{"id":4093,"date":"2022-04-14T18:15:53","date_gmt":"2022-04-14T18:15:53","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/skills-review-for-functions-of-several-variables\/"},"modified":"2022-11-09T16:35:25","modified_gmt":"2022-11-09T16:35:25","slug":"skills-review-for-functions-of-several-variables","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/skills-review-for-functions-of-several-variables\/","title":{"raw":"Skills Review for Functions of Several Variables and Limits and Continuity","rendered":"Skills Review for Functions of Several Variables and Limits and Continuity"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Complete a table values with solutions to an equation<\/li>\r\n \t<li>Evaluate trigonometric functions using the unit circle<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn the Functions of Several Variables and Limits and Continuity sections, we will define and develop skills needed to understand and work with functions that have more than one independent variable as well as learn how to find their limits. Here we will review how to evaluate functions including trigonometric functions.\r\n<h2>Complete a Table of Function Values<\/h2>\r\n<div>\r\n\r\nA table of values can be used to organize the\u00a0<em>y<\/em>-values or function values that result from plugging specific\u00a0<em>x<\/em>-values into a function's equation.\r\n\r\nSuppose we want to determine the various values of the equation [latex]f(x)=2x - 1[\/latex] at certain values of <em>x<\/em>. We can begin by substituting a value for <em>x<\/em> into the equation and determining the resulting value of the function. The table below\u00a0lists some values of <em>x<\/em> from \u20133 to 3 and the resulting function values.\r\n<table style=\"width: 411px; height: 84px;\" summary=\"This is a table with 8 rows and 3 columns. The first row has columns labeled: x, y = 2x-1, (x, y). The entries in the second row are: negative 3; y = 2 times negative 3 minus 1 = negative 7; (-3, -7). The entries in the third row are: negative 2; y = 2 times negative 2 minus 1 = negative 5; (-2, -5). The entries in the fourth row are: negative1; y = 2 times negative 1 minus 1 = negative 3; (-1, -3). The entries in the fifth row are: 0; y = 2 times 0 minus 1 = negative 1; (0, -1). The entries in the sixth row are: 1; y = 2 times 1 minus 1 = 1; (1, 1). The entries in the seventh row are: 2; y = 2 times 2 minus 1 = 3; (2, 3). The entries in the eight row are: 3, y = 2 times 3 minus 1 = 5; (3,5)\">\r\n<tbody>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 116.5px; height: 12px;\">[latex]x[\/latex]<\/td>\r\n<td style=\"width: 269.5px; height: 12px;\">[latex]f(x)=2x - 1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 116.5px; height: 12px;\">[latex]-3[\/latex]<\/td>\r\n<td style=\"width: 269.5px; height: 12px;\">[latex]f(-3)=2\\left(-3\\right)-1=-7[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 116.5px; height: 12px;\">[latex]-2[\/latex]<\/td>\r\n<td style=\"width: 269.5px; height: 12px;\">[latex]f(-2)=2\\left(-2\\right)-1=-5[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 116.5px; height: 12px;\">[latex]-1[\/latex]<\/td>\r\n<td style=\"width: 269.5px; height: 12px;\">[latex]f(-1)=2\\left(-1\\right)-1=-3[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 116.5px; height: 12px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 269.5px; height: 12px;\">[latex]f(0)=2\\left(0\\right)-1=-1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 116.5px; height: 