Substitution to Rewrite a Mathematical Equation<\/h2>\r\nSubstitution can be used to rewrite a mathematical expression in terms of another variable.\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Using Substitution to Rewrite a Mathematical Expression<\/h3>\r\nRewrite the equation [latex]x=4t+3[\/latex] in terms of [latex]y[\/latex] using [latex]y=2t-1[\/latex].\r\n\r\n[reveal-answer q=\"786744\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"786744\"]\r\n\r\nFirst, we will solve the equation [latex]y=2t-1[\/latex] for [latex]t[\/latex].\r\n<p style=\"text-align: center;\">[latex]y+1=2t[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\dfrac{y+1}{2}=t[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]t=\\dfrac{y+1}{2}[\/latex]<\/p>\r\nNow we can substitute [latex]t=\\dfrac{y+1}{2}[\/latex] into the equation\u00a0[latex]x=4t+3[\/latex].\r\n<p style=\"text-align: center;\">[latex]x=4(\\dfrac{y+1}{2})+3[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]x=2(y+1)+3[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]x=2y+2+3[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]x=2y+5[\/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\nRewrite the equation [latex]y=2t^{2}-1[\/latex] in terms of [latex]x[\/latex] using [latex]x=5-t[\/latex].\r\n\r\n[reveal-answer q=\"783260\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"783260\"]\r\n\r\n[latex]2x^2-20x+49[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Write the Equation of a Line<\/h2>\r\nTo write the equation of a line, the line's slope and a point the line goes through must be known. Perhaps the most familiar form of a linear equation is <strong>slope-intercept form<\/strong> written as [latex]y=mx+b[\/latex], where [latex]m=\\text{slope}[\/latex] and [latex]b=y\\text{-intercept}[\/latex]. Let us begin with the slope.\r\n\r\nOften, the starting point to writing the equation of a line is to use <strong>point-slope formula<\/strong>.\u00a0Given the slope and one point on a line, we can find the equation of the line using point-slope form shown below.\r\n<div style=\"text-align: center;\">[latex]y-{y}_{1}=m\\left(x-{x}_{1}\\right)[\/latex]<\/div>\r\nWe need only one point and the slope of the line to use the formula. After substituting the slope and the coordinates of one point into the formula, we simplify it and write it in slope-intercept form.\r\n<div class=\"textbox\">\r\n<h3>A General Note: The Point-Slope Formula<\/h3>\r\nGiven one point and the slope, using point-slope form will lead to the equation of a line:\r\n<div style=\"text-align: center;\">[latex]y-{y}_{1}=m\\left(x-{x}_{1}\\right)[\/latex]<\/div>\r\n<\/div>\r\n<div style=\"text-align: left;\">\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding the Equation of a Line Given the Slope and One Point<\/h3>\r\nWrite the equation of the line with slope [latex]m=-3[\/latex] and passing through the point [latex]\\left(4,8\\right)[\/latex]. Write the final equation in slope-intercept form.\r\n[reveal-answer q=\"201330\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"201330\"]\r\n\r\nUsing point-slope form, substitute [latex]-3[\/latex] for <em>m <\/em>and the point [latex]\\left(4,8\\right)[\/latex] for [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}y-{y}_{1}=m\\left(x-{x}_{1}\\right)\\hfill \\\\ y - 8=-3\\left(x - 4\\right)\\hfill \\\\ y - 8=-3x+12\\hfill \\\\ y=-3x+20\\hfill \\end{array}[\/latex]<\/div>\r\n<div>\r\n<div>\r\n<h4>Analysis of the Solution<\/h4>\r\n<\/div>\r\n<div>\r\n\r\nNote that any point on the line can be used to find the equation. If done correctly, the same final equation will be obtained.\r\n\r\n<\/div>\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nGiven [latex]m=4[\/latex], find the equation of the line in slope-intercept form passing through the point [latex]\\left(2,5\\right)[\/latex].\r\n\r\n[reveal-answer q=\"634647\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"634647\"]\r\n\r\n[latex]y=4x - 3[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]110942[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Evaluate algebraic expressions<\/li>\n<li>Evaluate trigonometric functions using the unit circle<\/li>\n<li>Use substitution to rewrite a mathematical equation<\/li>\n<li>Write the equation of a line using slope and a point on the line<\/li>\n<\/ul>\n<\/div>\n<p>In the Parametric Equations and Calculus of Parametric Equations sections, we will be working with parametric equations, graphing them, and using calculus to differentiate and integrate them. Here we will review how to evaluate both algebraic and trigonometric expressions, how to use substitution to rewrite a mathematical equation, and how to write the equation of a line.