{"id":4107,"date":"2022-04-14T18:15:55","date_gmt":"2022-04-14T18:15:55","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/skills-review-for-applications\/"},"modified":"2022-11-09T16:50:59","modified_gmt":"2022-11-09T16:50:59","slug":"skills-review-for-applications","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/skills-review-for-applications\/","title":{"raw":"Skills Review for Applications","rendered":"Skills Review for Applications"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Define direct variation and solve problems involving direct variation<\/li>\r\n \t<li>Evaluate inverse trigonometric functions<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn the Applications section, we will learn about various applications represented by second-order differential equations. Here we will review how to write direct variation equations and evaluate the inverse tangent function.\r\n<h2>Write Direct Variation Equations<\/h2>\r\nA used-car company has just offered their best candidate, Nicole, a position in sales. The position offers 16% commission on her sales. Her earnings depend on the amount of her sales. For instance, if she sells a vehicle for $4,600, she will earn $736. She wants to evaluate the offer, but she is not sure how. In this section, we will look at relationships, such as this one, between earnings, sales, and commission rate.\r\n\r\nIn the example above, Nicole\u2019s earnings can be found by multiplying her sales by her commission. The formula [latex]e = 0.16s[\/latex] tells us her earnings, [latex]e[\/latex], come from the product of 0.16, her commission, and the sale price of the vehicle, [latex]s[\/latex]. If we create a table, we observe that as the sales price increases, the earnings increase as well, which should be intuitive.\r\n<table summary=\"..\">\r\n<thead>\r\n<tr>\r\n<th>[latex]s[\/latex], sales prices<\/th>\r\n<th>[latex]e = 0.16s[\/latex]<\/th>\r\n<th>Interpretation<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>$4,600<\/td>\r\n<td>[latex]e=0.16(4,600)=736[\/latex]<\/td>\r\n<td>A sale of a $4,600 vehicle results in $736 earnings.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>$9,200<\/td>\r\n<td>[latex]e=0.16(9,200)=1,472[\/latex]<\/td>\r\n<td>A sale of a $9,200 vehicle results in $1472 earnings.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>$18,400<\/td>\r\n<td>[latex]e=0.16(18,400)=2,944[\/latex]<\/td>\r\n<td>A sale of a $18,400 vehicle results in $2944 earnings.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNotice that earnings are a multiple of sales. As sales increase, earnings increase in a predictable way. Double the sales of the vehicle from $4,600 to $9,200, and we double the earnings from $736 to $1,472. As the input increases, the output increases as a multiple of the input. A relationship in which one quantity is a constant multiplied by another quantity is called <strong>direct variation<\/strong>. Each variable in this type of relationship <strong>varies directly <\/strong>with the other.\r\n\r\nThe graph below\u00a0represents the data for Nicole\u2019s potential earnings. We say that earnings vary directly with the sales price of the car. The formula [latex]y=k{x}^{n}[\/latex] is used for direct variation. The value [latex]k[\/latex] is a nonzero constant greater than zero and is called the <strong>constant of variation<\/strong>. In this case, [latex]k=0.16[\/latex]\u00a0and [latex]n=1[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02222950\/CNX_Precalc_Figure_03_09_0012.jpg\" alt=\"Graph of y=(0.16)x where the horizontal axis is labeled,\" width=\"487\" height=\"459\" \/>\r\n<div class=\"textbox\">\r\n<h3>A General Note: Direct Variation<\/h3>\r\nIf [latex]x[\/latex]<em>\u00a0<\/em>and [latex]y[\/latex]\u00a0are related by an equation of the form\r\n<p style=\"text-align: center;\">[latex]y=k{x}^{n}[\/latex]<\/p>\r\nthen we say that the relationship is <strong>direct variation<\/strong> and [latex]y[\/latex]\u00a0<strong>varies directly<\/strong> with the [latex]n[\/latex]th power of [latex]x[\/latex]. In direct variation relationships, there is a nonzero constant ratio [latex]k=\\dfrac{y}{{x}^{n}}[\/latex], where [latex]k[\/latex]\u00a0is called the <strong>constant of variation<\/strong>, which help defines the relationship between the variables.\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a description of a direct variation problem, solve for an unknown<strong>\r\n<\/strong><\/h3>\r\n<ol id=\"fs-id1165137724401\">\r\n \t<li>Identify the input, [latex]x[\/latex], and the output, [latex]y[\/latex].<\/li>\r\n \t<li>Determine the constant of variation. You may need to divide [latex]y[\/latex]\u00a0by the specified power of [latex]x[\/latex]\u00a0to determine the constant of variation.<\/li>\r\n \t<li>Use the constant of variation to write an equation for the relationship.<\/li>\r\n \t<li>Substitute known values into the equation to find the unknown.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Writing a Direct Variation Equation<\/h3>\r\nThe quantity [latex]y[\/latex]\u00a0varies directly with the cube of [latex]x[\/latex]. If [latex]y=25[\/latex]\u00a0when [latex]x=2[\/latex], write the equation that represents this relationship. Then, find [latex]y[\/latex]\u00a0when [latex]x[\/latex]\u00a0is 6.