12px;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 269.5px; height: 12px;\">[latex]f(2)=2\\left(2\\right)-1=3[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 116.5px; height: 12px;\">[latex]3[\/latex]<\/td>\r\n<td style=\"width: 269.5px; height: 12px;\">[latex]f(3)=2\\left(3\\right)-1=5[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe can look for trends among our function values by looking at the table. For example, in this case, as\u00a0<em>x<\/em>-values increase, so do the\u00a0<em>y<\/em>-values. Also, it seems reasonable to assume, based on the table, an\u00a0<em>x<\/em>-value of 1 would result in a function value of 1. You can verify this for yourself by plugging 1 into [latex]f(x)=2x - 1[\/latex].\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Completing a table of function values<\/h3>\r\nCreate a table of function values for [latex]f(x)=-x+2[\/latex]. Use various integers\u00a0from -5 to 5 as the\u00a0<em>x<\/em>-values you plug into the function.\r\n[reveal-answer q=\"792137\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"792137\"]\r\n\r\n&nbsp;\r\n<table style=\"width: 385px;\" summary=\"The table shows 8 rows and 3 columns. The entries in the first row are: x; y = negative x plus 2; and (x, y). The entries in the second row are: negative 5; y = the opposite of negative 5 plus 2 = 7; (-5, 7). The entries in the third row are: negative 3; y = the opposite of negative 3 plus 2 = 5; (-3, 5). The entries in the fourth row are: -1; y = the opposite of negative 1 plus 2 = 3; (-1, 3). The entries in the fifth row are: 0; y = opposite of zero plus 2 = 2; (0, 2). The entries in the sixth row are: 1; y = the opposite of 1 plus 2 = 1; (1, 1). The entries in the seventh row are: 3; y = the opposite of 3 plus 2 = negative 1; (3, -1). The entries in the eighth row are: 5; y = the opposite of 5 plus 2 = negative 3; (5, -3).\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 117.656px;\">[latex]x[\/latex]<\/td>\r\n<td style=\"width: 244.656px;\">[latex]f(x)=-x+2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 117.656px;\">[latex]-5[\/latex]<\/td>\r\n<td style=\"width: 244.656px;\">[latex]f(-5)=-\\left(-5\\right)+2=7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 117.656px;\">[latex]-3[\/latex]<\/td>\r\n<td style=\"width: 244.656px;\">[latex]f(-3)=-\\left(-3\\right)+2=5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 117.656px;\">[latex]-1[\/latex]<\/td>\r\n<td style=\"width: 244.656px;\">[latex]f(-1)=-\\left(-1\\right)+2=3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 117.656px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 244.656px;\">[latex]f(0)=-\\left(0\\right)+2=2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 117.656px;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 244.656px;\">[latex]f(1)=-\\left(1\\right)+2=1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 117.656px;\">[latex]3[\/latex]<\/td>\r\n<td style=\"width: 244.656px;\">[latex]f(3)=-\\left(3\\right)+2=-1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 117.656px;\">[latex]5[\/latex]<\/td>\r\n<td style=\"width: 244.656px;\">[latex]f(5)=-\\left(5\\right)+2=-3[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\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]219319[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]219320[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Evaluate Trigonometric Functions Using the Unit Circle<\/h2>\r\n<strong><em>(also in Module 1, Skills Review