{"id":608,"date":"2019-03-07T13:27:04","date_gmt":"2019-03-07T13:27:04","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/chapter\/right-triangle-trigonometry\/"},"modified":"2019-05-14T13:53:59","modified_gmt":"2019-05-14T13:53:59","slug":"right-triangle-trigonometry","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/chapter\/right-triangle-trigonometry\/","title":{"raw":"Right Triangle Trigonometry","rendered":"Right Triangle Trigonometry"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\nIn this section, you will:\r\n<ul>\r\n \t<li>Use right triangles to evaluate trigonometric functions.<\/li>\r\n \t<li>Find function values for[latex]\\text{ }30\u00b0\\left(\\frac{\\pi }{6}\\right),\\text{ }[\/latex][latex]45\u00b0\\left(\\frac{\\pi }{4}\\right),\\text{ }[\/latex]and[latex]\\text{ }60\u00b0\\left(\\frac{\\pi }{3}\\right).[\/latex]<\/li>\r\n \t<li>Use cofunctions of complementary angles.<\/li>\r\n \t<li>Use the de\ufb01nitions of trigonometric functions of any angle.<\/li>\r\n \t<li>Use right triangle trigonometry to solve applied problems.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p id=\"fs-id1165137387312\">We have previously defined the sine and cosine of an angle in terms of the coordinates of a point on the unit circle intersected by the terminal side of the angle:<\/p>\r\n\r\n<div id=\"fs-id1165135501042\" class=\"unnumbered\">[latex]\\begin{array}{c}\\mathrm{cos}\\text{ }t=x\\\\ \\mathrm{sin}\\text{ }t=y\\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1165137668254\">In this section, we will see another way to define trigonometric functions using properties of <span class=\"no-emphasis\">right triangles<\/span>.<\/p>\r\n\r\n<div id=\"fs-id1165137547549\" class=\"bc-section section\">\r\n<h3>Using Right Triangles to Evaluate Trigonometric Functions<\/h3>\r\n<p id=\"fs-id1165134108581\">In earlier sections, we used a unit circle to define the <span class=\"no-emphasis\">trigonometric functions<\/span>. In this section, we will extend those definitions so that we can apply them to right triangles. The value of the sine or cosine function of[latex]\\text{ }t\\text{ }[\/latex]is its value at[latex]\\text{ }t\\text{ }[\/latex]radians. First, we need to create our right triangle. <a class=\"autogenerated-content\" href=\"#Figure_05_04_001\">(Figure)<\/a> shows a point on a <span class=\"no-emphasis\">unit circle<\/span> of radius 1. If we drop a vertical line segment from the point[latex]\\text{ }\\left(x,y\\right)\\text{ }[\/latex]to the <em>x<\/em>-axis, we have a right triangle whose vertical side has length[latex]\\text{ }y\\text{ }[\/latex]and whose horizontal side has length[latex]\\text{ }x.\\text{ }[\/latex]We can use this right triangle to redefine sine, cosine, and the other trigonometric functions as ratios of the sides of a right triangle.<\/p>\r\n\r\n<div id=\"Figure_05_04_001\" class=\"small\"><span id=\"fs-id1165137602828\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132528\/CNX_Precalc_Figure_05_04_001.jpg\" alt=\"Graph of quarter circle with radius of 1 and angle of t. Point of (x,y) is at intersection of terminal side of angle and edge of circle.\" \/><\/span><\/div>\r\n<p id=\"fs-id1165137470559\">We know<\/p>\r\n\r\n<div id=\"fs-id1165137417839\" class=\"unnumbered\">[latex]\\mathrm{cos}\\text{ }t=\\frac{x}{1}=x[\/latex]<\/div>\r\nLikewise, we know\r\n<div id=\"fs-id1165135426436\" class=\"unnumbered\">[latex]\\mathrm{sin}\\text{ }t=\\frac{y}{1}=y[\/latex]<\/div>\r\n<p id=\"fs-id1165135693752\">These ratios still apply to the sides of a right triangle when no unit circle is involved and when the triangle is not in standard position and is not being graphed using[latex]\\text{ }\\left(x,y\\right)\\text{ }[\/latex]coordinates. To be able to use these ratios freely, we will give the sides more general names: Instead of[latex]\\text{ }x,[\/latex]we will call the side between the given angle and the right angle the adjacent side to angle[latex]\\text{ }t.\\text{ }[\/latex](Adjacent means \u201cnext to.\u201d) Instead of[latex]\\text{ }y,[\/latex]we will call the side most distant from the given angle the opposite side from angle[latex]\\text{}t.\\text{ }[\/latex]And instead of[latex]\\text{ }1,[\/latex]we will call the side of a right triangle opposite the right angle the hypotenuse. These sides are labeled in <a class=\"autogenerated-content\" href=\"#Figure_05_04_002\">(Figure)<\/a>.<\/p>\r\n\r\n<div id=\"Figure_05_04_002\" class=\"small\">\r\n<div class=\"wp-caption-text\"><\/div>\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132531\/CNX_Precalc_Figure_05_04_002.jpg\" alt=\"A right triangle with hypotenuse, opposite, and adjacent sides labeled.\" width=\"487\" height=\"137\" \/> The sides of a right triangle in relation to angle [latex]t.[\/latex][\/caption]<\/div>\r\n<div id=\"fs-id1165137784426\" class=\"bc-section section\">\r\n<h4>Understanding Right Triangle Relationships<\/h4>\r\n<p id=\"fs-id1165135189859\">Given a right triangle with an acute angle of[latex]\\text{ }t,[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1165135177707\" class=\"unnumbered\">[latex]\\begin{array}{l}\\mathrm{sin}\\left(t\\right)=\\frac{\\text{opposite}}{\\text{hypotenuse}}\\hfill \\\\ \\mathrm{cos}\\left(t\\right)=\\frac{\\text{adjacent}}{\\text{hypotenuse}}\\hfill \\\\ \\mathrm{tan}\\left(t\\right)=\\frac{\\text{opposite}}{\\text{adjacent}}\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1165137559825\">A common mnemonic for remembering these relationships is SohCahToa, formed from the first letters of \u201c<u>S<\/u>ine is <u>o<\/u>pposite over <u>h<\/u>ypotenuse, <u>C<\/u>osine is <u>a<\/u>djacent over <u>h<\/u>ypotenuse, <u>T<\/u>angent is <u>o<\/u>pposite over <u>a<\/u>djacent.\u201d<\/p>\r\n\r\n<div id=\"fs-id1165135448331\" class=\"precalculus howto examples\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165137832115\"><strong>Given the side lengths of a right triangle and one of the acute angles, find the sine, cosine, and tangent of that angle.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165137465487\" type=\"1\">\r\n \t<li>Find the sine as the ratio of the opposite side to the hypotenuse.<\/li>\r\n \t<li>Find the cosine as the ratio of the adjacent side to the hypotenuse.<\/li>\r\n \t<li>Find the tangent as the ratio of the opposite side to the adjacent side.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_05_04_01\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137597073\">\r\n<div id=\"fs-id1165137410098\">\r\n<h3>Example 1: Evaluating a Trigonometric Function of a Right Triangle<\/h3>\r\n<p id=\"fs-id1165135630976\">Given the triangle shown in <a class=\"autogenerated-content\" href=\"#Figure_05_04_003\">(Figure)<\/a>, find the value of[latex]\\text{ }\\mathrm{cos}\\text{ }\\alpha .[\/latex]<\/p>\r\n\r\n<div id=\"Figure_05_04_003\" class=\"small\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132533\/CNX_Precalc_Figure_05_04_003.jpg\" alt=\"A right triangle with sid lengths of 8, 15, and 17. Angle alpha also labeled.\" \/><\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137398722\">[reveal-answer q=\"fs-id1165137398722\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137398722\"]\r\n<p id=\"fs-id1165137647892\">The side adjacent to the angle is 15, and the hypotenuse of the triangle is 17, so:<\/p>\r\n\r\n<div id=\"fs-id1165137770356\" class=\"unnumbered\">[latex]\\begin{array}{l}\\mathrm{cos}\\left(\\alpha \\right)=\\frac{\\text{adjacent}}{\\text{hypotenuse}}\\hfill \\\\ \\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }=\\frac{15}{17}\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137453511\" class=\"precalculus tryit\">\r\n<h3>Try it #1<\/h3>\r\n<div id=\"ti_05_04_01\">\r\n<div id=\"fs-id1165134570073\">\r\n<p id=\"fs-id1165137811034\">Given the triangle shown in <a class=\"autogenerated-content\" href=\"#Figure_05_04_004\">(Figure)<\/a>, find the value of[latex]\\text{ }\\text{sin}\\text{ }t.[\/latex]<\/p>\r\n\r\n<div id=\"Figure_05_04_004\" class=\"small\"><span id=\"fs-id1165135191134\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132536\/CNX_Precalc_Figure_05_04_004.jpg\" alt=\"A right triangle with sides of 7, 24, and 25. Also labeled is angle t.\" \/><\/span><\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137602319\">[reveal-answer q=\"fs-id1165137602319\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137602319\"]\r\n<p id=\"fs-id1165137422653\">[latex]\\frac{7}{25}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134199461\" class=\"bc-section section\">\r\n<h4>Relating Angles and Their Functions<\/h4>\r\n<p id=\"fs-id1165135241150\">When working with right triangles, the same rules apply regardless of the orientation of the triangle. In fact, we can evaluate the six trigonometric functions of either of the two acute angles in the triangle in <a class=\"autogenerated-content\" href=\"#Figure_05_04_005\">(Figure)<\/a>. The side opposite one acute angle is the side adjacent to the other acute angle, and vice versa.<\/p>\r\n\r\n<div id=\"Figure_05_04_005\" class=\"small\">\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\/3896\/2019\/03\/07132539\/CNX_Precalc_Figure_05_04_005.jpg\" alt=\"Right triangle with angles alpha and beta. Sides are labeled hypotenuse, adjacent to alpha\/opposite to beta, and adjacent to beta\/opposite alpha.\" width=\"487\" height=\"181\" \/> The side adjacent to one angle is opposite the other.[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1165137459869\">We will be asked to find all six trigonometric functions for a given angle in a triangle. Our strategy is to find the sine, cosine, and tangent of the angles first. Then, we can find the other trigonometric functions easily because we know that the reciprocal of sine is cosecant, the reciprocal of cosine is secant, and the reciprocal of tangent is cotangent.<\/p>\r\n\r\n<div id=\"fs-id1165137806953\" class=\"precalculus howto examples\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165137427122\"><strong>Given the side lengths of a right triangle, evaluate the six trigonometric functions of one of the acute angles.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165137762694\" type=\"1\">\r\n \t<li>If needed, draw the right triangle and label the angle provided.<\/li>\r\n \t<li>Identify the angle, the adjacent side, the side opposite the angle, and the hypotenuse of the right triangle.<\/li>\r\n \t<li>Find the required function:\r\n<ul id=\"fs-id1165135532521\">\r\n \t<li>sine as the ratio of the opposite side to the hypotenuse<\/li>\r\n \t<li>cosine as the ratio of the adjacent side to the hypotenuse<\/li>\r\n \t<li>tangent as the ratio of the opposite side to the adjacent side<\/li>\r\n \t<li>secant as the ratio of the hypotenuse to the adjacent side<\/li>\r\n \t<li>cosecant as the ratio of the hypotenuse to the opposite side<\/li>\r\n \t<li>cotangent as the ratio of the adjacent side to the opposite side<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_05_04_02\" class=\"textbox examples\">\r\n<div id=\"fs-id1165135187471\">\r\n<div id=\"fs-id1165137836968\">\r\n<h3>Example 2: Evaluating Trigonometric Functions of Angles Not in Standard Position<\/h3>\r\n<p id=\"fs-id1165137724941\">Using the triangle shown in <a class=\"autogenerated-content\" href=\"#Figure_05_04_006\">(Figure)<\/a>, evaluate [latex]\\mathrm{sin}\\text{ }\\alpha ,[\/latex] [latex]\\mathrm{cos}\\text{ }\\alpha ,[\/latex] [latex]\\mathrm{tan}\\text{ }\\alpha ,[\/latex][latex]\\mathrm{sec}\\text{ }\\alpha ,[\/latex] [latex]\\mathrm{csc}\\text{ }\\alpha ,[\/latex] and [latex]\\text{ }\\mathrm{cot}\\text{ }\\alpha .[\/latex]<\/p>\r\n\r\n<div id=\"Figure_05_04_006\" class=\"small\"><span id=\"fs-id1165137542988\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132541\/CNX_Precalc_Figure_05_04_006.jpg\" alt=\"Right triangle with sides of 3, 4, and 5. Angle alpha is also labeled.\" \/><\/span><\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137447828\">[reveal-answer q=\"fs-id1165137447828\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137447828\"]\r\n<div id=\"fs-id1165137571205\" class=\"unnumbered\">[latex]\\begin{array}{l}\\mathrm{sin}\\text{ }\\alpha =\\frac{\\text{opposite }\\alpha }{\\text{hypotenuse}}=\\frac{4}{5}\\hfill \\\\ \\mathrm{cos}\\text{ }\\alpha =\\frac{\\text{adjacent to }\\alpha }{\\text{hypotenuse}}=\\frac{3}{5}\\hfill \\\\ \\mathrm{tan}\\text{ }\\alpha =\\frac{\\text{opposite }\\alpha }{\\text{adjacent to }\\alpha }=\\frac{4}{3}\\hfill \\\\ \\mathrm{sec}\\text{ }\\alpha =\\frac{\\text{hypotenuse}}{\\text{adjacent to }\\alpha }=\\frac{5}{3}\\hfill \\\\ \\mathrm{csc}\\text{ }\\alpha =\\frac{\\text{hypotenuse}}{\\text{opposite }\\alpha }=\\frac{5}{4}\\hfill \\\\ \\mathrm{cot}\\text{ }\\alpha =\\frac{\\text{adjacent to }\\alpha }{\\text{opposite }\\alpha }=\\frac{3}{4}\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137548404\" class=\"precalculus tryit\">\r\n<h3>Try it #2<\/h3>\r\n<div id=\"ti_05_04_02\">\r\n<div id=\"fs-id1165137809949\">\r\n<p id=\"fs-id1165137809950\">Using the triangle shown in <a class=\"autogenerated-content\" href=\"#Figure_05_04_007\">(Figure)<\/a>, evaluate[latex]\\text{ }\\mathrm{sin}\\text{ }t,[\/latex] [latex]\\mathrm{cos}\\text{ }t,[\/latex] [latex]\\mathrm{tan}\\text{ }t,[\/latex][latex]\\mathrm{sec}\\text{ }t,[\/latex] [latex]\\mathrm{csc}\\text{ }t,[\/latex]and[latex]\\text{ }\\mathrm{cot}\\text{ }t.[\/latex]<\/p>\r\n\r\n<div id=\"Figure_05_04_007\" class=\"small\"><span id=\"fs-id1165137847139\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132544\/CNX_Precalc_Figure_05_04_007.jpg\" alt=\"Right triangle with sides 33, 56, and 65. Angle t is also labeled.\" \/><\/span><\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135452495\">[reveal-answer q=\"fs-id1165135452495\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135452495\"]\r\n<p id=\"fs-id1165135452496\">[latex]\\begin{array}{l}sin t=\\frac{33}{65},\\mathrm{cos} t=\\frac{56}{65},tan t=\\frac{33}{56},\\hfill \\\\ \\text{ }\\mathrm{sec} t=\\frac{65}{56},\\mathrm{csc} t=\\frac{65}{33},\\mathrm{cot} t=\\frac{56}{33}\\hfill \\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137810878\" class=\"bc-section section\">\r\n<h4>Finding Trigonometric Functions of Special Angles Using Side Lengths<\/h4>\r\n<p id=\"fs-id1165137531688\">We have already discussed the trigonometric functions as they relate to the <span class=\"no-emphasis\">special angles<\/span> on the unit circle. Now, we can use those relationships to evaluate triangles that contain those special angles. We do this because when we evaluate the special angles in trigonometric functions, they have relatively friendly values, values that contain either no or just one square root in the ratio. Therefore, these are the angles often used in math and science problems. We will use multiples of[latex]\\text{ }30\u00b0,[\/latex] [latex]60\u00b0,[\/latex] and[latex]\\text{ }45\u00b0,[\/latex] however, remember that when dealing with right triangles, we are limited to angles between[latex]\\text{ }0\u00b0\\text{ and 90\u00b0}\\text{.}[\/latex]<\/p>\r\n<p id=\"fs-id1165135193209\">Suppose we have a[latex]\\text{ }30\u00b0,60\u00b0,90\u00b0\\text{ }[\/latex]triangle, which can also be described as a[latex]\\text{ }\\frac{\\pi }{6},\\text{\u200b} \\frac{\\pi }{3},\\frac{\\pi }{2}\\text{ }[\/latex]triangle. The sides have lengths in the relation[latex]\\text{ }s,\\sqrt{3}s,2s.\\text{ }[\/latex]The sides of a[latex]\\text{ }45\u00b0,45\u00b0,90\u00b0[\/latex]triangle, which can also be described as a[latex]\\text{ }\\frac{\\pi }{4},\\frac{\\pi }{4},\\frac{\\pi }{2}\\text{ }[\/latex]triangle, have lengths in the relation[latex]\\text{ }s,s,\\sqrt{2}s.\\text{ }[\/latex]These relations are shown in <a class=\"autogenerated-content\" href=\"#Figure_05_04_008\">(Figure)<\/a>.<\/p>\r\n\r\n<div id=\"Figure_05_04_008\" class=\"wp-caption aligncenter\">\r\n<div class=\"wp-caption-text\">Side lengths of special triangles<\/div>\r\n<span id=\"fs-id1165137598316\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132547\/CNX_Precalc_Figure_05_04_008.jpg\" alt=\"Two side by side graphs of circles with inscribed angles. First circle has angle of pi\/3 inscribed. Second circle has angle of pi\/4 inscribed.\" \/><\/span>\r\n\r\n<\/div>\r\n<p id=\"fs-id1165135433018\">We can then use the ratios of the side lengths to evaluate trigonometric functions of special angles.<\/p>\r\n\r\n<div id=\"fs-id1165137592822\" class=\"precalculus howto examples\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165135242753\"><strong>Given trigonometric functions of a special angle, evaluate using side lengths.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165137451842\" type=\"1\">\r\n \t<li>Use the side lengths shown in <a class=\"autogenerated-content\" href=\"#Figure_05_04_008\">(Figure)<\/a> for the special angle you wish to evaluate.<\/li>\r\n \t<li>Use the ratio of side lengths appropriate to the function you wish to evaluate.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_05_04_03\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137762917\">\r\n<div id=\"fs-id1165137602526\">\r\n<h3>Example 3: Evaluating Trigonometric Functions of Special Angles Using Side Lengths<\/h3>\r\n<p id=\"fs-id1165137583573\">Find the exact value of the trigonometric functions of[latex]\\text{ }\\frac{\\pi }{3},[\/latex] using side lengths.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137758344\">[reveal-answer q=\"fs-id1165137758344\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137758344\"]\r\n<div id=\"fs-id1165137526289\" class=\"unnumbered\">[latex]\\begin{array}{l}\\mathrm{sin}\\left(\\frac{\\pi }{3}\\right)=\\frac{\\text{opp}}{\\text{hyp}}=\\frac{\\sqrt{3}s}{2s}=\\frac{\\sqrt{3}}{2}\\hfill \\\\ \\mathrm{cos}\\left(\\frac{\\pi }{3}\\right)=\\frac{\\text{adj}}{\\text{hyp}}=\\frac{s}{2s}=\\frac{1}{2}\\hfill \\\\ \\mathrm{tan}\\left(\\frac{\\pi }{3}\\right)=\\frac{\\text{opp}}{\\text{adj}}=\\frac{\\sqrt{3}s}{s}=\\sqrt{3}\\hfill \\\\ \\mathrm{sec}\\left(\\frac{\\pi }{3}\\right)=\\frac{\\text{hyp}}{\\text{adj}}=\\frac{2s}{s}=2\\hfill \\\\ \\mathrm{csc}\\left(\\frac{\\pi }{3}\\right)=\\frac{\\text{hyp}}{\\text{opp}}=\\frac{2s}{\\sqrt{3}s}=\\frac{2}{\\sqrt{3}}=\\frac{2\\sqrt{3}}{3}\\hfill \\\\ \\mathrm{cot}\\left(\\frac{\\pi }{3}\\right)=\\frac{\\text{adj}}{\\text{opp}}=\\frac{s}{\\sqrt{3}s}=\\frac{1}{\\sqrt{3}}=\\frac{\\sqrt{3}}{3}\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135190054\" class=\"precalculus tryit\">\r\n<h3>Try it #3<\/h3>\r\n<div id=\"ti_05_04_03\">\r\n<div id=\"fs-id1165135209398\">\r\n<p id=\"fs-id1165135209399\">Find the exact value of the trigonometric functions of[latex]\\text{ }\\frac{\\pi }{4},[\/latex] using side lengths.