\r\n\r\n[reveal-answer q=\"647220\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"647220\"]\r\n\r\nThe general formula for direct variation with a cube is [latex]y=k{x}^{3}[\/latex]. The constant can be found by dividing [latex]y[\/latex]\u00a0by the cube of [latex]x[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{align} k&amp;=\\dfrac{y}{{x}^{3}} \\\\[1mm] &amp;=\\dfrac{25}{{2}^{3}}\\\\[1mm] &amp;=\\dfrac{25}{8}\\end{align}[\/latex]<\/p>\r\nNow use the constant to write an equation that represents this relationship.\r\n<p style=\"text-align: center;\">[latex]y=\\dfrac{25}{8}{x}^{3}[\/latex]<\/p>\r\nSubstitute [latex]x=6[\/latex]\u00a0and solve for [latex]y[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{align}y&amp;=\\dfrac{25}{8}{\\left(6\\right)}^{3} \\\\[1mm] &amp;=675\\hfill \\end{align}[\/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\nThe quantity [latex]y[\/latex]\u00a0varies directly with the square of [latex]y[\/latex]. If [latex]y=24[\/latex]\u00a0when [latex]x=3[\/latex], find [latex]y[\/latex]\u00a0when [latex]x[\/latex]\u00a0is 4.\r\n\r\n[reveal-answer q=\"536994\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"536994\"]\r\n\r\n[latex]\\dfrac{128}{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[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=91391&amp;theme=oea&amp;iframe_resize_id=mom5[\/embed]\r\n\r\n<\/div>\r\nWatch this video to see a quick lesson about direct variation. You will see more worked examples.\r\n\r\nhttps:\/\/youtu.be\/plFOq4JaEyI\r\n<h2>Evaluate Inverse Trigonometric Functions<\/h2>\r\n<strong><em>(also in Module 1, Skills Review for Polar Coordinates)<\/em><\/strong>\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\nNote that sometimes you will not be able to find an exact angle measure for a given inverse trigonometric function. Instead, you can use a calculator. First, be sure you put the calculator in the correct mode (degrees or radians). Then, select the inverse trigonometric function of interest along with the corresponding values that go inside the function. Your calculator will then give you the angle measure in degrees (if in degree mode) or in radians (if in radian mode).","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Define direct variation and solve problems involving direct variation<\/li>\n<li>Evaluate inverse trigonometric functions<\/li>\n<\/ul>\n<\/div>\n<p>In the Applications section, we will learn about various applications represented by second-order differential equations. Here we will review how to write direct variation equations and evaluate the inverse tangent function.<\/p>\n<h2>Write Direct Variation Equations<\/h2>\n<p>A used-car company has just offered their best candidate, Nicole, a position in sales. The position offers 16% commission on her sales. Her earnings depend on the amount of her sales. For instance, if she sells a vehicle for $4,600, she will earn $736. She wants to evaluate the offer, but she is not sure how. In this section, we will look at relationships, such as this one, between earnings, sales, and commission rate.<\/p>\n<p>In the example above, Nicole\u2019s earnings can be found by multiplying her sales by her commission. The formula [latex]e = 0.16s[\/latex] tells us her earnings, [latex]e[\/latex], come from the product of 0.16, her commission, and the sale price of the vehicle, [latex]s[\/latex]. If we create a table, we observe that as the sales price increases, the earnings increase as well, which should be intuitive.<\/p>\n<table summary=\"..\">\n<thead>\n<tr>\n<th>[latex]s[\/latex], sales prices<\/th>\n<th>[latex]e = 0.16s[\/latex]<\/th>\n<th>Interpretation<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>$4,600<\/td>\n<td>[latex]e=0.16(4,600)=736[\/latex]<\/td>\n<td>A sale of a $4,600 vehicle results in $736 earnings.<\/td>\n<\/tr>\n<tr>\n<td>$9,200<\/td>\n<td>[latex]e=0.16(9,200)=1,472[\/latex]<\/td>\n<td>A sale of a $9,200 vehicle results in $1472 earnings.<\/td>\n<\/tr>\n<tr>\n<td>$18,400<\/td>\n<td>[latex]e=0.16(18,400)=2,944[\/latex]<\/td>\n<td>A sale of a $18,400 vehicle results in $2944 earnings.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Notice that earnings are a multiple of sales. As sales increase, earnings increase in a predictable way. Double the sales of the vehicle from $4,600 to $9,200, and we double the earnings from $736 to $1,472. As the input increases, the output increases as a multiple of the input. A relationship in which one quantity is a constant multiplied by another quantity is called <strong>direct variation<\/strong>. Each variable in this type of relationship <strong>varies directly <\/strong>with the other.