for Polar Coordinates)<\/em><\/strong>\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 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<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\" \/>\r\n<div class=\"textbox\">\r\n<h3>A General Note: Evaluating Tangent, Secant, Cosecant, and Cotangent Functions<\/h3>\r\nIf [latex] \\theta[\/latex] is an angle measure, then, using the unit circle,\r\n<p style=\"text-align: center;\">[latex]\\begin{gathered}\\tan \\theta=\\frac{sin \\theta}{cos \\theta}\\\\ \\sec \\theta=\\frac{1}{cos \\theta}\\\\ \\csc t=\\frac{1}{sin \\theta}\\\\ \\cot \\theta=\\frac{cos \\theta}{sin \\theta}\\end{gathered}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Using the Unit Circle to Find the Value of Trigonometric Functions<\/h3>\r\nFind [latex]\\sin \\theta,\\cos \\theta,\\tan \\theta,\\sec \\theta,\\csc \\theta[\/latex], and [latex]\\cot \\theta[\/latex] when [latex] \\theta=\\frac{\\pi }{3}[\/latex].\r\n\r\n[reveal-answer q=\"151548\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"151548\"]\r\n\r\n[latex]\\begin{align}&amp;\\sin \\frac{\\pi }{3}=\\frac{\\sqrt{3}}{2}\\\\ &amp;\\cos \\frac{\\pi }{3}=\\frac{1}{2}\\\\ &amp;\\tan \\frac{\\pi }{3}=\\sqrt{3} \\text{ (From}\\frac{sin \\theta}{cos \\theta})\\\\ &amp;\\sec \\frac{\\pi }{3}=2 \\text{ (From}\\frac{1}{cos \\theta})\\\\ &amp;\\csc \\frac{\\pi }{3}=\\frac{2\\sqrt{3}}{3}\\text{ (From}\\frac{1}{sin \\theta} \\text{ - do not forget to rationalize})\\\\ &amp;\\cot \\frac{\\pi }{3}=\\frac{\\sqrt{3}}{3}(\\text{ (From}\\frac{cos \\theta}{sin \\theta} \\text{ - do not forget to rationalize})\\end{align}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]173354[\/ohm_question]\r\n\r\n<\/div>\r\nBelow are the values of all six trigonometric functions evaluated at common angle measures from Quadrant I.\r\n<table id=\"Table_05_03_01\" summary=\"..\">\r\n<tbody>\r\n<tr>\r\n<td><strong>Angle<\/strong><\/td>\r\n<td><strong> [latex]0[\/latex] <\/strong><\/td>\r\n<td><strong> [latex]\\frac{\\pi }{6},\\text{ or }{30}^{\\circ}[\/latex] <\/strong><\/td>\r\n<td><strong> [latex]\\frac{\\pi }{4},\\text{ or } {45}^{\\circ }[\/latex] <\/strong><\/td>\r\n<td><strong> [latex]\\frac{\\pi }{3},\\text{ or }{60}^{\\circ }[\/latex] <\/strong><\/td>\r\n<td><strong> [latex]\\frac{\\pi }{2},\\text{ or }{90}^{\\circ }[\/latex] <\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Cosine<\/strong><\/td>\r\n<td>1<\/td>\r\n<td>[latex]\\frac{\\sqrt{3}}{2}[\/latex]<\/td>\r\n<td>[latex]\\frac{\\sqrt{2}}{2}[\/latex]<\/td>\r\n<td>[latex]\\frac{1}{2}[\/latex]<\/td>\r\n<td>0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Sine<\/strong><\/td>\r\n<td>0<\/td>\r\n<td>[latex]\\frac{1}{2}[\/latex]<\/td>\r\n<td>[latex]\\frac{\\sqrt{2}}{2}[\/latex]<\/td>\r\n<td>[latex]\\frac{\\sqrt{3}}{2}[\/latex]<\/td>\r\n<td>1<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Tangent<\/strong><\/td>\r\n<td>0<\/td>\r\n<td>[latex]\\frac{\\sqrt{3}}{3}[\/latex]<\/td>\r\n<td>1<\/td>\r\n<td>[latex]\\sqrt{3}[\/latex]<\/td>\r\n<td>Undefined<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Secant<\/strong><\/td>\r\n<td>1<\/td>\r\n<td>[latex]\\frac{2\\sqrt{3}}{3}[\/latex]<\/td>\r\n<td>[latex]\\sqrt{2}[\/latex]<\/td>\r\n<td>2<\/td>\r\n<td>Undefined<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Cosecant<\/strong><\/td>\r\n<td>Undefined<\/td>\r\n<td>2<\/td>\r\n<td>[latex]\\sqrt{2}[\/latex]<\/td>\r\n<td>[latex]\\frac{2\\sqrt{3}}{3}[\/latex]<\/td>\r\n<td>1<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Cotangent<\/strong><\/td>\r\n<td>Undefined<\/td>\r\n<td>[latex]\\sqrt{3}[\/latex]<\/td>\r\n<td>1<\/td>\r\n<td>[latex]\\frac{\\sqrt{3}}{3}[\/latex]<\/td>\r\n<td>0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Complete