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137619647\">[reveal-answer q=\"fs-id1165137619647\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137619647\"]\r\n<p id=\"fs-id1165137619648\">[latex]\\mathrm{sin}\\left(\\frac{\\pi }{4}\\right)=\\frac{\\sqrt{2}}{2},\\mathrm{cos}\\left(\\frac{\\pi }{4}\\right)=\\frac{\\sqrt{2}}{2},\\mathrm{tan}\\left(\\frac{\\pi }{4}\\right)=1,[\/latex]<\/p>\r\n\r\n<div>[\/hidden-answer]<\/div>\r\n[latex]\\mathrm{sec}\\left(\\frac{\\pi }{4}\\right)=\\sqrt{2},csc\\left(\\frac{\\pi }{4}\\right)=\\sqrt{2},\\mathrm{cot}\\left(\\frac{\\pi }{4}\\right)=1[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137409403\" class=\"bc-section section\">\r\n<h4>Using Equal Cofunction of Complements<\/h4>\r\n<p id=\"fs-id1165137565216\">If we look more closely at the relationship between the sine and cosine of the special angles relative to the unit circle, we will notice a pattern. In a right triangle with angles of[latex]\\text{ }\\frac{\\pi }{6}\\text{ }[\/latex]and[latex]\\text{ }\\frac{\\pi }{3},[\/latex] we see that the sine of[latex]\\text{ }\\frac{\\pi }{3},[\/latex] namely[latex]\\text{ }\\frac{\\sqrt{3}}{2},[\/latex] is also the cosine of[latex]\\text{ }\\frac{\\pi }{6},[\/latex] while the sine of[latex]\\text{ }\\frac{\\pi }{6},[\/latex] namely[latex]\\text{ }\\frac{1}{2},[\/latex] is also the cosine of[latex]\\text{ }\\frac{\\pi }{3}.[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1165137847268\" class=\"unnumbered\">[latex]\\begin{array}{l}\\begin{array}{l}\\hfill \\\\ \\mathrm{sin}\\text{ }\\frac{\\pi }{3}=\\mathrm{cos}\\text{ }\\frac{\\pi }{6}=\\frac{\\sqrt{3}s}{2s}=\\frac{\\sqrt{3}}{2}\\hfill \\end{array}\\hfill \\\\ \\mathrm{sin}\\text{ }\\frac{\\pi }{6}=\\mathrm{cos}\\text{ }\\frac{\\pi }{3}=\\frac{s}{2s}=\\frac{1}{2}\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1165137594972\">See <a class=\"autogenerated-content\" href=\"#Figure_05_04_009\">(Figure)<\/a><\/p>\r\n\r\n<div id=\"Figure_05_04_009\" class=\"small\">[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132550\/CNX_Precalc_Figure_05_04_009.jpg\" alt=\"A graph of circle with angle pi\/3 inscribed.\" width=\"487\" height=\"371\" \/> The sine of[latex]\\text{ }\\frac{\\pi }{3}\\text{ }[\/latex]equals the cosine of[latex]\\text{ }\\frac{\\pi }{6}\\text{ }[\/latex]and vice versa.[\/caption]<\/div>\r\n<p id=\"fs-id1165137612239\">This result should not be surprising because, as we see from <a class=\"autogenerated-content\" href=\"#Figure_05_04_009\">(Figure)<\/a>, the side opposite the angle of[latex]\\text{ }\\frac{\\pi }{3}\\text{ }[\/latex]is also the side adjacent to[latex]\\text{ }\\frac{\\pi }{6},[\/latex] so[latex]\\text{ }\\mathrm{sin}\\left(\\frac{\\pi }{3}\\right)\\text{ }[\/latex]and[latex]\\text{ }\\mathrm{cos}\\left(\\frac{\\pi }{6}\\right)\\text{ }[\/latex]are exactly the same ratio of the same two sides,[latex]\\text{ }\\sqrt{3}s\\text{ }[\/latex]and[latex]\\text{ }2s.\\text{ }[\/latex]Similarly,[latex]\\text{ }\\mathrm{cos}\\left(\\frac{\\pi }{3}\\right)\\text{ }[\/latex]and[latex]\\text{ }\\mathrm{sin}\\left(\\frac{\\pi }{6}\\right)\\text{ }[\/latex]are also the same ratio using the same two sides,[latex]\\text{ }s\\text{ }[\/latex]and[latex]\\text{ }2s.[\/latex]<\/p>\r\n<p id=\"fs-id1165137732326\">The interrelationship between the sines and cosines of[latex]\\text{ }\\frac{\\pi }{6}\\text{ }[\/latex]and[latex]\\text{ }\\frac{\\pi }{3}\\text{ }[\/latex]also holds for the two acute angles in any right triangle, since in every case, the ratio of the same two sides would constitute the sine of one angle and the cosine of the other. Since the three angles of a triangle add to[latex]\\text{ }\\pi ,[\/latex] and the right angle is[latex]\\text{ }\\frac{\\pi }{2},[\/latex] the remaining two angles must also add up to[latex]\\text{ }\\frac{\\pi }{2}.\\text{ }[\/latex]That means that a right triangle can be formed with any two angles that add to[latex]\\text{ }\\frac{\\pi }{2}[\/latex]\u2014in other words, any two complementary angles. So we may state a <em>cofunction identity<\/em>: If any two angles are complementary, the sine of one is the cosine of the other, and vice versa. This identity is illustrated in <a class=\"autogenerated-content\" href=\"#Figure_05_04_010\">(Figure)<\/a>.<\/p>\r\n\r\n<div id=\"Figure_05_04_010\" class=\"small\">\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\/3896\/2019\/03\/07132553\/CNX_Precalc_Figure_05_04_010.jpg\" alt=\"Right triangle with angles alpha and beta. Equivalence between sin alpha and cos beta. Equivalence between sin beta and cos alpha.\" width=\"487\" height=\"304\" \/> Cofunction identity of sine and cosine of complementary angles[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1165135181553\">Using this identity, we can state without calculating, for instance, that the sine of[latex]\\text{ }\\frac{\\pi }{12}\\text{ }[\/latex]equals the cosine of[latex]\\text{ }\\frac{5\\pi }{12},[\/latex] and that the sine of[latex]\\text{ }\\frac{5\\pi }{12}\\text{ }[\/latex]equals the cosine of[latex]\\text{ }\\frac{\\pi }{12}.\\text{ }[\/latex]We can also state that if, for a certain angle[latex]\\text{ }t,[\/latex] [latex]\\mathrm{cos}\\text{ }t=\\frac{5}{13},[\/latex] then[latex]\\text{ }\\mathrm{sin}\\left(\\frac{\\pi }{2}-t\\right)=\\frac{5}{13}\\text{ }[\/latex]as well.<\/p>\r\n\r\n<div id=\"fs-id1165137738292\">\r\n<h3>Cofunction Identities<\/h3>\r\n<p id=\"fs-id1165135178097\">The <span class=\"no-emphasis\">cofunction identities<\/span> in radians are listed in <a class=\"autogenerated-content\" href=\"#Table_05_04_01\">(Figure)<\/a>.<\/p>\r\n\r\n<table id=\"Table_05_04_01\" summary=\"..\"><caption><strong>Table 1<\/strong><\/caption>\r\n<tbody>\r\n<tr>\r\n<td class=\"border\">[latex]\\mathrm{cos}\\text{ }t=\\mathrm{sin}\\left(\\frac{\\pi }{2}-t\\right)[\/latex]<\/td>\r\n<td class=\"border\">[latex]\\mathrm{sin}\\text{ }t=\\mathrm{cos}\\left(\\frac{\\pi }{2}-t\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\">[latex]\\mathrm{tan}\\text{ }t=\\mathrm{cot}\\left(\\frac{\\pi }{2}-t\\right)[\/latex]<\/td>\r\n<td class=\"border\">[latex]\\mathrm{cot}\\text{ }t=\\mathrm{tan}\\left(\\frac{\\pi }{2}-t\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\">[latex]\\mathrm{sec}\\text{ }t=\\mathrm{csc}\\left(\\frac{\\pi }{2}-t\\right)[\/latex]<\/td>\r\n<td class=\"border\">[latex]\\mathrm{csc}\\text{ }t=\\mathrm{sec}\\left(\\frac{\\pi }{2}-t\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div id=\"fs-id1165137735167\" class=\"precalculus howto examples\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165137450768\"><strong>Given the sine and cosine of an angle, find the sine or cosine of its complement.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165135333178\" type=\"1\">\r\n \t<li>To find the sine of the complementary angle, find the cosine of the original angle.<\/li>\r\n \t<li>To find the cosine of the complementary angle, find the sine of the original angle.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_05_04_04\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137811906\">\r\n<div id=\"fs-id1165133103968\">\r\n<h3>Example 4: Using Cofunction Identities<\/h3>\r\n<p id=\"fs-id1165134148512\">If[latex]\\text{ }\\mathrm{sin}\\text{ }t=\\frac{5}{12},[\/latex]find[latex]\\text{ }\\left(\\mathrm{cos}\\frac{\\pi }{2}-t\\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137668960\">[reveal-answer q=\"fs-id1165137668960\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137668960\"]\r\n<p id=\"fs-id1165137410039\">According to the cofunction identities for sine and cosine,<\/p>\r\n\r\n<div id=\"fs-id1165137786693\" class=\"unnumbered\">[latex]\\mathrm{sin}\\text{ }t=\\mathrm{cos}\\left(\\frac{\\pi }{2}-t\\right).[\/latex]<\/div>\r\n<p id=\"fs-id1165135361352\">So<\/p>\r\n\r\n<div id=\"fs-id1165135596476\" class=\"unnumbered\">[latex]\\mathrm{cos}\\left(\\frac{\\pi }{2}-t\\right)=\\frac{5}{12}.[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137530693\" class=\"precalculus tryit\">\r\n<h3>Try it #4<\/h3>\r\n<div id=\"ti_05_04_04\">\r\n<div id=\"fs-id1165134192991\">\r\n<p id=\"fs-id1165134192992\">If[latex]\\text{ }\\mathrm{csc}\\left(\\frac{\\pi }{6}\\right)=2,[\/latex] find[latex]\\text{ }\\mathrm{sec}\\left(\\frac{\\pi }{3}\\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137387550\">[reveal-answer q=\"fs-id1165137387550\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137387550\"]\r\n<p id=\"fs-id1165137387551\">2<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137639938\" class=\"bc-section section\">\r\n<h4>Using Trigonometric Functions<\/h4>\r\n<p id=\"fs-id1165137629080\">In previous examples, we evaluated the sine and cosine in triangles where we knew all three sides. But the real power of right-triangle trigonometry emerges when we look at triangles in which we know an angle but do not know all the sides.<\/p>\r\n\r\n<div id=\"fs-id1165137456630\" class=\"precalculus howto examples\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165137475023\"><strong>Given a right triangle, the length of one side, and the measure of one acute angle, find the remaining sides.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165137564333\" type=\"1\">\r\n \t<li>For each side, select the trigonometric function that has the unknown side as either the numerator or the denominator. The known side will in turn be the denominator or the numerator.<\/li>\r\n \t<li>Write an equation setting the function value of the known angle equal to the ratio of the corresponding sides.<\/li>\r\n \t<li>Using the value of the trigonometric function and the known side length, solve for the missing side length.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_05_04_05\" class=\"textbox examples\">\r\n<div id=\"fs-id1165135192870\">\r\n<div id=\"fs-id1165135353061\">\r\n<h3>Example 5: Finding Missing Side Lengths Using Trigonometric Ratios<\/h3>\r\n<p id=\"fs-id1165137417073\">Find the unknown sides of the triangle in <a class=\"autogenerated-content\" href=\"#Figure_05_04_011\">(Figure)<\/a>.<\/p>\r\n\r\n<div id=\"Figure_05_04_011\" class=\"small\"><span id=\"fs-id1165133189417\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132556\/CNX_Precalc_Figure_05_04_011.jpg\" alt=\"A right triangle with sides a, c, and 7. Angle of 30 degrees is also labeled.\" \/><\/span><\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135503936\">[reveal-answer q=\"fs-id1165135503936\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135503936\"]\r\n<p id=\"fs-id1165137611783\">We know the angle and the opposite side, so we can use the tangent to find the adjacent side.<\/p>\r\n\r\n<div id=\"fs-id1165135432983\" class=\"unnumbered\">[latex]\\mathrm{tan}\\left(30\u00b0\\right)=\\frac{7}{a}[\/latex]<\/div>\r\n<p id=\"fs-id1165137574920\">We rearrange to solve for[latex]\\text{ }a.[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1165134054044\" class=\"unnumbered\">[latex]\\begin{array}{l}a=\\frac{7}{\\mathrm{tan}\\left(30\u00b0\\right)}\\hfill \\\\ \\text{ }\\text{ }\\text{ }\\approx 12.1\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1165137460411\">We can use the sine to find the hypotenuse.<\/p>\r\n\r\n<div id=\"fs-id1165135386417\" class=\"unnumbered\">[latex]\\mathrm{sin}\\left(30\u00b0\\right)=\\frac{7}{c}[\/latex]<\/div>\r\n<p id=\"fs-id1165137665408\">Again, we rearrange to solve for[latex]\\text{ }c.[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1165135186816\" class=\"unnumbered\">[latex]\\begin{array}{l}c=\\frac{7}{\\mathrm{sin}\\left(30\u00b0\\right)}\\hfill \\\\ \\text{ }\\text{ }\\text{ }\\approx 14\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137851227\" class=\"precalculus tryit\">\r\n<h3>Try it #5<\/h3>\r\n<div id=\"ti_05_04_05\">\r\n<div id=\"fs-id1165137727334\">\r\n<p id=\"fs-id1165137742301\">A right triangle has one angle of[latex]\\text{ }\\frac{\\pi }{3}\\text{ }[\/latex] and a hypotenuse of 20. Find the unknown sides and angle of the triangle.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137542469\">[reveal-answer q=\"fs-id1165137542469\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137542469\"][latex]\\text{adjacent}=10;\\text{ }[\/latex][latex]\\text{opposite}=10\\sqrt{3}\\text{ }[\/latex]; missing angle is[latex]\\text{ }\\frac{\\pi }{6}[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137658230\" class=\"bc-section section\">\r\n<h4>Using Right Triangle Trigonometry to Solve Applied Problems<\/h4>\r\n<p id=\"fs-id1165137437126\">Right-triangle trigonometry has many practical applications. For example, the ability to compute the lengths of sides of a triangle makes it possible to find the height of a tall object without climbing to the top or having to extend a tape measure along its height. We do so by measuring a distance from the base of the object to a point on the ground some distance away, where we can look up to the top of the tall object at an angle. The angle of elevation of an object above an observer relative to the observer is the angle between the horizontal and the line from the object to the observer's eye. The right triangle this position creates has sides that represent the unknown height, the measured distance from the base, and the angled line of sight from the ground to the top of the object. Knowing the measured distance to the base of the object and the angle of the line of sight, we can use trigonometric functions to calculate the unknown height. Similarly, we can form a triangle from the top of a tall object by looking downward. The angle of depression of an object below an observer relative to the observer is the angle between the horizontal and the line from the object to the observer's eye. See <a class=\"autogenerated-content\" href=\"#Figure_05_04_013\">(Figure)<\/a>.<\/p>\r\n\r\n<div id=\"Figure_05_04_013\" class=\"small\"><span id=\"fs-id1165137730420\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132559\/CNX_Precalc_Figure_05_04_013.jpg\" alt=\"Diagram of a radio tower with line segments extending from the top and base of the tower to a point on the ground some distance away. The two lines and the tower form a right triangle. The angle near the top of the tower is the angle of depression. The angle on the ground at a distance from the tower is the angle of elevation.\" \/><\/span><\/div>\r\n<div id=\"fs-id1165137473849\" class=\"precalculus howto examples\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165137723583\"><strong>Given a tall object, measure its height indirectly.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165135195794\" type=\"1\">\r\n \t<li>Make a sketch of the problem situation to keep track of known and unknown information.<\/li>\r\n \t<li>Lay out a measured distance from the base of the object to a point where the top of the object is clearly visible.<\/li>\r\n \t<li>At the other end of the measured distance, look up to the top of the object. Measure the angle the line of sight makes with the horizontal.<\/li>\r\n \t<li>Write an equation relating the unknown height, the measured distance, and the tangent of the angle of the line of sight.<\/li>\r\n \t<li>Solve the equation for the unknown height.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_05_04_06\" class=\"textbox examples\">\r\n<div id=\"fs-id1165135176667\">\r\n<div id=\"fs-id1165137553918\">\r\n<h3>Example 6: Measuring a Distance Indirectly<\/h3>\r\n<p id=\"fs-id1165137736590\">To find the height of a tree, a person walks to a point 30 feet from the base of the tree. She measures an angle of [latex]57\u00b0\\text{ }[\/latex]between a line of sight to the top of the tree and the ground, as shown in <a class=\"autogenerated-content\" href=\"#Figure_05_04_012\">(Figure)<\/a>. Find the height of the tree.<\/p>\r\n\r\n<div id=\"Figure_05_04_012\" class=\"small\"><span id=\"fs-id1165137804033\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132602\/CNX_Precalc_Figure_05_04_012.jpg\" alt=\"A tree with angle of 57 degrees from vantage point. Vantage point is 30 feet from tree.\" \/><\/span><\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134278695\">[reveal-answer q=\"fs-id1165134278695\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134278695\"]\r\n<p id=\"fs-id1165135210031\">We know that the angle of elevation is[latex]\\text{ }57\u00b0\\text{ }[\/latex]and the adjacent side is 30 ft long. The opposite side is the unknown height.<\/p>\r\n<p id=\"fs-id1165134047639\">The trigonometric function relating the side opposite to an angle and the side adjacent to the angle is the tangent. So we will state our information in terms of the tangent of [latex]57\u00b0,[\/latex] letting[latex]\\text{ }h\\text{ }[\/latex]be the unknown height.<\/p>\r\n\r\n<div id=\"fs-id1165134089530\" class=\"unnumbered\">[latex]\\begin{array}{ll}\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\mathrm{tan}\\text{ }\\theta =\\frac{\\text{opposite}}{\\text{adjacent}}\\hfill &amp; \\hfill \\\\ \\text{tan}\\left(57\u00b0\\right)=\\frac{h}{30}\\hfill &amp; \\text{Solve for }h.\\hfill \\\\ \\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }h=30\\mathrm{tan}\\left(57\u00b0\\right)\\hfill &amp; \\text{Multiply}.\\hfill \\\\ \\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }h\\approx 46.2\\hfill &amp; \\text{Use a calculator}.\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1165137550726\">The tree is approximately 46 feet tall.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137533956\" class=\"precalculus tryit\">\r\n<h3>Try it #6<\/h3>\r\n<div id=\"ti_05_04_06\">\r\n<div id=\"fs-id1165137855261\">\r\n<p id=\"fs-id1165137855262\">How long a ladder is needed to reach a windowsill 50 feet above the ground if the ladder rests against the building making an angle of[latex]\\text{ }\\frac{5\\pi }{12}\\text{ }[\/latex]with the ground? Round to the nearest foot.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137727529\">[reveal-answer q=\"fs-id1165137727529\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137727529\"]\r\n<p id=\"fs-id1165135403387\">About 52 ft<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137762260\" class=\"precalculus media\">\r\n<p id=\"fs-id1165137442106\">Access these online resources for additional instruction and practice with right triangle trigonometry.