<\/p>\n<h2>Evaluate Algebraic Expressions<\/h2>\n<p>An <strong>algebraic expression<\/strong> is a collection of constants and variables joined together by the algebraic operations of addition, subtraction, multiplication, and division.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Evaluating Algebraic Expressions<\/h3>\n<p>Evaluate each expression for the given values.<\/p>\n<ol>\n<li>[latex]x+5[\/latex] for [latex]x=-5[\/latex]<\/li>\n<li>[latex]\\frac{t}{2t - 1}[\/latex] for [latex]t=10[\/latex]<\/li>\n<li>[latex]\\dfrac{4}{3}\\pi {r}^{3}[\/latex] for [latex]r=5[\/latex]<\/li>\n<li>[latex]a+ab+b[\/latex] for [latex]a=11,b=-8[\/latex]<\/li>\n<li>[latex]\\sqrt{2{m}^{3}{n}^{2}}[\/latex] for [latex]m=2,n=3[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q182854\">Show Solution<\/span><\/p>\n<div id=\"q182854\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>Substitute [latex]-5[\/latex] for [latex]x[\/latex].\n<div style=\"text-align: center;\">[latex]\\begin{align}x+5 &=\\left(-5\\right)+5 \\\\ &=0\\end{align}[\/latex]<\/div>\n<\/li>\n<li>Substitute 10 for [latex]t[\/latex].\n<div style=\"text-align: center;\">[latex]\\begin{align}\\frac{t}{2t-1} & =\\frac{\\left(10\\right)}{2\\left(10\\right)-1} \\\\ & =\\frac{10}{20-1} \\\\ & =\\frac{10}{19}\\end{align}[\/latex]<\/div>\n<\/li>\n<li>Substitute 5 for [latex]r[\/latex].\n<div style=\"text-align: center;\">[latex]\\begin{align}\\frac{4}{3}\\pi r^{3} & =\\frac{4}{3}\\pi\\left(5\\right)^{3} \\\\ & =\\frac{4}{3}\\pi\\left(125\\right) \\\\ & =\\frac{500}{3}\\pi\\end{align}[\/latex]<\/div>\n<\/li>\n<li>Substitute 11 for [latex]a[\/latex] and \u20138 for [latex]b[\/latex].\n<div style=\"text-align: center;\">[latex]\\begin{align}a+ab+b & =\\left(11\\right)+\\left(11\\right)\\left(-8\\right)+\\left(-8\\right) \\\\ & =11-8-8 \\\\ & =-85\\end{align}[\/latex]<\/div>\n<\/li>\n<li>Substitute 2 for [latex]m[\/latex] and 3 for [latex]n[\/latex].\n<div style=\"text-align: center;\">[latex]\\begin{align}\\sqrt{2m^{3}n^{2}} & =\\sqrt{2\\left(2\\right)^{3}\\left(3\\right)^{2}} \\\\ & =\\sqrt{2\\left(8\\right)\\left(9\\right)} \\\\ & =\\sqrt{144} \\\\ & =12\\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><iframe loading=\"lazy\" id=\"ohm483\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=483&#38;theme=oea&#38;iframe_resize_id=ohm483&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm92388\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92388&#38;theme=oea&#38;iframe_resize_id=ohm92388&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm109700\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=109700&#38;theme=oea&#38;iframe_resize_id=ohm109700&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Evaluate Trigonometric Functions Using the Unit Circle<\/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<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>Use Substitution to Rewrite a Mathematical Equation<\/h2>\n<p>Substitution can be used to rewrite a mathematical expression in terms of another variable.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Using Substitution to Rewrite a Mathematical Expression<\/h3>\n<p>Rewrite the equation [latex]x=4t+3[\/latex] in terms of [latex]y[\/latex] using [latex]y=2t-1[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q786744\">Show Solution<\/span><\/p>\n<div id=\"q786744\" class=\"hidden-answer\" style=\"display: none\">\n<p>First, we will solve the equation [latex]y=2t-1[\/latex] for [latex]t[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]y+1=2t[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{y+1}{2}=t[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]t=\\dfrac{y+1}{2}[\/latex]<\/p>\n<p>Now we can substitute [latex]t=\\dfrac{y+1}{2}[\/latex] into the