a table values with solutions to an equation<\/li>\n<li>Evaluate trigonometric functions using the unit circle<\/li>\n<\/ul>\n<\/div>\n<p>In the Functions of Several Variables and Limits and Continuity sections, we will define and develop skills needed to understand and work with functions that have more than one independent variable as well as learn how to find their limits. Here we will review how to evaluate functions including trigonometric functions.<\/p>\n<h2>Complete a Table of Function Values<\/h2>\n<div>\n<p>A table of values can be used to organize the\u00a0<em>y<\/em>-values or function values that result from plugging specific\u00a0<em>x<\/em>-values into a function&#8217;s equation.<\/p>\n<p>Suppose we want to determine the various values of the equation [latex]f(x)=2x - 1[\/latex] at certain values of <em>x<\/em>. We can begin by substituting a value for <em>x<\/em> into the equation and determining the resulting value of the function. The table below\u00a0lists some values of <em>x<\/em> from \u20133 to 3 and the resulting function values.<\/p>\n<table style=\"width: 411px; height: 84px;\" summary=\"This is a table with 8 rows and 3 columns. The first row has columns labeled: x, y = 2x-1, (x, y). The entries in the second row are: negative 3; y = 2 times negative 3 minus 1 = negative 7; (-3, -7). The entries in the third row are: negative 2; y = 2 times negative 2 minus 1 = negative 5; (-2, -5). The entries in the fourth row are: negative1; y = 2 times negative 1 minus 1 = negative 3; (-1, -3). The entries in the fifth row are: 0; y = 2 times 0 minus 1 = negative 1; (0, -1). The entries in the sixth row are: 1; y = 2 times 1 minus 1 = 1; (1, 1). The entries in the seventh row are: 2; y = 2 times 2 minus 1 = 3; (2, 3). The entries in the eight row are: 3, y = 2 times 3 minus 1 = 5; (3,5)\">\n<tbody>\n<tr style=\"height: 12px;\">\n<td style=\"width: 116.5px; height: 12px;\">[latex]x[\/latex]<\/td>\n<td style=\"width: 269.5px; height: 12px;\">[latex]f(x)=2x - 1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 116.5px; height: 12px;\">[latex]-3[\/latex]<\/td>\n<td style=\"width: 269.5px; height: 12px;\">[latex]f(-3)=2\\left(-3\\right)-1=-7[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 116.5px; height: 12px;\">[latex]-2[\/latex]<\/td>\n<td style=\"width: 269.5px; height: 12px;\">[latex]f(-2)=2\\left(-2\\right)-1=-5[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 116.5px; height: 12px;\">[latex]-1[\/latex]<\/td>\n<td style=\"width: 269.5px; height: 12px;\">[latex]f(-1)=2\\left(-1\\right)-1=-3[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 116.5px; height: 12px;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 269.5px; height: 12px;\">[latex]f(0)=2\\left(0\\right)-1=-1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 116.5px; height: 12px;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 269.5px; height: 12px;\">[latex]f(2)=2\\left(2\\right)-1=3[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 116.5px; height: 12px;\">[latex]3[\/latex]<\/td>\n<td style=\"width: 269.5px; height: 12px;\">[latex]f(3)=2\\left(3\\right)-1=5[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We can look for trends among our function values by looking at the table. For example, in this case, as\u00a0<em>x<\/em>-values increase, so do the\u00a0<em>y<\/em>-values. Also, it seems reasonable to assume, based on the table, an\u00a0<em>x<\/em>-value of 1 would result in a function value of 1. You can verify this for yourself by plugging 1 into [latex]f(x)=2x - 1[\/latex].