<\/p>\r\n\r\n<ul id=\"fs-id1165137656591\">\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/findtrigcal\">Finding Trig Functions on Calculator<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/trigrttri\">Finding Trig Functions Using a Right Triangle<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/reltrigtri\">Relate Trig Functions to Sides of a Right Triangle<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/sixtrigfunc\">Determine Six Trig Functions from a Triangle<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/rttriside\">Determine Length of Right Triangle Side<\/a><\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<p id=\"eip-845\">Visit <a href=\"http:\/\/openstax.org\/l\/PreCalcLPC05\">this website<\/a> for additional practice questions from Learningpod.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135186669\" class=\"key-equations\">\r\n<h3>Key Equations<\/h3>\r\n<table id=\"eip-id1165137409421\" summary=\"..\">\r\n<tbody>\r\n<tr>\r\n<td class=\"border\">Cofunction Identities<\/td>\r\n<td class=\"border\">[latex]\\begin{array}{l}\\begin{array}{l}\\\\ \\mathrm{cos}\\text{ }t=\\mathrm{sin}\\left(\\frac{\\pi }{2}-t\\right)\\end{array}\\hfill \\\\ \\mathrm{sin}\\text{ }t=\\mathrm{cos}\\left(\\frac{\\pi }{2}-t\\right)\\hfill \\\\ \\mathrm{tan}\\text{ }t=\\mathrm{cot}\\left(\\frac{\\pi }{2}-t\\right)\\hfill \\\\ \\mathrm{cot}\\text{ }t=\\mathrm{tan}\\left(\\frac{\\pi }{2}-t\\right)\\hfill \\\\ \\mathrm{sec}\\text{ }t=\\mathrm{csc}\\left(\\frac{\\pi }{2}-t\\right)\\hfill \\\\ \\mathrm{csc}\\text{ }t=\\mathrm{sec}\\left(\\frac{\\pi }{2}-t\\right)\\hfill \\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div id=\"fs-id1165137481899\" class=\"textbox key-takeaways\">\r\n<h3>Key Concepts<\/h3>\r\n<ul id=\"fs-id1165137415357\">\r\n \t<li>We can define trigonometric functions as ratios of the side lengths of a right triangle. See <a class=\"autogenerated-content\" href=\"#Example_05_04_01\">(Figure)<\/a>.<\/li>\r\n \t<li>The same side lengths can be used to evaluate the trigonometric functions of either acute angle in a right triangle. See <a class=\"autogenerated-content\" href=\"#Example_05_04_02\">(Figure)<\/a>.<\/li>\r\n \t<li>We can evaluate the trigonometric functions of special angles, knowing the side lengths of the triangles in which they occur. See <a class=\"autogenerated-content\" href=\"#Example_05_04_03\">(Figure)<\/a>.<\/li>\r\n \t<li>Any two complementary angles could be the two acute angles of a right triangle.<\/li>\r\n \t<li>If two angles are complementary, the cofunction identities state that the sine of one equals the cosine of the other and vice versa. See <a class=\"autogenerated-content\" href=\"#Example_05_04_04\">(Figure)<\/a>.<\/li>\r\n \t<li>We can use trigonometric functions of an angle to find unknown side lengths.<\/li>\r\n \t<li>Select the trigonometric function representing the ratio of the unknown side to the known side. See <a class=\"autogenerated-content\" href=\"#Example_05_04_05\">(Figure)<\/a>.<\/li>\r\n \t<li>Right-triangle trigonometry permits the measurement of inaccessible heights and distances.<\/li>\r\n \t<li>The unknown height or distance can be found by creating a right triangle in which the unknown height or distance is one of the sides, and another side and angle are known. See <a class=\"autogenerated-content\" href=\"#Example_05_04_06\">(Figure)<\/a>.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1165135458650\" class=\"textbox exercises\">\r\n<h3>Section Exercises<\/h3>\r\n<div id=\"fs-id1165137736563\" class=\"bc-section section\">\r\n<h4>Verbal<\/h4>\r\n<div id=\"fs-id1165137460283\">\r\n<div id=\"fs-id1165137460284\">\r\n<p id=\"fs-id1165133112791\">For the given right triangle, label the adjacent side, opposite side, and hypotenuse for the indicated angle.<\/p>\r\n\r\n<div><\/div>\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132604\/CNX_Precalc_Figure_05_04_201.jpg\" alt=\"A right triangle.\" \/>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137803574\">[reveal-answer q=\"fs-id1165137803574\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137803574\"]<span id=\"fs-id1165137401049\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132606\/CNX_Precalc_Figure_05_04_202.jpg\" alt=\"A right triangle with side opposite, adjacent, and hypotenuse labeled.\" \/><\/span>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137472665\">\r\n<div id=\"fs-id1165137452030\">\r\n<p id=\"fs-id1165137452031\">When a right triangle with a hypotenuse of 1 is placed in the unit circle, which sides of the triangle correspond to the <em>x<\/em>- and <em>y<\/em>-coordinates?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137404958\">\r\n<div id=\"fs-id1165137404959\">\r\n<p id=\"fs-id1165137698057\">The tangent of an angle compares which sides of the right triangle?<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135315582\">[reveal-answer q=\"fs-id1165135315582\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135315582\"]\r\n<p id=\"fs-id1165137400574\">The tangent of an angle is the ratio of the opposite side to the adjacent side.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137462272\">\r\n<div id=\"fs-id1165135536523\">\r\n<p id=\"fs-id1165135536524\">What is the relationship between the two acute angles in a right triangle?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137854900\">\r\n<div id=\"fs-id1165137854901\">\r\n<p id=\"fs-id1165137725210\">Explain the cofunction identity.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135209966\">[reveal-answer q=\"fs-id1165135209966\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135209966\"]\r\n<p id=\"fs-id1165137675894\">For example, the sine of an angle is equal to the cosine of its complement; the cosine of an angle is equal to the sine of its complement.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135406939\" class=\"bc-section section\">\r\n<h4>Algebraic<\/h4>\r\n<p id=\"fs-id1165135501987\">For the following exercises, use cofunctions of complementary angles.<\/p>\r\n\r\n<div id=\"fs-id1165137761553\">\r\n<div id=\"fs-id1165137459885\">\r\n<p id=\"fs-id1165137459886\">[latex]\\mathrm{cos}\\left(\\text{34\u00b0}\\right)=\\mathrm{sin}\\left(\\text{__\u00b0}\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137935703\">\r\n<div id=\"fs-id1165137935704\">\r\n<p id=\"fs-id1165137755665\">[latex]\\mathrm{cos}\\left(\\frac{\\pi }{3}\\right)=\\mathrm{sin}\\text{(___)}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137805390\">[reveal-answer q=\"fs-id1165137805390\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137805390\"]\r\n<p id=\"fs-id1165131974796\">[latex]\\frac{\\pi }{6}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137911234\">\r\n<div id=\"fs-id1165135465250\">\r\n<p id=\"fs-id1165135465251\">[latex]\\mathrm{csc}\\left(\\text{21\u00b0}\\right)=\\mathrm{sec}\\left(\\text{___\u00b0}\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137768200\">\r\n<div id=\"fs-id1165137768201\">\r\n<p id=\"fs-id1165134276136\">[latex]\\mathrm{tan}\\left(\\frac{\\pi }{4}\\right)=\\mathrm{cot}\\left(\\text{__}\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137470154\">[reveal-answer q=\"fs-id1165137470154\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137470154\"]\r\n<p id=\"fs-id1165137470155\">[latex]\\frac{\\pi }{4}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165137455693\">For the following exercises, find the lengths of the missing sides if side[latex]\\text{ }a\\text{ }[\/latex]is opposite angle[latex]\\text{ }A,[\/latex] side[latex]\\text{ }b\\text{ }[\/latex] is opposite angle[latex]\\text{ }B,[\/latex] and side[latex]\\text{ }c\\text{ }[\/latex]is the hypotenuse.<\/p>\r\n\r\n<div id=\"fs-id1165137549317\">\r\n<div id=\"fs-id1165137549318\">\r\n<p id=\"fs-id1165135393375\">[latex]\\mathrm{cos}\\text{ }B=\\frac{4}{5},a=10[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135453101\">\r\n<div id=\"fs-id1165135453102\">\r\n<p id=\"fs-id1165135453103\">[latex]\\mathrm{sin}\\text{ }B=\\frac{1}{2}, a=20[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137832156\">[reveal-answer q=\"fs-id1165137832156\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137832156\"]\r\n<p id=\"fs-id1165135187865\">[latex]b=\\frac{20\\sqrt{3}}{3},c=\\frac{40\\sqrt{3}}{3}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137549742\">\r\n<div id=\"fs-id1165137549743\">\r\n<p id=\"fs-id1165137549744\">[latex]\\mathrm{tan}\\text{ }A=\\frac{5}{12},b=6[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137766757\">\r\n<div id=\"fs-id1165137894557\">\r\n<p id=\"fs-id1165137894558\">[latex]\\mathrm{tan}\\text{ }A=100,b=100[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137657401\">[reveal-answer q=\"fs-id1165137657401\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137657401\"]\r\n<p id=\"fs-id1165137657402\">[latex]a=10,000,c=10,000.5[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137409084\">\r\n<div id=\"fs-id1165137409085\">\r\n<p id=\"fs-id1165135196985\">[latex]\\mathrm{sin}\\text{ }B=\\frac{1}{\\sqrt{3}}, a=2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135407091\">\r\n<div id=\"fs-id1165135407092\">\r\n<p id=\"fs-id1165135407093\">[latex]a=5,\\text{ }\\measuredangle \\text{ }A={60}^{\\circ }[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137803393\">[reveal-answer q=\"fs-id1165137803393\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137803393\"]\r\n<p id=\"fs-id1165137803394\">[latex]b=\\frac{5\\sqrt{3}}{3},c=\\frac{10\\sqrt{3}}{3}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135186918\">\r\n<div id=\"fs-id1165135186919\">\r\n<p id=\"fs-id1165137447359\">[latex]c=12,\\text{ }\\measuredangle \\text{ }A={45}^{\\circ }[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137450999\" class=\"bc-section section\">\r\n<h4>Graphical<\/h4>\r\n<p id=\"fs-id1165135191519\">For the following exercises, use <a class=\"autogenerated-content\" href=\"#Figure_05_04_203\">(Figure)<\/a> to evaluate each trigonometric function of angle[latex]\\text{ }A.[\/latex]<\/p>\r\n\r\n<div id=\"Figure_05_04_203\" class=\"wp-caption aligncenter\"><span id=\"fs-id1165132960807\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132609\/CNX_Precalc_Figure_05_04_203.jpg\" alt=\"A right triangle with sides 4 and 10 and angle of A labeled.\" \/><\/span><\/div>\r\n<div id=\"fs-id1165137397852\">\r\n<div id=\"fs-id1165135500824\">\r\n<p id=\"fs-id1165135500826\">[latex]\\mathrm{sin}\\text{ }A[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135434893\">[reveal-answer q=\"fs-id1165135434893\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135434893\"]\r\n<p id=\"fs-id1165137639470\">[latex]\\frac{5\\sqrt{29}}{29}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137834908\">\r\n<div id=\"fs-id1165137834909\">\r\n<p id=\"fs-id1165135485865\">[latex]\\mathrm{cos}\\text{ }A[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135567456\">\r\n<div id=\"fs-id1165135567457\">\r\n<p id=\"fs-id1165137531079\">[latex]\\mathrm{tan}\\text{ }A[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137732079\">[reveal-answer q=\"fs-id1165137732079\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137732079\"]\r\n<p id=\"fs-id1165137732080\">[latex]\\frac{5}{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137647703\">\r\n<div id=\"fs-id1165137647704\">\r\n<p id=\"fs-id1165135241318\">[latex]\\mathrm{csc}\\text{ }A[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137597798\">\r\n<div id=\"fs-id1165137597799\">\r\n<p id=\"fs-id1165137803588\">[latex]\\mathrm{sec}\\text{ }A[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137693464\">[reveal-answer q=\"fs-id1165137693464\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137693464\"]\r\n<p id=\"fs-id1165137693465\">[latex]\\frac{\\sqrt{29}}{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137920736\">\r\n<div id=\"fs-id1165137920737\">\r\n<p id=\"fs-id1165135241382\">[latex]\\mathrm{cot}\\text{ }A[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165137452925\">For the following exercises, use <a class=\"autogenerated-content\" href=\"#Figure_05_04_204\">(Figure)<\/a> to evaluate each trigonometric function of angle[latex]\\text{ }A.[\/latex]<\/p>\r\n\r\n<div id=\"Figure_05_04_204\" class=\"wp-caption aligncenter\"><span id=\"fs-id1165135160637\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132612\/CNX_Precalc_Figure_05_04_204.jpg\" alt=\"A right triangle with sides of 10 and 8 and angle of A labeled.\" \/><\/span><\/div>\r\n<div id=\"fs-id1165137749319\">\r\n<div id=\"fs-id1165135613637\">\r\n<p id=\"fs-id1165135613638\">[latex]\\mathrm{sin}\\text{ }A[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137683057\">[reveal-answer q=\"fs-id1165137683057\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137683057\"]\r\n<p id=\"fs-id1165137419300\">[latex]\\frac{5\\sqrt{41}}{41}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137601187\">\r\n<div id=\"fs-id1165137601188\">\r\n<p id=\"fs-id1165135173373\">[latex]\\mathrm{cos}\\text{ }A[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137462646\">\r\n<div id=\"fs-id1165137462647\">\r\n<p id=\"fs-id1165137847132\">[latex]\\mathrm{tan}\\text{ }A[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135190741\">[reveal-answer q=\"fs-id1165135190741\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135190741\"]\r\n<p id=\"fs-id1165135190742\">[latex]\\frac{5}{4}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137415361\">\r\n<div id=\"fs-id1165137415362\">\r\n<p id=\"fs-id1165137415363\">[latex]\\mathrm{csc}\\text{ }A[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137596480\">\r\n<div id=\"fs-id1165137933799\">\r\n<p id=\"fs-id1165137933800\">[latex]\\mathrm{sec}\\text{ }A[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137779088\">[reveal-answer q=\"fs-id1165137779088\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137779088\"]\r\n<p id=\"fs-id1165137779089\">[latex]\\frac{\\sqrt{41}}{4}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137810135\">\r\n<div id=\"fs-id1165137810136\">\r\n<p id=\"fs-id1165137810138\">[latex]\\mathrm{cot}\\text{ }A[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165135168306\">For the following exercises, solve for the unknown sides of the given triangle.<\/p>\r\n\r\n<div id=\"fs-id1165135168309\">\r\n<div id=\"fs-id1165137805206\"><span id=\"fs-id1165135613627\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132615\/CNX_Precalc_Figure_05_04_205.jpg\" alt=\"A right triangle with sides of 7, b, and c labeled. Angles of B and 30 degrees also labeled.\" \/><\/span><\/div>\r\n<div id=\"fs-id1165137832242\">[reveal-answer q=\"fs-id1165137832242\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137832242\"]\r\n<p id=\"fs-id1165137832243\">[latex]c=14, b=7\\sqrt{3}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134094614\">\r\n<div id=\"fs-id1165134094615\"><span id=\"fs-id1165135550447\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132618\/CNX_Precalc_Figure_05_04_206.jpg\" alt=\"A right triangle with sides of 10, a, and c. Angles of 60 degrees and A also labeled.\" \/><\/span><\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134085640\">\r\n<div id=\"fs-id1165134085642\"><span id=\"fs-id1165137766966\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132621\/CNX_Precalc_Figure_05_04_207.jpg\" alt=\"A right triangle with corners labeled A, B, and C. Hypotenuse has length of 15 times square root of 2. Angle B is 45 degrees.\" \/><\/span><\/div>\r\n<div>\r\n<p id=\"fs-id1165135645975\">[latex]a=15, b=15[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137736877\" class=\"bc-section section\">\r\n<h4>Technology<\/h4>\r\n<p id=\"fs-id1165137476432\">For the following exercises, use a calculator to find the length of each side to four decimal places.<\/p>\r\n\r\n<div id=\"fs-id1165137874745\">\r\n<div id=\"fs-id1165137874746\"><span id=\"fs-id1165137530086\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132623\/CNX_Precalc_Figure_05_04_208.jpg\" alt=\"A right triangle with sides of 10, a, and c. Angles of A and 62 degrees are also labeled.\" \/><\/span><\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137401776\">\r\n<div id=\"fs-id1165137401777\"><span id=\"fs-id1165137730500\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132626\/CNX_Precalc_Figure_05_04_209.jpg\" alt=\"A right triangle with sides of 7, b, and c. Angles of 35 degrees and B are also labeled.\" \/><\/span><\/div>\r\n<div id=\"fs-id1165134298850\">[reveal-answer q=\"fs-id1165134298850\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134298850\"]\r\n<p id=\"fs-id1165134298851\">[latex]b=9.9970, c=12.2041[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135193172\">\r\n<div id=\"fs-id1165135193173\"><span id=\"fs-id1165137401263\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132629\/CNX_Precalc_Figure_05_04_210.jpg\" alt=\"A right triangle with sides of a, b, and 10 labeled. Angles of 65 degrees and B are also labeled.\" \/><\/span><\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135159938\">\r\n<div id=\"fs-id1165137667450\"><span id=\"fs-id1165135369236\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132632\/CNX_Precalc_Figure_05_04_211.jpg\" alt=\"A right triangle with sides a, b, and 12. Angles of 10 degrees and B are also labeled.\" \/><\/span><\/div>\r\n<div id=\"fs-id1165137460061\">[reveal-answer q=\"fs-id1165137460061\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137460061\"]\r\n<p id=\"fs-id1165137460062\">[latex]a=2.0838, b=11.8177[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137922573\">\r\n<div id=\"fs-id1165137922574\"><span id=\"fs-id1165135190611\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132634\/CNX_Precalc_Figure_05_04_212.jpg\" alt=\"A right triangle with corners labeled A, B, and C. Sides labeled b, c, and 16.5. Angle of 81 degrees also labeled.