equation\u00a0[latex]x=4t+3[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]x=4(\\dfrac{y+1}{2})+3[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]x=2(y+1)+3[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]x=2y+2+3[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]x=2y+5[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Rewrite the equation [latex]y=2t^{2}-1[\/latex] in terms of [latex]x[\/latex] using [latex]x=5-t[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q783260\">Show Solution<\/span><\/p>\n<div id=\"q783260\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]2x^2-20x+49[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Write the Equation of a Line<\/h2>\n<p>To write the equation of a line, the line&#8217;s slope and a point the line goes through must be known. Perhaps the most familiar form of a linear equation is <strong>slope-intercept form<\/strong> written as [latex]y=mx+b[\/latex], where [latex]m=\\text{slope}[\/latex] and [latex]b=y\\text{-intercept}[\/latex]. Let us begin with the slope.<\/p>\n<p>Often, the starting point to writing the equation of a line is to use <strong>point-slope formula<\/strong>.\u00a0Given the slope and one point on a line, we can find the equation of the line using point-slope form shown below.<\/p>\n<div style=\"text-align: center;\">[latex]y-{y}_{1}=m\\left(x-{x}_{1}\\right)[\/latex]<\/div>\n<p>We need only one point and the slope of the line to use the formula. After substituting the slope and the coordinates of one point into the formula, we simplify it and write it in slope-intercept form.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: The Point-Slope Formula<\/h3>\n<p>Given one point and the slope, using point-slope form will lead to the equation of a line:<\/p>\n<div style=\"text-align: center;\">[latex]y-{y}_{1}=m\\left(x-{x}_{1}\\right)[\/latex]<\/div>\n<\/div>\n<div style=\"text-align: left;\">\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Equation of a Line Given the Slope and One Point<\/h3>\n<p>Write the equation of the line with slope [latex]m=-3[\/latex] and passing through the point [latex]\\left(4,8\\right)[\/latex]. Write the final equation in slope-intercept form.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q201330\">Show Solution<\/span><\/p>\n<div id=\"q201330\" class=\"hidden-answer\" style=\"display: none\">\n<p>Using point-slope form, substitute [latex]-3[\/latex] for <em>m <\/em>and the point [latex]\\left(4,8\\right)[\/latex] for [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}y-{y}_{1}=m\\left(x-{x}_{1}\\right)\\hfill \\\\ y - 8=-3\\left(x - 4\\right)\\hfill \\\\ y - 8=-3x+12\\hfill \\\\ y=-3x+20\\hfill \\end{array}[\/latex]<\/div>\n<div>\n<div>\n<h4>Analysis of the Solution<\/h4>\n<\/div>\n<div>\n<p>Note that any point on the line can be used to find the equation. If done correctly, the same final equation will be obtained.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Given [latex]m=4[\/latex], find the equation of the line in slope-intercept form passing through the point [latex]\\left(2,5\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q634647\">Show Solution<\/span><\/p>\n<div id=\"q634647\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]y=4x - 3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm110942\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=110942&theme=oea&iframe_resize_id=ohm110942&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\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-2338\">\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>Modification and Revision. <strong>Provided by<\/strong>: Lumen Learning. <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\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra Corequisite. <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/\">https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/<\/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>Precalculus. <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.lumenlearning.com\/precalculus\/\">https:\/\/courses.lumenlearning.com\/precalculus\/<\/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":6,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Modification and Revision\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra Corequisite\",\"author\":\"\",\"organization\":\"Lumen 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