<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Completing a table of function values<\/h3>\n<p>Create a table of function values for [latex]f(x)=-x+2[\/latex]. Use various integers\u00a0from -5 to 5 as the\u00a0<em>x<\/em>-values you plug into the function.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q792137\">Show Solution<\/span><\/p>\n<div id=\"q792137\" class=\"hidden-answer\" style=\"display: none\">\n<p>&nbsp;<\/p>\n<table style=\"width: 385px;\" summary=\"The table shows 8 rows and 3 columns. The entries in the first row are: x; y = negative x plus 2; and (x, y). The entries in the second row are: negative 5; y = the opposite of negative 5 plus 2 = 7; (-5, 7). The entries in the third row are: negative 3; y = the opposite of negative 3 plus 2 = 5; (-3, 5). The entries in the fourth row are: -1; y = the opposite of negative 1 plus 2 = 3; (-1, 3). The entries in the fifth row are: 0; y = opposite of zero plus 2 = 2; (0, 2). The entries in the sixth row are: 1; y = the opposite of 1 plus 2 = 1; (1, 1). The entries in the seventh row are: 3; y = the opposite of 3 plus 2 = negative 1; (3, -1). The entries in the eighth row are: 5; y = the opposite of 5 plus 2 = negative 3; (5, -3).\">\n<tbody>\n<tr>\n<td style=\"width: 117.656px;\">[latex]x[\/latex]<\/td>\n<td style=\"width: 244.656px;\">[latex]f(x)=-x+2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 117.656px;\">[latex]-5[\/latex]<\/td>\n<td style=\"width: 244.656px;\">[latex]f(-5)=-\\left(-5\\right)+2=7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 117.656px;\">[latex]-3[\/latex]<\/td>\n<td style=\"width: 244.656px;\">[latex]f(-3)=-\\left(-3\\right)+2=5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 117.656px;\">[latex]-1[\/latex]<\/td>\n<td style=\"width: 244.656px;\">[latex]f(-1)=-\\left(-1\\right)+2=3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 117.656px;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 244.656px;\">[latex]f(0)=-\\left(0\\right)+2=2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 117.656px;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 244.656px;\">[latex]f(1)=-\\left(1\\right)+2=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 117.656px;\">[latex]3[\/latex]<\/td>\n<td style=\"width: 244.656px;\">[latex]f(3)=-\\left(3\\right)+2=-1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 117.656px;\">[latex]5[\/latex]<\/td>\n<td style=\"width: 244.656px;\">[latex]f(5)=-\\left(5\\right)+2=-3[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\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=\"ohm219319\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=219319&theme=oea&iframe_resize_id=ohm219319&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm219320\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=219320&theme=oea&iframe_resize_id=ohm219320&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Evaluate Trigonometric Functions Using the Unit Circle<\/h2>\n<p><strong><em>(also in Module 1, Skills Review for Polar Coordinates)<\/em><\/strong><br \/>\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 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><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\" \/><\/p>\n<div