\" \/><\/span><\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134357798\">\r\n<div id=\"fs-id1165134357799\">\r\n<p id=\"fs-id1165137573269\">[latex]b=15,\\text{ }\\measuredangle B={15}^{\\circ }[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137705539\">[reveal-answer q=\"fs-id1165137705539\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137705539\"]\r\n<p id=\"fs-id1165137705540\">[latex]a=55.9808,c=57.9555[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135475897\">\r\n<div id=\"fs-id1165135475898\">\r\n<p id=\"fs-id1165135475899\">[latex]c=200,\\text{ }\\measuredangle B={5}^{\\circ }[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137810074\">\r\n<div id=\"fs-id1165137810075\">\r\n<p id=\"fs-id1165137810076\">[latex]c=50,\\text{ }\\measuredangle B={21}^{\\circ }[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137767696\">[reveal-answer q=\"fs-id1165137767696\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137767696\"]\r\n<p id=\"fs-id1165137767697\">[latex]a=46.6790,b=17.9184[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135260744\">\r\n<div id=\"fs-id1165137806834\">\r\n<p id=\"fs-id1165137806835\">[latex]a=30,\\text{ }\\measuredangle A={27}^{\\circ }[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135331740\">\r\n<div id=\"fs-id1165135496307\">\r\n<p id=\"fs-id1165135496308\">[latex]b=3.5,\\text{ }\\measuredangle A={78}^{\\circ }[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135390953\">[reveal-answer q=\"fs-id1165135390953\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135390953\"]\r\n<p id=\"fs-id1165135307915\">[latex]a=16.4662,c=16.8341[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137603593\" class=\"bc-section section\">\r\n<h4>Extensions<\/h4>\r\n<div id=\"fs-id1165137427200\">\r\n<div id=\"fs-id1165137404949\">\r\n<p id=\"fs-id1165137404950\">Find[latex]\\text{ }x.[\/latex]<\/p>\r\n<span id=\"fs-id1165137738251\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132637\/CNX_Precalc_Figure_05_04_213.jpg\" alt=\"A triangle with angles of 63 degrees and 39 degrees and side x. Bisector in triangle with length of 82.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135173378\">\r\n<div id=\"fs-id1165135173379\">\r\n<p id=\"fs-id1165135173380\">Find[latex]\\text{ }x.[\/latex]<\/p>\r\n<span id=\"fs-id1165137768593\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132640\/CNX_Precalc_Figure_05_04_214.jpg\" alt=\"A triangle with angles of 36 degrees and 50 degrees and side x. Bisector in triangle with length of 85.\" \/><\/span>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137679296\">[reveal-answer q=\"fs-id1165137679296\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137679296\"]\r\n<p id=\"fs-id1165137679297\">188.3159<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135176637\">\r\n<div id=\"fs-id1165135176638\">\r\n<p id=\"fs-id1165135176639\">Find[latex]\\text{ }x.[\/latex]<\/p>\r\n<span id=\"fs-id1165137771945\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132643\/CNX_Precalc_Figure_05_04_215.jpg\" alt=\"A right triangle with side of 115 and angle of 35 degrees. Within right triangle there is another right triangle with angle of 56 degrees. Side length difference between two triangles is x.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137432679\">\r\n<div id=\"fs-id1165137432680\">\r\n<p id=\"fs-id1165137432681\">Find[latex]\\text{ }x.[\/latex]<\/p>\r\n<span id=\"fs-id1165137469034\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132646\/CNX_Precalc_Figure_05_04_216.jpg\" alt=\"A right triangle with side of 119 and angle of 26 degrees. Within right triangle there is another right triangle with angle of 70 degrees instead of 26 degrees. Difference in side length between two triangles is x.\" \/><\/span>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135160172\">[reveal-answer q=\"fs-id1165135160172\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135160172\"]\r\n<p id=\"fs-id1165137435323\">200.6737<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137435326\">\r\n<div id=\"fs-id1165137444512\">\r\n<p id=\"fs-id1165137444513\">A radio tower is located 400 feet from a building. From a window in the building, a person determines that the angle of elevation to the top of the tower is[latex]\\text{ }36\u00b0,[\/latex] and that the angle of depression to the bottom of the tower is[latex]\\text{ }23\u00b0.\\text{ }[\/latex]How tall is the tower?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137433654\">\r\n<div id=\"fs-id1165135353109\">\r\n<p id=\"fs-id1165135353110\">A radio tower is located 325 feet from a building. From a window in the building, a person determines that the angle of elevation to the top of the tower is[latex]\\text{ }43\u00b0,[\/latex] and that the angle of depression to the bottom of the tower is[latex]\\text{ }31\u00b0.\\text{ }[\/latex]How tall is the tower?<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135512511\">[reveal-answer q=\"fs-id1165135512511\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135512511\"]\r\n<p id=\"fs-id1165137677809\">498.3471 ft<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137677812\">\r\n<div id=\"fs-id1165135368447\">\r\n<p id=\"fs-id1165135368448\">A 200-foot tall monument is located in the distance. From a window in a building, a person determines that the angle of elevation to the top of the monument is[latex]\\text{ }15\u00b0,[\/latex] and that the angle of depression to the bottom of the tower is[latex]\\text{ }2\u00b0.\\text{ }[\/latex]How far is the person from the monument?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137648500\">\r\n<div id=\"fs-id1165137694171\">\r\n<p id=\"fs-id1165137694172\">A 400-foot tall monument is located in the distance. From a window in a building, a person determines that the angle of elevation to the top of the monument is[latex]\\text{ }18\u00b0,[\/latex] and that the angle of depression to the bottom of the monument is[latex]\\text{ }3\u00b0.\\text{ }[\/latex]How far is the person from the monument?<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137645701\">[reveal-answer q=\"fs-id1165137645701\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137645701\"]\r\n<p id=\"fs-id1165135181440\">1060.09 ft<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135181443\">\r\n<div id=\"fs-id1165137414694\">\r\n<p id=\"fs-id1165137414695\">There is an antenna on the top of a building. From a location 300 feet from the base of the building, the angle of elevation to the top of the building is measured to be[latex]\\text{ }40\u00b0.\\text{ }[\/latex]From the same location, the angle of elevation to the top of the antenna is measured to be[latex]\\text{ }43\u00b0.\\text{ }[\/latex]Find the height of the antenna.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137436702\">\r\n<div id=\"fs-id1165137849558\">\r\n<p id=\"fs-id1165137849560\">There is lightning rod on the top of a building. From a location 500 feet from the base of the building, the angle of elevation to the top of the building is measured to be[latex]\\text{ }36\u00b0.\\text{ }[\/latex]From the same location, the angle of elevation to the top of the lightning rod is measured to be[latex]\\text{ }38\u00b0.\\text{ }[\/latex]Find the height of the lightning rod.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137767139\">[reveal-answer q=\"fs-id1165137767139\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137767139\"]\r\n<p id=\"fs-id1165135189997\">27.372 ft<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137730425\" class=\"bc-section section\">\r\n<h4>Real-World Applications<\/h4>\r\n<div id=\"fs-id1165137451128\">\r\n<div id=\"fs-id1165137451129\">\r\n<p id=\"fs-id1165137451130\">A 33-ft ladder leans against a building so that the angle between the ground and the ladder is[latex]\\text{ }80\u00b0.\\text{ }[\/latex]How high does the ladder reach up the side of the building?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137437700\">\r\n<div id=\"fs-id1165137437702\">\r\n<p id=\"fs-id1165137437703\">A 23-ft ladder leans against a building so that the angle between the ground and the ladder is[latex]\\text{ }80\u00b0.\\text{ }[\/latex]How high does the ladder reach up the side of the building?<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137437289\">[reveal-answer q=\"fs-id1165137437289\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137437289\"]\r\n<p id=\"fs-id1165137437290\">22.6506 ft<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137452345\">\r\n<div id=\"fs-id1165137452346\">\r\n<p id=\"fs-id1165137935681\">The angle of elevation to the top of a building in New York is found to be 9 degrees from the ground at a distance of 1 mile from the base of the building. Using this information, find the height of the building.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137665678\">\r\n<div>\r\n<p id=\"fs-id1165137665680\">The angle of elevation to the top of a building in Seattle is found to be 2 degrees from the ground at a distance of 2 miles from the base of the building. Using this information, find the height of the building.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135250591\">[reveal-answer q=\"fs-id1165135250591\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135250591\"]\r\n<p id=\"fs-id1165135250592\">368.7633 ft<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135690138\">\r\n<div id=\"fs-id1165135690140\">\r\n<p id=\"fs-id1165135690141\">Assuming that a 370-foot tall giant redwood grows vertically, if I walk a certain distance from the tree and measure the angle of elevation to the top of the tree to be[latex]\\text{ }60\u00b0,[\/latex] how far from the base of the tree am I?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137627082\" class=\"review-exercises\">\r\n<h3>Review Exercises<\/h3>\r\n<div id=\"fs-id1165135160073\" class=\"bc-section section\">\r\n<h4><a class=\"target-chapter\" href=\"\/contents\/6a416d21-7302-4df4-a53d-06ac8a166e31\">Angles<\/a><\/h4>\r\n<p id=\"fs-id1165135186396\">For the following exercises, convert the angle measures to degrees.<\/p>\r\n\r\n<div id=\"fs-id1165137604792\">\r\n<div id=\"fs-id1165137604793\">\r\n<p id=\"fs-id1165137604794\">[latex]\\frac{\\pi }{4} [\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135173589\">[reveal-answer q=\"fs-id1165135173589\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135173589\"]\r\n<p id=\"fs-id1165135173590\">[latex]45\u00b0[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134223337\">\r\n<div id=\"fs-id1165134223338\">\r\n<p id=\"fs-id1165134223340\">[latex]-\\frac{5\\pi }{3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165137445371\">For the following exercises, convert the angle measures to radians.<\/p>\r\n\r\n<div id=\"fs-id1165137923488\">\r\n<div id=\"fs-id1165137923489\">\r\n<p id=\"fs-id1165137923490\">-210\u00b0<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135192319\">[reveal-answer q=\"fs-id1165135192319\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135192319\"]\r\n<p id=\"fs-id1165135192320\">[latex]-\\frac{7\\pi }{6}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137626943\">\r\n<div id=\"fs-id1165137812372\">\r\n<p id=\"fs-id1165137812373\">180\u00b0<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137544293\">\r\n<div id=\"fs-id1165137544294\">\r\n<p id=\"fs-id1165137652804\">Find the length of an arc in a circle of radius 7 meters subtended by the central angle of 85\u00b0.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135527005\">[reveal-answer q=\"fs-id1165135527005\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135527005\"]\r\n<p id=\"fs-id1165135527006\">10.385 meters<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137833897\">\r\n<div id=\"fs-id1165137833898\">\r\n<p id=\"fs-id1165137901222\">Find the area of the sector of a circle with diameter 32 feet and an angle of[latex]\\text{ }\\frac{3\\pi }{5}\\text{ }[\/latex]radians.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165135160645\">For the following exercises, find the angle between 0\u00b0 and 360\u00b0 that is coterminal with the given angle.<\/p>\r\n\r\n<div id=\"fs-id1165137506838\">\r\n<div id=\"fs-id1165137506839\">\r\n<p id=\"fs-id1165137506840\">[latex]420\u00b0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137735631\">[reveal-answer q=\"fs-id1165137735631\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137735631\"]\r\n<p id=\"fs-id1165137735632\">[latex]60\u00b0[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137534948\">\r\n<div id=\"fs-id1165137534949\">\r\n<p id=\"fs-id1165134069177\">[latex]-80\u00b0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165135149019\">For the following exercises, find the angle between 0 and[latex]\\text{ }2\\pi \\text{ }[\/latex]in radians that is coterminal with the given angle.<\/p>\r\n\r\n<div id=\"fs-id1165137605274\">\r\n<div id=\"fs-id1165134342624\">\r\n<p id=\"fs-id1165134342625\">[latex]-\\text{ }\\frac{20\\pi }{11}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137444823\">[reveal-answer q=\"fs-id1165137444823\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137444823\"]\r\n<p id=\"fs-id1165137444824\">[latex]\\frac{2\\pi }{11}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137727049\">\r\n<div id=\"fs-id1165137727050\">\r\n<p id=\"fs-id1165137727051\">[latex]\\frac{14\\pi }{5}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165137415610\">For the following exercises, draw the angle provided in standard position on the Cartesian plane.<\/p>\r\n\r\n<div id=\"fs-id1165137415613\">\r\n<div id=\"fs-id1165137460755\">\r\n<p id=\"fs-id1165137460756\">-210\u00b0<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135209568\">[reveal-answer q=\"fs-id1165135209568\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135209568\"]<span id=\"fs-id1165135403289\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132649\/CNX_Precalc_Figure_05_04_217.jpg\" alt=\"A graph of a circle with a negative angle inscribed.\" \/><\/span>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137771820\">\r\n<div id=\"fs-id1165137771822\">\r\n<p id=\"fs-id1165135497744\">75\u00b0<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137611468\">\r\n<div id=\"fs-id1165137611469\">\r\n<p id=\"fs-id1165137611470\">[latex]\\frac{5\\pi }{4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137785040\">[reveal-answer q=\"fs-id1165137785040\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137785040\"]<span id=\"fs-id1165135169236\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132651\/CNX_Precalc_Figure_05_04_219.jpg\" alt=\"A graph of a circle with an angle inscribed.\" \/><\/span>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137451538\">\r\n<div id=\"fs-id1165135299863\">\r\n<p id=\"fs-id1165135299864\">[latex]-\\frac{\\pi }{3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137842349\">\r\n<div id=\"fs-id1165137842350\">\r\n<p id=\"fs-id1165137842352\">Find the linear speed of a point on the equator of the earth if the earth has a radius of 3,960 miles and the earth rotates on its axis every 24 hours. Express answer in miles per hour.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135194111\">[reveal-answer q=\"fs-id1165135194111\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135194111\"]\r\n<p id=\"fs-id1165137475894\">1036.73 miles per hour<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137475897\">\r\n<div id=\"fs-id1165137745166\">\r\n<p id=\"fs-id1165137745167\">A car wheel with a diameter of 18 inches spins at the rate of 10 revolutions per second. What is the car's speed in miles per hour?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137598589\" class=\"bc-section section\">\r\n<h4><a class=\"target-chapter\" href=\"\/contents\/17a6d7f5-c90c-4047-abac-0376d582549b\">Unit Circle: Sine and Cosine Functions<\/a><\/h4>\r\n<div id=\"fs-id1165137804818\">\r\n<div id=\"fs-id1165137470140\">\r\n<p id=\"fs-id1165137470141\">Find the exact value of[latex]\\text{ }\\mathrm{sin}\\text{ }\\frac{\\pi }{3}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137642361\">[reveal-answer q=\"fs-id1165137642361\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137642361\"]\r\n<p id=\"fs-id1165137642362\">[latex]\\frac{\\sqrt{3}}{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137647355\">\r\n<div id=\"fs-id1165137647356\">\r\n<p id=\"fs-id1165135708042\">Find the exact value of[latex]\\text{ }\\mathrm{cos}\\text{ }\\frac{\\pi }{4}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137566636\">\r\n<div id=\"fs-id1165137566637\">\r\n<p id=\"fs-id1165137566638\">Find the exact value of[latex]\\text{ }\\mathrm{cos}\\text{ }\\pi .[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135173474\">[reveal-answer q=\"fs-id1165135173474\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135173474\"]\r\n<p id=\"fs-id1165135173475\">\u20131<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165133281394\">\r\n<div id=\"fs-id1165133281395\">\r\n<p id=\"fs-id1165133281396\">State the reference angle for[latex]\\text{ }300\u00b0.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137465126\">\r\n<div id=\"fs-id1165137465127\">\r\n<p id=\"fs-id1165137465128\">State the reference angle for[latex]\\text{ }\\frac{3\\pi }{4}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137756068\">[reveal-answer q=\"fs-id1165137756068\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137756068\"]\r\n<p id=\"fs-id1165137723588\">[latex]\\frac{\\pi }{4}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137426933\">\r\n<div id=\"fs-id1165137426934\">\r\n<p id=\"fs-id1165137426935\">Compute cosine of[latex]\\text{ }330\u00b0.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137602620\">\r\n<div id=\"fs-id1165137602621\">\r\n<p id=\"fs-id1165137602622\">Compute sine of[latex]\\text{ }\\frac{5\\pi }{4}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137748992\">[reveal-answer q=\"fs-id1165137748992\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137748992\"]\r\n<p id=\"fs-id1165137506605\">[latex]-\\frac{\\sqrt{2}}{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137573390\">\r\n<div id=\"fs-id1165137573391\">\r\n<p id=\"fs-id1165137573392\">State the domain of the sine and cosine functions.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137574969\">\r\n<div id=\"fs-id1165137574970\">\r\n<p id=\"fs-id1165137748521\">State the range of the sine and cosine functions.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137748524\">[reveal-answer q=\"fs-id1165137748524\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137748524\"]\r\n<p id=\"fs-id1165137583678\">[latex]\\left[\u20131,1\\right][\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137668652\" class=\"bc-section section\">\r\n<h4><a class=\"target-chapter\" href=\"\/contents\/df1a76e7-7b9c-4f58-8a81-afd48cedcf0b\">The Other Trigonometric Functions<\/a><\/h4>\r\n<p id=\"fs-id1165137657202\">For the following exercises, find the exact value of the given expression.