<\/p>\n<p>The graph below\u00a0represents the data for Nicole\u2019s potential earnings. We say that earnings vary directly with the sales price of the car. The formula [latex]y=k{x}^{n}[\/latex] is used for direct variation. The value [latex]k[\/latex] is a nonzero constant greater than zero and is called the <strong>constant of variation<\/strong>. In this case, [latex]k=0.16[\/latex]\u00a0and [latex]n=1[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02222950\/CNX_Precalc_Figure_03_09_0012.jpg\" alt=\"Graph of y=(0.16)x where the horizontal axis is labeled,\" width=\"487\" height=\"459\" \/><\/p>\n<div class=\"textbox\">\n<h3>A General Note: Direct Variation<\/h3>\n<p>If [latex]x[\/latex]<em>\u00a0<\/em>and [latex]y[\/latex]\u00a0are related by an equation of the form<\/p>\n<p style=\"text-align: center;\">[latex]y=k{x}^{n}[\/latex]<\/p>\n<p>then we say that the relationship is <strong>direct variation<\/strong> and [latex]y[\/latex]\u00a0<strong>varies directly<\/strong> with the [latex]n[\/latex]th power of [latex]x[\/latex]. In direct variation relationships, there is a nonzero constant ratio [latex]k=\\dfrac{y}{{x}^{n}}[\/latex], where [latex]k[\/latex]\u00a0is called the <strong>constant of variation<\/strong>, which help defines the relationship between the variables.<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a description of a direct variation problem, solve for an unknown<strong><br \/>\n<\/strong><\/h3>\n<ol id=\"fs-id1165137724401\">\n<li>Identify the input, [latex]x[\/latex], and the output, [latex]y[\/latex].<\/li>\n<li>Determine the constant of variation. You may need to divide [latex]y[\/latex]\u00a0by the specified power of [latex]x[\/latex]\u00a0to determine the constant of variation.<\/li>\n<li>Use the constant of variation to write an equation for the relationship.<\/li>\n<li>Substitute known values into the equation to find the unknown.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Writing a Direct Variation Equation<\/h3>\n<p>The quantity [latex]y[\/latex]\u00a0varies directly with the cube of [latex]x[\/latex]. If [latex]y=25[\/latex]\u00a0when [latex]x=2[\/latex], write the equation that represents this relationship. Then, find [latex]y[\/latex]\u00a0when [latex]x[\/latex]\u00a0is 6.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q647220\">Show Solution<\/span><\/p>\n<div id=\"q647220\" class=\"hidden-answer\" style=\"display: none\">\n<p>The general formula for direct variation with a cube is [latex]y=k{x}^{3}[\/latex]. The constant can be found by dividing [latex]y[\/latex]\u00a0by the cube of [latex]x[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align} k&=\\dfrac{y}{{x}^{3}} \\\\[1mm] &=\\dfrac{25}{{2}^{3}}\\\\[1mm] &=\\dfrac{25}{8}\\end{align}[\/latex]<\/p>\n<p>Now use the constant to write an equation that represents this relationship.<\/p>\n<p style=\"text-align: center;\">[latex]y=\\dfrac{25}{8}{x}^{3}[\/latex]<\/p>\n<p>Substitute [latex]x=6[\/latex]\u00a0and solve for [latex]y[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}y&=\\dfrac{25}{8}{\\left(6\\right)}^{3} \\\\[1mm] &=675\\hfill \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>The quantity [latex]y[\/latex]\u00a0varies directly with the square of [latex]y[\/latex]. If [latex]y=24[\/latex]\u00a0when [latex]x=3[\/latex], find [latex]y[\/latex]\u00a0when [latex]x[\/latex]\u00a0is 4.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q536994\">Show Solution<\/span><\/p>\n<div id=\"q536994\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\dfrac{128}{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=\"ohm91391\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=91391&#38;theme=oea&#38;iframe_resize_id=ohm91391&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Watch this video to see a quick lesson about direct variation. You will see more worked examples.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Direct Variation Applications\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/plFOq4JaEyI?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Evaluate Inverse Trigonometric Functions<\/h2>\n<p><strong><em>(also in Module 1, Skills Review for Polar Coordinates)<\/em><\/strong><br \/>\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.<\/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<p>Note that sometimes you will not be able to find an exact angle measure for a given inverse trigonometric function. Instead, you can use a calculator. First, be sure you put the calculator in the correct mode (degrees or radians). Then, select the inverse trigonometric function of interest along with the corresponding values that go inside the function. Your calculator will then give you the angle measure in degrees (if in degree mode) or in radians (if in radian mode).<\/p>\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-4107\">\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-4107","chapter","type-chapter","status-publish","hentry"],"part":4150,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/4107","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":4,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/4107\/revisions"}],"predecessor-version":[{"id":6496,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/4107\/revisions\/6496"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/4150"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/4107\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=4107"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=4107"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=4107"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=4107"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}