class=\"textbox\">\n<h3>A General Note: Evaluating Tangent, Secant, Cosecant, and Cotangent Functions<\/h3>\n<p>If [latex]\\theta[\/latex] is an angle measure, then, using the unit circle,<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{gathered}\\tan \\theta=\\frac{sin \\theta}{cos \\theta}\\\\ \\sec \\theta=\\frac{1}{cos \\theta}\\\\ \\csc t=\\frac{1}{sin \\theta}\\\\ \\cot \\theta=\\frac{cos \\theta}{sin \\theta}\\end{gathered}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using the Unit Circle to Find the Value of Trigonometric Functions<\/h3>\n<p>Find [latex]\\sin \\theta,\\cos \\theta,\\tan \\theta,\\sec \\theta,\\csc \\theta[\/latex], and [latex]\\cot \\theta[\/latex] when [latex]\\theta=\\frac{\\pi }{3}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q151548\">Show Solution<\/span><\/p>\n<div id=\"q151548\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\begin{align}&\\sin \\frac{\\pi }{3}=\\frac{\\sqrt{3}}{2}\\\\ &\\cos \\frac{\\pi }{3}=\\frac{1}{2}\\\\ &\\tan \\frac{\\pi }{3}=\\sqrt{3} \\text{ (From}\\frac{sin \\theta}{cos \\theta})\\\\ &\\sec \\frac{\\pi }{3}=2 \\text{ (From}\\frac{1}{cos \\theta})\\\\ &\\csc \\frac{\\pi }{3}=\\frac{2\\sqrt{3}}{3}\\text{ (From}\\frac{1}{sin \\theta} \\text{ - do not forget to rationalize})\\\\ &\\cot \\frac{\\pi }{3}=\\frac{\\sqrt{3}}{3}(\\text{ (From}\\frac{cos \\theta}{sin \\theta} \\text{ - do not forget to rationalize})\\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm173354\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=173354&theme=oea&iframe_resize_id=ohm173354\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Below are the values of all six trigonometric functions evaluated at common angle measures from Quadrant I.<\/p>\n<table id=\"Table_05_03_01\" summary=\"..\">\n<tbody>\n<tr>\n<td><strong>Angle<\/strong><\/td>\n<td><strong> [latex]0[\/latex] <\/strong><\/td>\n<td><strong> [latex]\\frac{\\pi }{6},\\text{ or }{30}^{\\circ}[\/latex] <\/strong><\/td>\n<td><strong> [latex]\\frac{\\pi }{4},\\text{ or } {45}^{\\circ }[\/latex] <\/strong><\/td>\n<td><strong> [latex]\\frac{\\pi }{3},\\text{ or }{60}^{\\circ }[\/latex] <\/strong><\/td>\n<td><strong> [latex]\\frac{\\pi }{2},\\text{ or }{90}^{\\circ }[\/latex] <\/strong><\/td>\n<\/tr>\n<tr>\n<td><strong>Cosine<\/strong><\/td>\n<td>1<\/td>\n<td>[latex]\\frac{\\sqrt{3}}{2}[\/latex]<\/td>\n<td>[latex]\\frac{\\sqrt{2}}{2}[\/latex]<\/td>\n<td>[latex]\\frac{1}{2}[\/latex]<\/td>\n<td>0<\/td>\n<\/tr>\n<tr>\n<td><strong>Sine<\/strong><\/td>\n<td>0<\/td>\n<td>[latex]\\frac{1}{2}[\/latex]<\/td>\n<td>[latex]\\frac{\\sqrt{2}}{2}[\/latex]<\/td>\n<td>[latex]\\frac{\\sqrt{3}}{2}[\/latex]<\/td>\n<td>1<\/td>\n<\/tr>\n<tr>\n<td><strong>Tangent<\/strong><\/td>\n<td>0<\/td>\n<td>[latex]\\frac{\\sqrt{3}}{3}[\/latex]<\/td>\n<td>1<\/td>\n<td>[latex]\\sqrt{3}[\/latex]<\/td>\n<td>Undefined<\/td>\n<\/tr>\n<tr>\n<td><strong>Secant<\/strong><\/td>\n<td>1<\/td>\n<td>[latex]\\frac{2\\sqrt{3}}{3}[\/latex]<\/td>\n<td>[latex]\\sqrt{2}[\/latex]<\/td>\n<td>2<\/td>\n<td>Undefined<\/td>\n<\/tr>\n<tr>\n<td><strong>Cosecant<\/strong><\/td>\n<td>Undefined<\/td>\n<td>2<\/td>\n<td>[latex]\\sqrt{2}[\/latex]<\/td>\n<td>[latex]\\frac{2\\sqrt{3}}{3}[\/latex]<\/td>\n<td>1<\/td>\n<\/tr>\n<tr>\n<td><strong>Cotangent<\/strong><\/td>\n<td>Undefined<\/td>\n<td>[latex]\\sqrt{3}[\/latex]<\/td>\n<td>1<\/td>\n<td>[latex]\\frac{\\sqrt{3}}{3}[\/latex]<\/td>\n<td>0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\t\t\t 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