<\/p>\r\n\r\n<div id=\"fs-id1165137920680\">\r\n<div id=\"fs-id1165137920681\">\r\n<p id=\"fs-id1165134284487\">[latex]\\mathrm{cos}\\text{ }\\frac{\\pi }{6}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137871054\">\r\n<div id=\"fs-id1165135512426\">\r\n<p id=\"fs-id1165135512427\">[latex]\\mathrm{tan}\\text{ }\\frac{\\pi }{4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137676096\">[reveal-answer q=\"fs-id1165137676096\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137676096\"]\r\n<p id=\"fs-id1165137676098\">1<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137596056\">\r\n<div id=\"fs-id1165137596057\">\r\n<p id=\"fs-id1165137596058\">[latex]\\mathrm{csc}\\text{ }\\frac{\\pi }{3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137656500\">\r\n<div id=\"fs-id1165137656501\">\r\n<p id=\"fs-id1165137656502\">[latex]\\mathrm{sec}\\text{ }\\frac{\\pi }{4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137730531\">[reveal-answer q=\"fs-id1165137730531\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137730531\"]\r\n<p id=\"fs-id1165137730532\">[latex]\\sqrt{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165134381589\">For the following exercises, use reference angles to evaluate the given expression.<\/p>\r\n\r\n<div id=\"fs-id1165137862404\">\r\n<div id=\"fs-id1165137862406\">\r\n<p id=\"fs-id1165137862407\">[latex]\\mathrm{sec}\\text{ }\\frac{11\\pi }{3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137588722\">\r\n<div id=\"fs-id1165137588723\">\r\n<p id=\"fs-id1165137836532\">[latex]\\mathrm{sec}\\text{ }315\u00b0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137549719\">[reveal-answer q=\"fs-id1165137549719\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137549719\"]\r\n<p id=\"fs-id1165137637844\">[latex]\\sqrt{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135205897\">\r\n<div id=\"fs-id1165135205898\">\r\n<p id=\"fs-id1165137580092\">If[latex]\\text{ }\\mathrm{sec}\\left(t\\right)=-2.5\\text{ }[\/latex], what is the[latex]\\text{ }\\text{sec}\\left(-t\\right)?[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165132943553\">\r\n<div id=\"fs-id1165132943554\">\r\n<p id=\"fs-id1165137768826\">If[latex]\\text{ }\\text{tan}\\left(t\\right)=-0.6,[\/latex] what is the[latex]\\text{ }\\text{tan}\\left(-t\\right)?[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137780740\">[reveal-answer q=\"fs-id1165137780740\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137780740\"]\r\n<p id=\"fs-id1165137780741\">0.6<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135378773\">\r\n<div id=\"fs-id1165135378774\">\r\n<p id=\"fs-id1165135378775\">If[latex]\\text{ }\\text{tan}\\left(t\\right)=\\frac{1}{3},[\/latex] find[latex]\\text{ }\\text{tan}\\left(t-\\pi \\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165133356011\">\r\n<div id=\"fs-id1165133356012\">\r\n<p id=\"fs-id1165133356013\">If[latex]\\text{ }\\text{cos}\\left(t\\right)=\\frac{\\sqrt{2}}{2},[\/latex] find[latex]\\text{ }\\text{sin}\\left(t+2\\pi \\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137574568\">[reveal-answer q=\"fs-id1165137574568\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137574568\"]\r\n<p id=\"fs-id1165137574570\">[latex]\\frac{\\sqrt{2}}{2}\\text{ }[\/latex]or[latex]\\text{ }-\\frac{\\sqrt{2}}{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137758484\">\r\n<div id=\"fs-id1165137758485\">\r\n<p id=\"fs-id1165137871876\">Which trigonometric functions are even?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137871879\">\r\n<div id=\"fs-id1165137572480\">\r\n<p id=\"fs-id1165137572482\">Which trigonometric functions are odd?<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137471478\">[reveal-answer q=\"fs-id1165137471478\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137471478\"]\r\n<p id=\"fs-id1165137471479\">sine, cosecant, tangent, cotangent<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137911687\" class=\"bc-section section\">\r\n<h4><a class=\"target-chapter\" href=\"\/contents\/2d2ba0f3-4818-4665-9f37-7ec8c6d36e52\">Right Triangle Trigonometry<\/a><\/h4>\r\n<p id=\"fs-id1165137425556\">For the following exercises, use side lengths to evaluate.<\/p>\r\n\r\n<div id=\"fs-id1165137571591\">\r\n<div id=\"fs-id1165137571592\">\r\n<p id=\"fs-id1165137571593\">[latex]\\mathrm{cos}\\text{ }\\frac{\\pi }{4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137803944\">\r\n<div id=\"fs-id1165137803945\">\r\n<p id=\"fs-id1165137658119\">[latex]\\mathrm{cot}\\text{ }\\frac{\\pi }{3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137663617\">[reveal-answer q=\"fs-id1165137663617\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137663617\"]\r\n<p id=\"fs-id1165137665107\">[latex]\\frac{\\sqrt{3}}{3}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165133047532\">\r\n<div id=\"fs-id1165135693727\">\r\n\r\n[latex]\\mathrm{tan}\\text{ }\\frac{\\pi }{6}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137898078\">\r\n<div id=\"fs-id1165137898079\">\r\n<p id=\"fs-id1165137898080\">[latex]\\mathrm{cos}\\left(\\frac{\\pi }{2}\\right)=\\mathrm{sin}\\left(\\text{__\u00b0}\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137605493\">[reveal-answer q=\"fs-id1165137605493\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137605493\"]\r\n<p id=\"fs-id1165135193114\">0<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135193118\">\r\n<div id=\"fs-id1165137653909\">\r\n<p id=\"fs-id1165137653910\">[latex]\\mathrm{csc}\\left(18\\text{\u00b0}\\right)=\\mathrm{sec}\\left(\\text{__\u00b0}\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165137724964\">For the following exercises, use the given information to find the lengths of the other two sides of the right triangle.<\/p>\r\n\r\n<div id=\"fs-id1165135528966\">\r\n<div id=\"fs-id1165135528967\">\r\n<p id=\"fs-id1165135560804\">[latex]\\mathrm{cos}\\text{ }B=\\frac{3}{5},a=6[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137433002\">[reveal-answer q=\"fs-id1165137433002\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137433002\"]\r\n<p id=\"fs-id1165137433003\">[latex]b=8,c=10[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137410615\">\r\n<div id=\"fs-id1165137410616\">\r\n<p id=\"fs-id1165137410617\">[latex]\\mathrm{tan}\\text{ }A=\\frac{5}{9},b=6[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165137644993\">For the following exercises, use <a class=\"autogenerated-content\" href=\"#Figure_05_04_221\">(Figure)<\/a> to evaluate each trigonometric function.<\/p>\r\n\r\n<div id=\"Figure_05_04_221\" class=\"small\"><span id=\"fs-id1165137532849\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132653\/CNX_Precalc_Figure_05_04_221.jpg\" alt=\"A right triangle with side lengths of 11 and 6. Corners A and B are also labeled.\" \/><\/span><\/div>\r\n<div id=\"fs-id1165135424671\">\r\n<div id=\"fs-id1165135424672\">\r\n\r\n[latex]\\mathrm{sin}\\text{ }A[\/latex]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137734516\">[reveal-answer q=\"fs-id1165137734516\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137734516\"]\r\n<p id=\"fs-id1165134122790\">[latex]\\frac{11\\sqrt{157}}{157}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137558054\">\r\n<div id=\"fs-id1165137442090\">\r\n<p id=\"fs-id1165137442091\">[latex]\\mathrm{tan}\\text{ }B[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165135192508\">For the following exercises, solve for the unknown sides of the given triangle.<\/p>\r\n\r\n<div id=\"fs-id1165135263670\">\r\n<div id=\"fs-id1165135263671\"><span id=\"fs-id1165137834282\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132656\/CNX_Precalc_Figure_05_04_222n.jpg\" alt=\"A right triangle with corners labeled A, B, and C. Hyptenuse has length of 4 times square root of 2. Other angles measure 45 degrees.\" \/><\/span><\/div>\r\n<div id=\"fs-id1165137575901\">[reveal-answer q=\"fs-id1165137575901\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137575901\"]\r\n<p id=\"fs-id1165137575902\">[latex]a=4, b=4[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135436551\">\r\n<div id=\"fs-id1165135436552\"><span id=\"fs-id1165137406146\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132658\/CNX_Precalc_Figure_05_04_223n.jpg\" alt=\"A right triangle with hypotenuse with length 5, and an angle of 30 degrees.\" \/><\/span><\/div>\r\n<\/div>\r\n<div id=\"fs-id1165133255048\">\r\n<div id=\"fs-id1165133255050\">\r\n<p id=\"fs-id1165133255051\">A 15-ft ladder leans against a building so that the angle between the ground and the ladder is[latex]\\text{ }70\u00b0.\\text{ }[\/latex]How high does the ladder reach up the side of the building?<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135332736\">[reveal-answer q=\"fs-id1165135332736\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135332736\"]\r\n<p id=\"fs-id1165137767865\">14.0954 ft<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137767869\">\r\n<div id=\"fs-id1165135536497\">\r\n<p id=\"fs-id1165135536498\">The angle of elevation to the top of a building in Baltimore is found to be 4 degrees from the ground at a distance of 1 mile from the base of the building. Using this information, find the height of the building.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137668291\" class=\"practice-test\">\r\n<h3>Practice Test<\/h3>\r\n<div id=\"fs-id1165137668059\">\r\n<div id=\"fs-id1165137735263\">\r\n<p id=\"fs-id1165137735264\">Convert[latex]\\text{ }\\frac{5\\pi }{6}\\text{ }[\/latex]radians to degrees.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137560606\">[reveal-answer q=\"fs-id1165137560606\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137560606\"]\r\n<p id=\"fs-id1165137560607\">[latex]150\u00b0[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135694590\">\r\n<div id=\"fs-id1165137767383\">\r\n<p id=\"fs-id1165137767384\">Convert[latex]\\text{ }-620\u00b0\\text{ }[\/latex]to radians.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137452465\">\r\n<div id=\"fs-id1165137452466\">\r\n<p id=\"fs-id1165137452467\">Find the length of a circular arc with a radius 12 centimeters subtended by the central angle of[latex]\\text{ }30\u00b0.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137862788\">[reveal-answer q=\"fs-id1165137862788\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137862788\"]\r\n<p id=\"fs-id1165137862790\">6.283 centimeters<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137401698\">\r\n<div id=\"fs-id1165137401699\">\r\n<p id=\"fs-id1165137401700\">Find the area of the sector with radius of 8 feet and an angle of[latex]\\text{ }\\frac{5\\pi }{4}\\text{ }[\/latex]radians.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135408494\">\r\n<div id=\"fs-id1165135183090\">\r\n<p id=\"fs-id1165135183091\">Find the angle between[latex]\\text{ }0\u00b0\\text{ }[\/latex]and[latex]\\text{ }\\text{360\u00b0}\\text{ }[\/latex]that is coterminal with[latex]\\text{ }375\u00b0.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135161403\">[reveal-answer q=\"fs-id1165135161403\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135161403\"]\r\n<p id=\"fs-id1165135161404\">[latex]15\u00b0[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137619343\">\r\n<div id=\"fs-id1165137768782\">\r\n<p id=\"fs-id1165137768783\">Find the angle between 0 and[latex]\\text{ }2\\pi \\text{ }[\/latex]in radians that is coterminal with[latex]\\text{ }-\\frac{4\\pi }{7}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137931243\">\r\n<div id=\"fs-id1165137931244\">\r\n<p id=\"fs-id1165137931245\">Draw the angle[latex]\\text{ }315\u00b0\\text{ }[\/latex]in standard position on the Cartesian plane.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137694225\">[reveal-answer q=\"fs-id1165137694225\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137694225\"]<span id=\"fs-id1165137592067\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132701\/CNX_Precalc_Figure_05_04_224.jpg\" alt=\"A graph of a circle with an angle inscribed.\" \/><\/span>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137851378\">\r\n<div id=\"fs-id1165137402102\">\r\n<p id=\"fs-id1165137402103\">Draw the angle[latex]\\text{ }-\\frac{\\pi }{6}\\text{ }[\/latex]in standard position on the Cartesian plane.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137653537\">\r\n<div id=\"fs-id1165137653538\">\r\n<p id=\"fs-id1165137653540\">A carnival has a Ferris wheel with a diameter of 80 feet. The time for the Ferris wheel to make one revolution is 75 seconds. What is the linear speed in feet per second of a point on the Ferris wheel? What is the angular speed in radians per second?<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137725827\">[reveal-answer q=\"fs-id1165137725827\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137725827\"]\r\n<p id=\"fs-id1165137557535\">3.351 feet per second,[latex]\\text{ }\\frac{2\\pi }{75}\\text{ }[\/latex]radians per second<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135160182\">\r\n<div id=\"fs-id1165135160183\">\r\n<p id=\"fs-id1165135160184\">Find the exact value of[latex]\\text{ }\\mathrm{sin}\\text{ }\\frac{\\pi }{6}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137575005\">\r\n<div id=\"fs-id1165137575006\">\r\n<p id=\"fs-id1165135639873\">Compute sine of[latex]\\text{ }240\u00b0.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137408475\">[reveal-answer q=\"fs-id1165137408475\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137408475\"]\r\n<p id=\"fs-id1165137408476\">[latex]-\\frac{\\sqrt{3}}{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135343004\">\r\n<div id=\"fs-id1165135593415\">\r\n<p id=\"fs-id1165135593416\">State the domain of the sine and cosine functions.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135593420\">\r\n<div id=\"fs-id1165135192267\">\r\n<p id=\"fs-id1165135192268\">State the range of the sine and cosine functions.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135192271\">[reveal-answer q=\"fs-id1165135192271\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135192271\"]\r\n<p id=\"fs-id1165134269017\">[latex]\\left[\u20131,1\\right][\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135344124\">\r\n<div id=\"fs-id1165137605201\">\r\n<p id=\"fs-id1165137605202\">Find the exact value of[latex]\\text{ }\\mathrm{cot}\\text{ }\\frac{\\pi }{4}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137722364\">\r\n<div id=\"fs-id1165137722365\">\r\n<p id=\"fs-id1165137722366\">Find the exact value of[latex]\\text{ }\\mathrm{tan}\\text{ }\\frac{\\pi }{3}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137823309\">[reveal-answer q=\"fs-id1165137823309\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137823309\"]\r\n<p id=\"fs-id1165137823310\">[latex]\\sqrt{3}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137664264\">\r\n<div id=\"fs-id1165137664265\">\r\n<p id=\"fs-id1165137603262\">Use reference angles to evaluate[latex]\\text{ }\\mathrm{csc}\\text{ }\\frac{7\\pi }{4}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137415501\">\r\n<div id=\"fs-id1165137415502\">\r\n<p id=\"fs-id1165137415504\">Use reference angles to evaluate[latex]\\text{ }\\mathrm{tan}\\text{ }210\u00b0.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135405233\">[reveal-answer q=\"fs-id1165135405233\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135405233\"]\r\n<p id=\"fs-id1165135405234\">[latex]\\frac{\\sqrt{3}}{3}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135181505\">\r\n<div id=\"fs-id1165135181506\">\r\n<p id=\"fs-id1165135181507\">If[latex]\\text{ }\\text{csc}\\text{ }t=0.68,[\/latex]what is the[latex]\\text{ }\\text{csc}\\left(-t\\right)?[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137634897\">\r\n<div id=\"fs-id1165137634898\">\r\n<p id=\"fs-id1165137634899\">If[latex]\\text{ }\\text{cos}\\text{ }\\text{t}=\\frac{\\sqrt{3}}{2},[\/latex]find[latex]\\text{ }\\text{cos}\\left(t-2\\pi \\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137447886\">[reveal-answer q=\"fs-id1165137447886\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137447886\"]\r\n<p id=\"fs-id1165137447887\">[latex]\\frac{\\sqrt{3}}{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137730237\">\r\n<div id=\"fs-id1165137432724\">\r\n<p id=\"fs-id1165137432725\">Which trigonometric functions are even?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137432728\">\r\n<div id=\"fs-id1165137891501\">\r\n<p id=\"fs-id1165137891502\">Find the missing angle:[latex]\\text{ }\\mathrm{cos}\\left(\\frac{\\pi }{6}\\right)=\\mathrm{sin}\\left(___\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137476439\">[reveal-answer q=\"fs-id1165137476439\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137476439\"]\r\n<p id=\"fs-id1165137476440\">[latex]\\frac{\\pi }{3}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div>\r\n<div id=\"fs-id1165135196915\">\r\n<p id=\"fs-id1165135196916\">Find the missing sides of the triangle[latex]\\text{ }ABC:\\mathrm{sin}\\text{ }B=\\frac{3}{4},c=12[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137661200\">\r\n<div id=\"fs-id1165137661201\">\r\n<p id=\"fs-id1165137661202\">Find the missing sides of the triangle.<\/p>\r\n<span id=\"fs-id1165137410882\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132703\/CNX_Precalc_Figure_05_04_226.jpg\" alt=\"A right triangle with hyptenuse length of 9 and angle measure of 60 degrees.\" \/><\/span>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165134113903\">[reveal-answer q=\"fs-id1165134113903\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134113903\"]\r\n<p id=\"fs-id1165134113904\">[latex]a=\\frac{9}{2},b=\\frac{9\\sqrt{3}}{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165132971678\">\r\n<div id=\"fs-id1165137480414\">\r\n<p id=\"fs-id1165137480415\">The angle of elevation to the top of a building in Chicago is found to be 9 degrees from the ground at a distance of 2000 feet from the base of the building. Using this information, find the height of the building.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Glossary<\/h3>\r\n<dl id=\"fs-id1165137446119\">\r\n \t<dt>adjacent side<\/dt>\r\n \t<dd id=\"fs-id1165137446123\">in a right triangle, the side between a given angle and the right angle<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137465232\">\r\n \t<dt>angle of depression<\/dt>\r\n \t<dd id=\"fs-id1165135175026\">the angle between the horizontal and the line from the object to the observer\u2019s eye, assuming the object is positioned lower than the observer<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137447602\">\r\n \t<dt>angle of elevation<\/dt>\r\n \t<dd id=\"fs-id1165135185281\">the angle between the horizontal and the line from the object to the observer\u2019s eye, assuming the object is positioned higher than the observer<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137558543\">\r\n \t<dt>opposite side<\/dt>\r\n \t<dd id=\"fs-id1165137558547\">in a right triangle, the side most distant from a given angle<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135477488\">\r\n \t<dt>hypotenuse<\/dt>\r\n \t<dd id=\"fs-id1165137588091\">the side of a right triangle opposite the right angle<\/dd>\r\n<\/dl>\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p>In this section, you will:<\/p>\n<ul>\n<li>Use right triangles to evaluate trigonometric functions.<\/li>\n<li>Find function values for[latex]\\text{ }30\u00b0\\left(\\frac{\\pi }{6}\\right),\\text{ }[\/latex][latex]45\u00b0\\left(\\frac{\\pi }{4}\\right),\\text{ }[\/latex]and[latex]\\text{ }60\u00b0\\left(\\frac{\\pi }{3}\\right).[\/latex]<\/li>\n<li>Use cofunctions of complementary angles.<\/li>\n<li>Use the de\ufb01nitions of trigonometric functions of any angle.<\/li>\n<li>Use right triangle trigonometry to solve applied problems.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1165137387312\">We have previously defined the sine and cosine of an angle in terms of the coordinates of a point on the unit circle intersected by the terminal side of the angle:<\/p>\n<div id=\"fs-id1165135501042\" class=\"unnumbered\">[latex]\\begin{array}{c}\\mathrm{cos}\\text{ }t=x\\\\ \\mathrm{sin}\\text{ }t=y\\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137668254\">In this section, we will see another way to define trigonometric functions using properties of <span class=\"no-emphasis\">right triangles<\/span>.<\/p>\n<div id=\"fs-id1165137547549\" class=\"bc-section section\">\n<h3>Using Right Triangles to Evaluate Trigonometric Functions<\/h3>\n<p id=\"fs-id1165134108581\">In earlier sections, we used a unit circle to define the <span class=\"no-emphasis\">trigonometric functions<\/span>. In this section, we will extend those definitions so that we can apply them to right triangles. The value of the sine or cosine function of[latex]\\text{ }t\\text{ }[\/latex]is its value at[latex]\\text{ }t\\text{ }[\/latex]radians. First, we need to create our right triangle. <a class=\"autogenerated-content\" href=\"#Figure_05_04_001\">(Figure)<\/a> shows a point on a <span class=\"no-emphasis\">unit circle<\/span> of radius 1. If we drop a vertical line segment from the point[latex]\\text{ }\\left(x,y\\right)\\text{ }[\/latex]to the <em>x<\/em>-axis, we have a right triangle whose vertical side has length[latex]\\text{ }y\\text{ }[\/latex]and whose horizontal side has length[latex]\\text{ }x.\\text{ }[\/latex]We can use this right triangle to redefine sine, cosine, and the other trigonometric functions as ratios of the sides of a right triangle.<\/p>\n<div id=\"Figure_05_04_001\" class=\"small\"><span id=\"fs-id1165137602828\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132528\/CNX_Precalc_Figure_05_04_001.jpg\" alt=\"Graph of quarter circle with radius of 1 and angle of t. Point of (x,y) is at intersection of terminal side of angle and edge of circle.\" \/><\/span><\/div>\n<p id=\"fs-id1165137470559\">We know<\/p>\n<div id=\"fs-id1165137417839\" class=\"unnumbered\">[latex]\\mathrm{cos}\\text{ }t=\\frac{x}{1}=x[\/latex]<\/div>\n<p>Likewise, we know<\/p>\n<div id=\"fs-id1165135426436\" class=\"unnumbered\">[latex]\\mathrm{sin}\\text{ }t=\\frac{y}{1}=y[\/latex]<\/div>\n<p id=\"fs-id1165135693752\">These ratios still apply to the sides of a right triangle when no unit circle is involved and when the triangle is not in standard position and is not being graphed using[latex]\\text{ }\\left(x,y\\right)\\text{ }[\/latex]coordinates. To be able to use these ratios freely, we will give the sides more general names: Instead of[latex]\\text{ }x,[\/latex]we will call the side between the given angle and the right angle the adjacent side to angle[latex]\\text{ }t.\\text{ }[\/latex](Adjacent means \u201cnext to.\u201d) Instead of[latex]\\text{ }y,[\/latex]we will call the side most distant from the given angle the opposite side from angle[latex]\\text{}t.\\text{ }[\/latex]And instead of[latex]\\text{ }1,[\/latex]we will call the side of a right triangle opposite the right angle the hypotenuse. These sides are labeled in <a class=\"autogenerated-content\" href=\"#Figure_05_04_002\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_05_04_002\" class=\"small\">\n<div class=\"wp-caption-text\"><\/div>\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\/3896\/2019\/03\/07132531\/CNX_Precalc_Figure_05_04_002.jpg\" alt=\"A right triangle with hypotenuse, opposite, and adjacent sides labeled.\" width=\"487\" height=\"137\" \/><\/p>\n<p class=\"wp-caption-text\">The sides of a right triangle in relation to angle [latex]t.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137784426\" class=\"bc-section section\">\n<h4>Understanding Right Triangle Relationships<\/h4>\n<p id=\"fs-id1165135189859\">Given a right triangle with an acute angle of[latex]\\text{ }t,[\/latex]<\/p>\n<div id=\"fs-id1165135177707\" class=\"unnumbered\">[latex]\\begin{array}{l}\\mathrm{sin}\\left(t\\right)=\\frac{\\text{opposite}}{\\text{hypotenuse}}\\hfill \\\\ \\mathrm{cos}\\left(t\\right)=\\frac{\\text{adjacent}}{\\text{hypotenuse}}\\hfill \\\\ \\mathrm{tan}\\left(t\\right)=\\frac{\\text{opposite}}{\\text{adjacent}}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137559825\">A common mnemonic for remembering these relationships is SohCahToa, formed from the first letters of \u201c<u>S<\/u>ine is <u>o<\/u>pposite over <u>h<\/u>ypotenuse, <u>C<\/u>osine is <u>a<\/u>djacent over <u>h<\/u>ypotenuse, <u>T<\/u>angent is <u>o<\/u>pposite over <u>a<\/u>djacent.\u201d<\/p>\n<div id=\"fs-id1165135448331\" class=\"precalculus howto examples\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137832115\"><strong>Given the side lengths of a right triangle and one of the acute angles, find the sine, cosine, and tangent of that angle.<\/strong><\/p>\n<ol id=\"fs-id1165137465487\" type=\"1\">\n<li>Find the sine as the ratio of the opposite side to the hypotenuse.<\/li>\n<li>Find the cosine as the ratio of the adjacent side to the hypotenuse.<\/li>\n<li>Find the tangent as the ratio of the opposite side to the adjacent side.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_05_04_01\" class=\"textbox examples\">\n<div id=\"fs-id1165137597073\">\n<div id=\"fs-id1165137410098\">\n<h3>Example 1: Evaluating a Trigonometric Function of a Right Triangle<\/h3>\n<p id=\"fs-id1165135630976\">Given the triangle shown in <a class=\"autogenerated-content\" href=\"#Figure_05_04_003\">(Figure)<\/a>, find the value of[latex]\\text{ }\\mathrm{cos}\\text{ }\\alpha .[\/latex]<\/p>\n<div id=\"Figure_05_04_003\" class=\"small\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132533\/CNX_Precalc_Figure_05_04_003.jpg\" alt=\"A right triangle with sid lengths of 8, 15, and 17. Angle alpha also labeled.\" \/><\/div>\n<\/div>\n<div id=\"fs-id1165137398722\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137398722\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137398722\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137647892\">The side adjacent to the angle is 15, and the hypotenuse of the triangle is 17, so:<\/p>\n<div id=\"fs-id1165137770356\" class=\"unnumbered\">[latex]\\begin{array}{l}\\mathrm{cos}\\left(\\alpha \\right)=\\frac{\\text{adjacent}}{\\text{hypotenuse}}\\hfill \\\\ \\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }=\\frac{15}{17}\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137453511\" class=\"precalculus tryit\">\n<h3>Try it #1<\/h3>\n<div id=\"ti_05_04_01\">\n<div id=\"fs-id1165134570073\">\n<p id=\"fs-id1165137811034\">Given the triangle shown in <a class=\"autogenerated-content\" href=\"#Figure_05_04_004\">(Figure)<\/a>, find the value of[latex]\\text{ }\\text{sin}\\text{ }t.[\/latex]<\/p>\n<div id=\"Figure_05_04_004\" class=\"small\"><span id=\"fs-id1165135191134\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132536\/CNX_Precalc_Figure_05_04_004.jpg\" alt=\"A right triangle with sides of 7, 24, and 25. Also labeled is angle t.\" \/><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165137602319\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137602319\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137602319\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137422653\">[latex]\\frac{7}{25}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134199461\" class=\"bc-section section\">\n<h4>Relating Angles and Their Functions<\/h4>\n<p id=\"fs-id1165135241150\">When working with right triangles, the same rules apply regardless of the orientation of the triangle. In fact, we can evaluate the six trigonometric functions of either of the two acute angles in the triangle in <a class=\"autogenerated-content\" href=\"#Figure_05_04_005\">(Figure)<\/a>. The side opposite one acute angle is the side adjacent to the other acute angle, and vice versa.<\/p>\n<div id=\"Figure_05_04_005\" class=\"small\">\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\/3896\/2019\/03\/07132539\/CNX_Precalc_Figure_05_04_005.jpg\" alt=\"Right triangle with angles alpha and beta. Sides are labeled hypotenuse, adjacent to alpha\/opposite to beta, and adjacent to beta\/opposite alpha.\" width=\"487\" height=\"181\" \/><\/p>\n<p class=\"wp-caption-text\">The side adjacent to one angle is opposite the other.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137459869\">We will be asked to find all six trigonometric functions for a given angle in a triangle. Our strategy is to find the sine, cosine, and tangent of the angles first. Then, we can find the other trigonometric functions easily because we know that the reciprocal of sine is cosecant, the reciprocal of cosine is secant, and the reciprocal of tangent is cotangent.<\/p>\n<div id=\"fs-id1165137806953\" class=\"precalculus howto examples\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137427122\"><strong>Given the side lengths of a right triangle, evaluate the six trigonometric functions of one of the acute angles.<\/strong><\/p>\n<ol id=\"fs-id1165137762694\" type=\"1\">\n<li>If needed, draw the right triangle and label the angle provided.<\/li>\n<li>Identify the angle, the adjacent side, the side opposite the angle, and the hypotenuse of the right triangle.<\/li>\n<li>Find the required function:\n<ul id=\"fs-id1165135532521\">\n<li>sine as the ratio of the opposite side to the hypotenuse<\/li>\n<li>cosine as the ratio of the adjacent side to the hypotenuse<\/li>\n<li>tangent as the ratio of the opposite side to the adjacent side<\/li>\n<li>secant as the ratio of the hypotenuse to the adjacent side<\/li>\n<li>cosecant as the ratio of the hypotenuse to the opposite side<\/li>\n<li>cotangent as the ratio of the adjacent side to the opposite side<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_05_04_02\" class=\"textbox examples\">\n<div id=\"fs-id1165135187471\">\n<div id=\"fs-id1165137836968\">\n<h3>Example 2: Evaluating Trigonometric Functions of Angles Not in Standard Position<\/h3>\n<p id=\"fs-id1165137724941\">Using the triangle shown in <a class=\"autogenerated-content\" href=\"#Figure_05_04_006\">(Figure)<\/a>, evaluate [latex]\\mathrm{sin}\\text{ }\\alpha ,[\/latex] [latex]\\mathrm{cos}\\text{ }\\alpha ,[\/latex] [latex]\\mathrm{tan}\\text{ }\\alpha ,[\/latex][latex]\\mathrm{sec}\\text{ }\\alpha ,[\/latex] [latex]\\mathrm{csc}\\text{ }\\alpha ,[\/latex] and [latex]\\text{ }\\mathrm{cot}\\text{ }\\alpha .[\/latex]<\/p>\n<div id=\"Figure_05_04_006\" class=\"small\"><span id=\"fs-id1165137542988\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132541\/CNX_Precalc_Figure_05_04_006.jpg\" alt=\"Right triangle with sides of 3, 4, and 5. Angle alpha is also labeled.\" \/><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165137447828\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137447828\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137447828\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165137571205\" class=\"unnumbered\">[latex]\\begin{array}{l}\\mathrm{sin}\\text{ }\\alpha =\\frac{\\text{opposite }\\alpha }{\\text{hypotenuse}}=\\frac{4}{5}\\hfill \\\\ \\mathrm{cos}\\text{ }\\alpha =\\frac{\\text{adjacent to }\\alpha }{\\text{hypotenuse}}=\\frac{3}{5}\\hfill \\\\ \\mathrm{tan}\\text{ }\\alpha =\\frac{\\text{opposite }\\alpha }{\\text{adjacent to }\\alpha }=\\frac{4}{3}\\hfill \\\\ \\mathrm{sec}\\text{ }\\alpha =\\frac{\\text{hypotenuse}}{\\text{adjacent to }\\alpha }=\\frac{5}{3}\\hfill \\\\ \\mathrm{csc}\\text{ }\\alpha =\\frac{\\text{hypotenuse}}{\\text{opposite }\\alpha }=\\frac{5}{4}\\hfill \\\\ \\mathrm{cot}\\text{ }\\alpha =\\frac{\\text{adjacent to }\\alpha }{\\text{opposite }\\alpha }=\\frac{3}{4}\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137548404\" class=\"precalculus tryit\">\n<h3>Try it #2<\/h3>\n<div id=\"ti_05_04_02\">\n<div id=\"fs-id1165137809949\">\n<p id=\"fs-id1165137809950\">Using the triangle shown in <a class=\"autogenerated-content\" href=\"#Figure_05_04_007\">(Figure)<\/a>, evaluate[latex]\\text{ }\\mathrm{sin}\\text{ }t,[\/latex] [latex]\\mathrm{cos}\\text{ }t,[\/latex] [latex]\\mathrm{tan}\\text{ }t,[\/latex][latex]\\mathrm{sec}\\text{ }t,[\/latex] [latex]\\mathrm{csc}\\text{ }t,[\/latex]and[latex]\\text{ }\\mathrm{cot}\\text{ }t.[\/latex]<\/p>\n<div id=\"Figure_05_04_007\" class=\"small\"><span id=\"fs-id1165137847139\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132544\/CNX_Precalc_Figure_05_04_007.jpg\" alt=\"Right triangle with sides 33, 56, and 65. Angle t is also labeled.\" \/><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165135452495\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135452495\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135452495\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135452496\">[latex]\\begin{array}{l}sin t=\\frac{33}{65},\\mathrm{cos} t=\\frac{56}{65},tan t=\\frac{33}{56},\\hfill \\\\ \\text{ }\\mathrm{sec} t=\\frac{65}{56},\\mathrm{csc} t=\\frac{65}{33},\\mathrm{cot} t=\\frac{56}{33}\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137810878\" class=\"bc-section section\">\n<h4>Finding Trigonometric Functions of Special Angles Using Side Lengths<\/h4>\n<p id=\"fs-id1165137531688\">We have already discussed the trigonometric functions as they relate to the <span class=\"no-emphasis\">special angles<\/span> on the unit circle. Now, we can use those relationships to evaluate triangles that contain those special angles. We do this because when we evaluate the special angles in trigonometric functions, they have relatively friendly values, values that contain either no or just one square root in the ratio. Therefore, these are the angles often used in math and science problems. We will use multiples of[latex]\\text{ }30\u00b0,[\/latex] [latex]60\u00b0,[\/latex] and[latex]\\text{ }45\u00b0,[\/latex] however, remember that when dealing with right triangles, we are limited to angles between[latex]\\text{ }0\u00b0\\text{ and 90\u00b0}\\text{.}[\/latex]<\/p>\n<p id=\"fs-id1165135193209\">Suppose we have a[latex]\\text{ }30\u00b0,60\u00b0,90\u00b0\\text{ }[\/latex]triangle, which can also be described as a[latex]\\text{ }\\frac{\\pi }{6},\\text{\u200b} \\frac{\\pi }{3},\\frac{\\pi }{2}\\text{ }[\/latex]triangle. The sides have lengths in the relation[latex]\\text{ }s,\\sqrt{3}s,2s.\\text{ }[\/latex]The sides of a[latex]\\text{ }45\u00b0,45\u00b0,90\u00b0[\/latex]triangle, which can also be described as a[latex]\\text{ }\\frac{\\pi }{4},\\frac{\\pi }{4},\\frac{\\pi }{2}\\text{ }[\/latex]triangle, have lengths in the relation[latex]\\text{ }s,s,\\sqrt{2}s.\\text{ }[\/latex]These relations are shown in <a class=\"autogenerated-content\" href=\"#Figure_05_04_008\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_05_04_008\" class=\"wp-caption aligncenter\">\n<div class=\"wp-caption-text\">Side lengths of special triangles<\/div>\n<p><span id=\"fs-id1165137598316\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132547\/CNX_Precalc_Figure_05_04_008.jpg\" alt=\"Two side by side graphs of circles with inscribed angles. First circle has angle of pi\/3 inscribed. Second circle has angle of pi\/4 inscribed.\" \/><\/span><\/p>\n<\/div>\n<p id=\"fs-id1165135433018\">We can then use the ratios of the side lengths to evaluate trigonometric functions of special angles.<\/p>\n<div id=\"fs-id1165137592822\" class=\"precalculus howto examples\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165135242753\"><strong>Given trigonometric functions of a special angle, evaluate using side lengths.<\/strong><\/p>\n<ol id=\"fs-id1165137451842\" type=\"1\">\n<li>Use the side lengths shown in <a class=\"autogenerated-content\" href=\"#Figure_05_04_008\">(Figure)<\/a> for the special angle you wish to evaluate.<\/li>\n<li>Use the ratio of side lengths appropriate to the function you wish to evaluate.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_05_04_03\" class=\"textbox examples\">\n<div id=\"fs-id1165137762917\">\n<div id=\"fs-id1165137602526\">\n<h3>Example 3: Evaluating Trigonometric Functions of Special Angles Using Side Lengths<\/h3>\n<p id=\"fs-id1165137583573\">Find the exact value of the trigonometric functions of[latex]\\text{ }\\frac{\\pi }{3},[\/latex] using side lengths.<\/p>\n<\/div>\n<div id=\"fs-id1165137758344\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137758344\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137758344\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165137526289\" class=\"unnumbered\">[latex]\\begin{array}{l}\\mathrm{sin}\\left(\\frac{\\pi }{3}\\right)=\\frac{\\text{opp}}{\\text{hyp}}=\\frac{\\sqrt{3}s}{2s}=\\frac{\\sqrt{3}}{2}\\hfill \\\\ \\mathrm{cos}\\left(\\frac{\\pi }{3}\\right)=\\frac{\\text{adj}}{\\text{hyp}}=\\frac{s}{2s}=\\frac{1}{2}\\hfill \\\\ \\mathrm{tan}\\left(\\frac{\\pi }{3}\\right)=\\frac{\\text{opp}}{\\text{adj}}=\\frac{\\sqrt{3}s}{s}=\\sqrt{3}\\hfill \\\\ \\mathrm{sec}\\left(\\frac{\\pi }{3}\\right)=\\frac{\\text{hyp}}{\\text{adj}}=\\frac{2s}{s}=2\\hfill \\\\ \\mathrm{csc}\\left(\\frac{\\pi }{3}\\right)=\\frac{\\text{hyp}}{\\text{opp}}=\\frac{2s}{\\sqrt{3}s}=\\frac{2}{\\sqrt{3}}=\\frac{2\\sqrt{3}}{3}\\hfill \\\\ \\mathrm{cot}\\left(\\frac{\\pi }{3}\\right)=\\frac{\\text{adj}}{\\text{opp}}=\\frac{s}{\\sqrt{3}s}=\\frac{1}{\\sqrt{3}}=\\frac{\\sqrt{3}}{3}\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135190054\" class=\"precalculus tryit\">\n<h3>Try it #3<\/h3>\n<div id=\"ti_05_04_03\">\n<div id=\"fs-id1165135209398\">\n<p id=\"fs-id1165135209399\">Find the exact value of the trigonometric functions of[latex]\\text{ }\\frac{\\pi }{4},[\/latex] using side lengths.<\/p>\n<\/div>\n<div id=\"fs-id1165137619647\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137619647\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137619647\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137619648\">[latex]\\mathrm{sin}\\left(\\frac{\\pi }{4}\\right)=\\frac{\\sqrt{2}}{2},\\mathrm{cos}\\left(\\frac{\\pi }{4}\\right)=\\frac{\\sqrt{2}}{2},\\mathrm{tan}\\left(\\frac{\\pi }{4}\\right)=1,[\/latex]<\/p>\n<div><\/div>\n<\/div>\n<\/div>\n<p>[latex]\\mathrm{sec}\\left(\\frac{\\pi }{4}\\right)=\\sqrt{2},csc\\left(\\frac{\\pi }{4}\\right)=\\sqrt{2},\\mathrm{cot}\\left(\\frac{\\pi }{4}\\right)=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137409403\" class=\"bc-section section\">\n<h4>Using Equal Cofunction of Complements<\/h4>\n<p id=\"fs-id1165137565216\">If we look more closely at the relationship between the sine and cosine of the special angles relative to the unit circle, we will notice a pattern. In a right triangle with angles of[latex]\\text{ }\\frac{\\pi }{6}\\text{ }[\/latex]and[latex]\\text{ }\\frac{\\pi }{3},[\/latex] we see that the sine of[latex]\\text{ }\\frac{\\pi }{3},[\/latex] namely[latex]\\text{ }\\frac{\\sqrt{3}}{2},[\/latex] is also the cosine of[latex]\\text{ }\\frac{\\pi }{6},[\/latex] while the sine of[latex]\\text{ }\\frac{\\pi }{6},[\/latex] namely[latex]\\text{ }\\frac{1}{2},[\/latex] is also the cosine of[latex]\\text{ }\\frac{\\pi }{3}.[\/latex]<\/p>\n<div id=\"fs-id1165137847268\" class=\"unnumbered\">[latex]\\begin{array}{l}\\begin{array}{l}\\hfill \\\\ \\mathrm{sin}\\text{ }\\frac{\\pi }{3}=\\mathrm{cos}\\text{ }\\frac{\\pi }{6}=\\frac{\\sqrt{3}s}{2s}=\\frac{\\sqrt{3}}{2}\\hfill \\end{array}\\hfill \\\\ \\mathrm{sin}\\text{ }\\frac{\\pi }{6}=\\mathrm{cos}\\text{ }\\frac{\\pi }{3}=\\frac{s}{2s}=\\frac{1}{2}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137594972\">See <a class=\"autogenerated-content\" href=\"#Figure_05_04_009\">(Figure)<\/a><\/p>\n<div id=\"Figure_05_04_009\" class=\"small\">\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\/3896\/2019\/03\/07132550\/CNX_Precalc_Figure_05_04_009.jpg\" alt=\"A graph of circle with angle pi\/3 inscribed.\" width=\"487\" height=\"371\" \/><\/p>\n<p class=\"wp-caption-text\">The sine of[latex]\\text{ }\\frac{\\pi }{3}\\text{ }[\/latex]equals the cosine of[latex]\\text{ }\\frac{\\pi }{6}\\text{ }[\/latex]and vice versa.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137612239\">This result should not be surprising because, as we see from <a class=\"autogenerated-content\" href=\"#Figure_05_04_009\">(Figure)<\/a>, the side opposite the angle of[latex]\\text{ }\\frac{\\pi }{3}\\text{ }[\/latex]is also the side adjacent to[latex]\\text{ }\\frac{\\pi }{6},[\/latex] so[latex]\\text{ }\\mathrm{sin}\\left(\\frac{\\pi }{3}\\right)\\text{ }[\/latex]and[latex]\\text{ }\\mathrm{cos}\\left(\\frac{\\pi }{6}\\right)\\text{ }[\/latex]are exactly the same ratio of the same two sides,[latex]\\text{ }\\sqrt{3}s\\text{ }[\/latex]and[latex]\\text{ }2s.\\text{ }[\/latex]Similarly,[latex]\\text{ }\\mathrm{cos}\\left(\\frac{\\pi }{3}\\right)\\text{ }[\/latex]and[latex]\\text{ }\\mathrm{sin}\\left(\\frac{\\pi }{6}\\right)\\text{ }[\/latex]are also the same ratio using the same two sides,[latex]\\text{ }s\\text{ }[\/latex]and[latex]\\text{ }2s.[\/latex]<\/p>\n<p id=\"fs-id1165137732326\">The interrelationship between the sines and cosines of[latex]\\text{ }\\frac{\\pi }{6}\\text{ }[\/latex]and[latex]\\text{ }\\frac{\\pi }{3}\\text{ }[\/latex]also holds for the two acute angles in any right triangle, since in every case, the ratio of the same two sides would constitute the sine of one angle and the cosine of the other. Since the three angles of a triangle add to[latex]\\text{ }\\pi ,[\/latex] and the right angle is[latex]\\text{ }\\frac{\\pi }{2},[\/latex] the remaining two angles must also add up to[latex]\\text{ }\\frac{\\pi }{2}.\\text{ }[\/latex]That means that a right triangle can be formed with any two angles that add to[latex]\\text{ }\\frac{\\pi }{2}[\/latex]\u2014in other words, any two complementary angles. So we may state a <em>cofunction identity<\/em>: If any two angles are complementary, the sine of one is the cosine of the other, and vice versa. This identity is illustrated in <a class=\"autogenerated-content\" href=\"#Figure_05_04_010\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_05_04_010\" class=\"small\">\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\/3896\/2019\/03\/07132553\/CNX_Precalc_Figure_05_04_010.jpg\" alt=\"Right triangle with angles alpha and beta. Equivalence between sin alpha and cos beta. Equivalence between sin beta and cos alpha.\" width=\"487\" height=\"304\" \/><\/p>\n<p class=\"wp-caption-text\">Cofunction identity of sine and cosine of complementary angles<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135181553\">Using this identity, we can state without calculating, for instance, that the sine of[latex]\\text{ }\\frac{\\pi }{12}\\text{ }[\/latex]equals the cosine of[latex]\\text{ }\\frac{5\\pi }{12},[\/latex] and that the sine of[latex]\\text{ }\\frac{5\\pi }{12}\\text{ }[\/latex]equals the cosine of[latex]\\text{ }\\frac{\\pi }{12}.\\text{ }[\/latex]We can also state that if, for a certain angle[latex]\\text{ }t,[\/latex] [latex]\\mathrm{cos}\\text{ }t=\\frac{5}{13},[\/latex] then[latex]\\text{ }\\mathrm{sin}\\left(\\frac{\\pi }{2}-t\\right)=\\frac{5}{13}\\text{ }[\/latex]as well.<\/p>\n<div id=\"fs-id1165137738292\">\n<h3>Cofunction Identities<\/h3>\n<p id=\"fs-id1165135178097\">The <span class=\"no-emphasis\">cofunction identities<\/span> in radians are listed in <a class=\"autogenerated-content\" href=\"#Table_05_04_01\">(Figure)<\/a>.<\/p>\n<table id=\"Table_05_04_01\" summary=\"..\">\n<caption><strong>Table 1<\/strong><\/caption>\n<tbody>\n<tr>\n<td class=\"border\">[latex]\\mathrm{cos}\\text{ }t=\\mathrm{sin}\\left(\\frac{\\pi }{2}-t\\right)[\/latex]<\/td>\n<td class=\"border\">[latex]\\mathrm{sin}\\text{ }t=\\mathrm{cos}\\left(\\frac{\\pi }{2}-t\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\">[latex]\\mathrm{tan}\\text{ }t=\\mathrm{cot}\\left(\\frac{\\pi }{2}-t\\right)[\/latex]<\/td>\n<td class=\"border\">[latex]\\mathrm{cot}\\text{ }t=\\mathrm{tan}\\left(\\frac{\\pi }{2}-t\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\">[latex]\\mathrm{sec}\\text{ }t=\\mathrm{csc}\\left(\\frac{\\pi }{2}-t\\right)[\/latex]<\/td>\n<td class=\"border\">[latex]\\mathrm{csc}\\text{ }t=\\mathrm{sec}\\left(\\frac{\\pi }{2}-t\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165137735167\" class=\"precalculus howto examples\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137450768\"><strong>Given the sine and cosine of an angle, find the sine or cosine of its complement.<\/strong><\/p>\n<ol id=\"fs-id1165135333178\" type=\"1\">\n<li>To find the sine of the complementary angle, find the cosine of the original angle.<\/li>\n<li>To find the cosine of the complementary angle, find the sine of the original angle.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_05_04_04\" class=\"textbox examples\">\n<div id=\"fs-id1165137811906\">\n<div id=\"fs-id1165133103968\">\n<h3>Example 4: Using Cofunction Identities<\/h3>\n<p id=\"fs-id1165134148512\">If[latex]\\text{ }\\mathrm{sin}\\text{ }t=\\frac{5}{12},[\/latex]find[latex]\\text{ }\\left(\\mathrm{cos}\\frac{\\pi }{2}-t\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137668960\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137668960\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137668960\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137410039\">According to the cofunction identities for sine and cosine,<\/p>\n<div id=\"fs-id1165137786693\" class=\"unnumbered\">[latex]\\mathrm{sin}\\text{ }t=\\mathrm{cos}\\left(\\frac{\\pi }{2}-t\\right).[\/latex]<\/div>\n<p id=\"fs-id1165135361352\">So<\/p>\n<div id=\"fs-id1165135596476\" class=\"unnumbered\">[latex]\\mathrm{cos}\\left(\\frac{\\pi }{2}-t\\right)=\\frac{5}{12}.[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137530693\" class=\"precalculus tryit\">\n<h3>Try it #4<\/h3>\n<div id=\"ti_05_04_04\">\n<div id=\"fs-id1165134192991\">\n<p id=\"fs-id1165134192992\">If[latex]\\text{ }\\mathrm{csc}\\left(\\frac{\\pi }{6}\\right)=2,[\/latex] find[latex]\\text{ }\\mathrm{sec}\\left(\\frac{\\pi }{3}\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137387550\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137387550\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137387550\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137387551\">2<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137639938\" class=\"bc-section section\">\n<h4>Using Trigonometric Functions<\/h4>\n<p id=\"fs-id1165137629080\">In previous examples, we evaluated the sine and cosine in triangles where we knew all three sides. But the real power of right-triangle trigonometry emerges when we look at triangles in which we know an angle but do not know all the sides.<\/p>\n<div id=\"fs-id1165137456630\" class=\"precalculus howto examples\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137475023\"><strong>Given a right triangle, the length of one side, and the measure of one acute angle, find the remaining sides.<\/strong><\/p>\n<ol id=\"fs-id1165137564333\" type=\"1\">\n<li>For each side, select the trigonometric function that has the unknown side as either the numerator or the denominator. The known side will in turn be the denominator or the numerator.<\/li>\n<li>Write an equation setting the function value of the known angle equal to the ratio of the corresponding sides.<\/li>\n<li>Using the value of the trigonometric function and the known side length, solve for the missing side length.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_05_04_05\" class=\"textbox examples\">\n<div id=\"fs-id1165135192870\">\n<div id=\"fs-id1165135353061\">\n<h3>Example 5: Finding Missing Side Lengths Using Trigonometric Ratios<\/h3>\n<p id=\"fs-id1165137417073\">Find the unknown sides of the triangle in <a class=\"autogenerated-content\" href=\"#Figure_05_04_011\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_05_04_011\" class=\"small\"><span id=\"fs-id1165133189417\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132556\/CNX_Precalc_Figure_05_04_011.jpg\" alt=\"A right triangle with sides a, c, and 7. Angle of 30 degrees is also labeled.\" \/><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165135503936\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135503936\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135503936\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137611783\">We know the angle and the opposite side, so we can use the tangent to find the adjacent side.<\/p>\n<div id=\"fs-id1165135432983\" class=\"unnumbered\">[latex]\\mathrm{tan}\\left(30\u00b0\\right)=\\frac{7}{a}[\/latex]<\/div>\n<p id=\"fs-id1165137574920\">We rearrange to solve for[latex]\\text{ }a.[\/latex]<\/p>\n<div id=\"fs-id1165134054044\" class=\"unnumbered\">[latex]\\begin{array}{l}a=\\frac{7}{\\mathrm{tan}\\left(30\u00b0\\right)}\\hfill \\\\ \\text{ }\\text{ }\\text{ }\\approx 12.1\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137460411\">We can use the sine to find the hypotenuse.<\/p>\n<div id=\"fs-id1165135386417\" class=\"unnumbered\">[latex]\\mathrm{sin}\\left(30\u00b0\\right)=\\frac{7}{c}[\/latex]<\/div>\n<p id=\"fs-id1165137665408\">Again, we rearrange to solve for[latex]\\text{ }c.[\/latex]<\/p>\n<div id=\"fs-id1165135186816\" class=\"unnumbered\">[latex]\\begin{array}{l}c=\\frac{7}{\\mathrm{sin}\\left(30\u00b0\\right)}\\hfill \\\\ \\text{ }\\text{ }\\text{ }\\approx 14\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137851227\" class=\"precalculus tryit\">\n<h3>Try it #5<\/h3>\n<div id=\"ti_05_04_05\">\n<div id=\"fs-id1165137727334\">\n<p id=\"fs-id1165137742301\">A right triangle has one angle of[latex]\\text{ }\\frac{\\pi }{3}\\text{ }[\/latex] and a hypotenuse of 20. Find the unknown sides and angle of the triangle.<\/p>\n<\/div>\n<div id=\"fs-id1165137542469\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137542469\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137542469\" class=\"hidden-answer\" style=\"display: none\">[latex]\\text{adjacent}=10;\\text{ }[\/latex][latex]\\text{opposite}=10\\sqrt{3}\\text{ }[\/latex]; missing angle is[latex]\\text{ }\\frac{\\pi }{6}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137658230\" class=\"bc-section section\">\n<h4>Using Right Triangle Trigonometry to Solve Applied Problems<\/h4>\n<p id=\"fs-id1165137437126\">Right-triangle trigonometry has many practical applications. For example, the ability to compute the lengths of sides of a triangle makes it possible to find the height of a tall object without climbing to the top or having to extend a tape measure along its height. We do so by measuring a distance from the base of the object to a point on the ground some distance away, where we can look up to the top of the tall object at an angle. The angle of elevation of an object above an observer relative to the observer is the angle between the horizontal and the line from the object to the observer&#8217;s eye. The right triangle this position creates has sides that represent the unknown height, the measured distance from the base, and the angled line of sight from the ground to the top of the object. Knowing the measured distance to the base of the object and the angle of the line of sight, we can use trigonometric functions to calculate the unknown height. Similarly, we can form a triangle from the top of a tall object by looking downward. The angle of depression of an object below an observer relative to the observer is the angle between the horizontal and the line from the object to the observer&#8217;s eye. See <a class=\"autogenerated-content\" href=\"#Figure_05_04_013\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_05_04_013\" class=\"small\"><span id=\"fs-id1165137730420\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132559\/CNX_Precalc_Figure_05_04_013.jpg\" alt=\"Diagram of a radio tower with line segments extending from the top and base of the tower to a point on the ground some distance away. The two lines and the tower form a right triangle. The angle near the top of the tower is the angle of depression. The angle on the ground at a distance from the tower is the angle of elevation.\" \/><\/span><\/div>\n<div id=\"fs-id1165137473849\" class=\"precalculus howto examples\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137723583\"><strong>Given a tall object, measure its height indirectly.<\/strong><\/p>\n<ol id=\"fs-id1165135195794\" type=\"1\">\n<li>Make a sketch of the problem situation to keep track of known and unknown information.<\/li>\n<li>Lay out a measured distance from the base of the object to a point where the top of the object is clearly visible.<\/li>\n<li>At the other end of the measured distance, look up to the top of the object. Measure the angle the line of sight makes with the horizontal.<\/li>\n<li>Write an equation relating the unknown height, the measured distance, and the tangent of the angle of the line of sight.<\/li>\n<li>Solve the equation for the unknown height.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_05_04_06\" class=\"textbox examples\">\n<div id=\"fs-id1165135176667\">\n<div id=\"fs-id1165137553918\">\n<h3>Example 6: Measuring a Distance Indirectly<\/h3>\n<p id=\"fs-id1165137736590\">To find the height of a tree, a person walks to a point 30 feet from the base of the tree. She measures an angle of [latex]57\u00b0\\text{ }[\/latex]between a line of sight to the top of the tree and the ground, as shown in <a class=\"autogenerated-content\" href=\"#Figure_05_04_012\">(Figure)<\/a>. Find the height of the tree.<\/p>\n<div id=\"Figure_05_04_012\" class=\"small\"><span id=\"fs-id1165137804033\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132602\/CNX_Precalc_Figure_05_04_012.jpg\" alt=\"A tree with angle of 57 degrees from vantage point. Vantage point is 30 feet from tree.\" \/><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165134278695\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165134278695\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165134278695\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135210031\">We know that the angle of elevation is[latex]\\text{ }57\u00b0\\text{ }[\/latex]and the adjacent side is 30 ft long. The opposite side is the unknown height.<\/p>\n<p id=\"fs-id1165134047639\">The trigonometric function relating the side opposite to an angle and the side adjacent to the angle is the tangent. So we will state our information in terms of the tangent of [latex]57\u00b0,[\/latex] letting[latex]\\text{ }h\\text{ }[\/latex]be the unknown height.<\/p>\n<div id=\"fs-id1165134089530\" class=\"unnumbered\">[latex]\\begin{array}{ll}\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\mathrm{tan}\\text{ }\\theta =\\frac{\\text{opposite}}{\\text{adjacent}}\\hfill & \\hfill \\\\ \\text{tan}\\left(57\u00b0\\right)=\\frac{h}{30}\\hfill & \\text{Solve for }h.\\hfill \\\\ \\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }h=30\\mathrm{tan}\\left(57\u00b0\\right)\\hfill & \\text{Multiply}.\\hfill \\\\ \\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }h\\approx 46.2\\hfill & \\text{Use a calculator}.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137550726\">The tree is approximately 46 feet tall.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137533956\" class=\"precalculus tryit\">\n<h3>Try it #6<\/h3>\n<div id=\"ti_05_04_06\">\n<div id=\"fs-id1165137855261\">\n<p id=\"fs-id1165137855262\">How long a ladder is needed to reach a windowsill 50 feet above the ground if the ladder rests against the building making an angle of[latex]\\text{ }\\frac{5\\pi }{12}\\text{ }[\/latex]with the ground? Round to the nearest foot.<\/p>\n<\/div>\n<div id=\"fs-id1165137727529\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137727529\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137727529\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135403387\">About 52 ft<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137762260\" class=\"precalculus media\">\n<p id=\"fs-id1165137442106\">Access these online resources for additional instruction and practice with right triangle trigonometry.<\/p>\n<ul id=\"fs-id1165137656591\">\n<li><a href=\"http:\/\/openstax.org\/l\/findtrigcal\">Finding Trig Functions on Calculator<\/a><\/li>\n<li><a href=\"http:\/\/openstax.org\/l\/trigrttri\">Finding Trig Functions Using a Right Triangle<\/a><\/li>\n<li><a href=\"http:\/\/openstax.org\/l\/reltrigtri\">Relate Trig Functions to Sides of a Right Triangle<\/a><\/li>\n<li><a href=\"http:\/\/openstax.org\/l\/sixtrigfunc\">Determine Six Trig Functions from a Triangle<\/a><\/li>\n<li><a href=\"http:\/\/openstax.org\/l\/rttriside\">Determine Length of Right Triangle Side<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<p id=\"eip-845\">Visit <a href=\"http:\/\/openstax.org\/l\/PreCalcLPC05\">this website<\/a> for additional practice questions from Learningpod.<\/p>\n<\/div>\n<div id=\"fs-id1165135186669\" class=\"key-equations\">\n<h3>Key Equations<\/h3>\n<table id=\"eip-id1165137409421\" summary=\"..\">\n<tbody>\n<tr>\n<td class=\"border\">Cofunction Identities<\/td>\n<td class=\"border\">[latex]\\begin{array}{l}\\begin{array}{l}\\\\ \\mathrm{cos}\\text{ }t=\\mathrm{sin}\\left(\\frac{\\pi }{2}-t\\right)\\end{array}\\hfill \\\\ \\mathrm{sin}\\text{ }t=\\mathrm{cos}\\left(\\frac{\\pi }{2}-t\\right)\\hfill \\\\ \\mathrm{tan}\\text{ }t=\\mathrm{cot}\\left(\\frac{\\pi }{2}-t\\right)\\hfill \\\\ \\mathrm{cot}\\text{ }t=\\mathrm{tan}\\left(\\frac{\\pi }{2}-t\\right)\\hfill \\\\ \\mathrm{sec}\\text{ }t=\\mathrm{csc}\\left(\\frac{\\pi }{2}-t\\right)\\hfill \\\\ \\mathrm{csc}\\text{ }t=\\mathrm{sec}\\left(\\frac{\\pi }{2}-t\\right)\\hfill \\end{array}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165137481899\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1165137415357\">\n<li>We can define trigonometric functions as ratios of the side lengths of a right triangle. See <a class=\"autogenerated-content\" href=\"#Example_05_04_01\">(Figure)<\/a>.<\/li>\n<li>The same side lengths can be used to evaluate the trigonometric functions of either acute angle in a right triangle. See <a class=\"autogenerated-content\" href=\"#Example_05_04_02\">(Figure)<\/a>.<\/li>\n<li>We can evaluate the trigonometric functions of special angles, knowing the side lengths of the triangles in which they occur. See <a class=\"autogenerated-content\" href=\"#Example_05_04_03\">(Figure)<\/a>.<\/li>\n<li>Any two complementary angles could be the two acute angles of a right triangle.<\/li>\n<li>If two angles are complementary, the cofunction identities state that the sine of one equals the cosine of the other and vice versa. See <a class=\"autogenerated-content\" href=\"#Example_05_04_04\">(Figure)<\/a>.<\/li>\n<li>We can use trigonometric functions of an angle to find unknown side lengths.<\/li>\n<li>Select the trigonometric function representing the ratio of the unknown side to the known side. See <a class=\"autogenerated-content\" href=\"#Example_05_04_05\">(Figure)<\/a>.<\/li>\n<li>Right-triangle trigonometry permits the measurement of inaccessible heights and distances.<\/li>\n<li>The unknown height or distance can be found by creating a right triangle in which the unknown height or distance is one of the sides, and another side and angle are known. See <a class=\"autogenerated-content\" href=\"#Example_05_04_06\">(Figure)<\/a>.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165135458650\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id1165137736563\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id1165137460283\">\n<div id=\"fs-id1165137460284\">\n<p id=\"fs-id1165133112791\">For the given right triangle, label the adjacent side, opposite side, and hypotenuse for the indicated angle.<\/p>\n<div><\/div>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132604\/CNX_Precalc_Figure_05_04_201.jpg\" alt=\"A right triangle.\" \/><\/p>\n<\/div>\n<div id=\"fs-id1165137803574\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137803574\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137803574\" class=\"hidden-answer\" style=\"display: none\"><span id=\"fs-id1165137401049\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132606\/CNX_Precalc_Figure_05_04_202.jpg\" alt=\"A right triangle with side opposite, adjacent, and hypotenuse labeled.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137472665\">\n<div id=\"fs-id1165137452030\">\n<p id=\"fs-id1165137452031\">When a right triangle with a hypotenuse of 1 is placed in the unit circle, which sides of the triangle correspond to the <em>x<\/em>&#8211; and <em>y<\/em>-coordinates?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137404958\">\n<div id=\"fs-id1165137404959\">\n<p id=\"fs-id1165137698057\">The tangent of an angle compares which sides of the right triangle?<\/p>\n<\/div>\n<div id=\"fs-id1165135315582\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135315582\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135315582\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137400574\">The tangent of an angle is the ratio of the opposite side to the adjacent side.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137462272\">\n<div id=\"fs-id1165135536523\">\n<p id=\"fs-id1165135536524\">What is the relationship between the two acute angles in a right triangle?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137854900\">\n<div id=\"fs-id1165137854901\">\n<p id=\"fs-id1165137725210\">Explain the cofunction identity.<\/p>\n<\/div>\n<div id=\"fs-id1165135209966\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135209966\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135209966\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137675894\">For example, the sine of an angle is equal to the cosine of its complement; the cosine of an angle is equal to the sine of its complement.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135406939\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\n<p id=\"fs-id1165135501987\">For the following exercises, use cofunctions of complementary angles.<\/p>\n<div id=\"fs-id1165137761553\">\n<div id=\"fs-id1165137459885\">\n<p id=\"fs-id1165137459886\">[latex]\\mathrm{cos}\\left(\\text{34\u00b0}\\right)=\\mathrm{sin}\\left(\\text{__\u00b0}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137935703\">\n<div id=\"fs-id1165137935704\">\n<p id=\"fs-id1165137755665\">[latex]\\mathrm{cos}\\left(\\frac{\\pi }{3}\\right)=\\mathrm{sin}\\text{(___)}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137805390\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137805390\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137805390\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165131974796\">[latex]\\frac{\\pi }{6}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137911234\">\n<div id=\"fs-id1165135465250\">\n<p id=\"fs-id1165135465251\">[latex]\\mathrm{csc}\\left(\\text{21\u00b0}\\right)=\\mathrm{sec}\\left(\\text{___\u00b0}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137768200\">\n<div id=\"fs-id1165137768201\">\n<p id=\"fs-id1165134276136\">[latex]\\mathrm{tan}\\left(\\frac{\\pi }{4}\\right)=\\mathrm{cot}\\left(\\text{__}\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137470154\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137470154\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137470154\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137470155\">[latex]\\frac{\\pi }{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137455693\">For the following exercises, find the lengths of the missing sides if side[latex]\\text{ }a\\text{ }[\/latex]is opposite angle[latex]\\text{ }A,[\/latex] side[latex]\\text{ }b\\text{ }[\/latex] is opposite angle[latex]\\text{ }B,[\/latex] and side[latex]\\text{ }c\\text{ }[\/latex]is the hypotenuse.<\/p>\n<div id=\"fs-id1165137549317\">\n<div id=\"fs-id1165137549318\">\n<p id=\"fs-id1165135393375\">[latex]\\mathrm{cos}\\text{ }B=\\frac{4}{5},a=10[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135453101\">\n<div id=\"fs-id1165135453102\">\n<p id=\"fs-id1165135453103\">[latex]\\mathrm{sin}\\text{ }B=\\frac{1}{2}, a=20[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137832156\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137832156\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137832156\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135187865\">[latex]b=\\frac{20\\sqrt{3}}{3},c=\\frac{40\\sqrt{3}}{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137549742\">\n<div id=\"fs-id1165137549743\">\n<p id=\"fs-id1165137549744\">[latex]\\mathrm{tan}\\text{ }A=\\frac{5}{12},b=6[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137766757\">\n<div id=\"fs-id1165137894557\">\n<p id=\"fs-id1165137894558\">[latex]\\mathrm{tan}\\text{ }A=100,b=100[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137657401\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137657401\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137657401\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137657402\">[latex]a=10,000,c=10,000.5[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137409084\">\n<div id=\"fs-id1165137409085\">\n<p id=\"fs-id1165135196985\">[latex]\\mathrm{sin}\\text{ }B=\\frac{1}{\\sqrt{3}}, a=2[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135407091\">\n<div id=\"fs-id1165135407092\">\n<p id=\"fs-id1165135407093\">[latex]a=5,\\text{ }\\measuredangle \\text{ }A={60}^{\\circ }[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137803393\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137803393\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137803393\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137803394\">[latex]b=\\frac{5\\sqrt{3}}{3},c=\\frac{10\\sqrt{3}}{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135186918\">\n<div id=\"fs-id1165135186919\">\n<p id=\"fs-id1165137447359\">[latex]c=12,\\text{ }\\measuredangle \\text{ }A={45}^{\\circ }[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137450999\" class=\"bc-section section\">\n<h4>Graphical<\/h4>\n<p id=\"fs-id1165135191519\">For the following exercises, use <a class=\"autogenerated-content\" href=\"#Figure_05_04_203\">(Figure)<\/a> to evaluate each trigonometric function of angle[latex]\\text{ }A.[\/latex]<\/p>\n<div id=\"Figure_05_04_203\" class=\"wp-caption aligncenter\"><span id=\"fs-id1165132960807\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132609\/CNX_Precalc_Figure_05_04_203.jpg\" alt=\"A right triangle with sides 4 and 10 and angle of A labeled.\" \/><\/span><\/div>\n<div id=\"fs-id1165137397852\">\n<div id=\"fs-id1165135500824\">\n<p id=\"fs-id1165135500826\">[latex]\\mathrm{sin}\\text{ }A[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135434893\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135434893\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135434893\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137639470\">[latex]\\frac{5\\sqrt{29}}{29}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137834908\">\n<div id=\"fs-id1165137834909\">\n<p id=\"fs-id1165135485865\">[latex]\\mathrm{cos}\\text{ }A[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135567456\">\n<div id=\"fs-id1165135567457\">\n<p id=\"fs-id1165137531079\">[latex]\\mathrm{tan}\\text{ }A[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137732079\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137732079\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137732079\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137732080\">[latex]\\frac{5}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137647703\">\n<div id=\"fs-id1165137647704\">\n<p id=\"fs-id1165135241318\">[latex]\\mathrm{csc}\\text{ }A[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137597798\">\n<div id=\"fs-id1165137597799\">\n<p id=\"fs-id1165137803588\">[latex]\\mathrm{sec}\\text{ }A[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137693464\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137693464\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137693464\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137693465\">[latex]\\frac{\\sqrt{29}}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137920736\">\n<div id=\"fs-id1165137920737\">\n<p id=\"fs-id1165135241382\">[latex]\\mathrm{cot}\\text{ }A[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137452925\">For the following exercises, use <a class=\"autogenerated-content\" href=\"#Figure_05_04_204\">(Figure)<\/a> to evaluate each trigonometric function of angle[latex]\\text{ }A.[\/latex]<\/p>\n<div id=\"Figure_05_04_204\" class=\"wp-caption aligncenter\"><span id=\"fs-id1165135160637\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132612\/CNX_Precalc_Figure_05_04_204.jpg\" alt=\"A right triangle with sides of 10 and 8 and angle of A labeled.\" \/><\/span><\/div>\n<div id=\"fs-id1165137749319\">\n<div id=\"fs-id1165135613637\">\n<p id=\"fs-id1165135613638\">[latex]\\mathrm{sin}\\text{ }A[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137683057\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137683057\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137683057\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137419300\">[latex]\\frac{5\\sqrt{41}}{41}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137601187\">\n<div id=\"fs-id1165137601188\">\n<p id=\"fs-id1165135173373\">[latex]\\mathrm{cos}\\text{ }A[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137462646\">\n<div id=\"fs-id1165137462647\">\n<p id=\"fs-id1165137847132\">[latex]\\mathrm{tan}\\text{ }A[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135190741\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135190741\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135190741\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135190742\">[latex]\\frac{5}{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137415361\">\n<div id=\"fs-id1165137415362\">\n<p id=\"fs-id1165137415363\">[latex]\\mathrm{csc}\\text{ }A[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137596480\">\n<div id=\"fs-id1165137933799\">\n<p id=\"fs-id1165137933800\">[latex]\\mathrm{sec}\\text{ }A[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137779088\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137779088\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137779088\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137779089\">[latex]\\frac{\\sqrt{41}}{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137810135\">\n<div id=\"fs-id1165137810136\">\n<p id=\"fs-id1165137810138\">[latex]\\mathrm{cot}\\text{ }A[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135168306\">For the following exercises, solve for the unknown sides of the given triangle.<\/p>\n<div id=\"fs-id1165135168309\">\n<div id=\"fs-id1165137805206\"><span id=\"fs-id1165135613627\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132615\/CNX_Precalc_Figure_05_04_205.jpg\" alt=\"A right triangle with sides of 7, b, and c labeled. Angles of B and 30 degrees also labeled.\" \/><\/span><\/div>\n<div id=\"fs-id1165137832242\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137832242\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137832242\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137832243\">[latex]c=14, b=7\\sqrt{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134094614\">\n<div id=\"fs-id1165134094615\"><span id=\"fs-id1165135550447\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132618\/CNX_Precalc_Figure_05_04_206.jpg\" alt=\"A right triangle with sides of 10, a, and c. Angles of 60 degrees and A also labeled.\" \/><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165134085640\">\n<div id=\"fs-id1165134085642\"><span id=\"fs-id1165137766966\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132621\/CNX_Precalc_Figure_05_04_207.jpg\" alt=\"A right triangle with corners labeled A, B, and C. Hypotenuse has length of 15 times square root of 2. Angle B is 45 degrees.\" \/><\/span><\/div>\n<div>\n<p id=\"fs-id1165135645975\">[latex]a=15, b=15[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137736877\" class=\"bc-section section\">\n<h4>Technology<\/h4>\n<p id=\"fs-id1165137476432\">For the following exercises, use a calculator to find the length of each side to four decimal places.<\/p>\n<div id=\"fs-id1165137874745\">\n<div id=\"fs-id1165137874746\"><span id=\"fs-id1165137530086\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132623\/CNX_Precalc_Figure_05_04_208.jpg\" alt=\"A right triangle with sides of 10, a, and c. Angles of A and 62 degrees are also labeled.\" \/><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165137401776\">\n<div id=\"fs-id1165137401777\"><span id=\"fs-id1165137730500\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132626\/CNX_Precalc_Figure_05_04_209.jpg\" alt=\"A right triangle with sides of 7, b, and c. Angles of 35 degrees and B are also labeled.\" \/><\/span><\/div>\n<div id=\"fs-id1165134298850\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165134298850\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165134298850\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165134298851\">[latex]b=9.9970, c=12.2041[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135193172\">\n<div id=\"fs-id1165135193173\"><span id=\"fs-id1165137401263\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132629\/CNX_Precalc_Figure_05_04_210.jpg\" alt=\"A right triangle with sides of a, b, and 10 labeled. Angles of 65 degrees and B are also labeled.\" \/><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165135159938\">\n<div id=\"fs-id1165137667450\"><span id=\"fs-id1165135369236\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132632\/CNX_Precalc_Figure_05_04_211.jpg\" alt=\"A right triangle with sides a, b, and 12. Angles of 10 degrees and B are also labeled.\" \/><\/span><\/div>\n<div id=\"fs-id1165137460061\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137460061\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137460061\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137460062\">[latex]a=2.0838, b=11.8177[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137922573\">\n<div id=\"fs-id1165137922574\"><span id=\"fs-id1165135190611\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132634\/CNX_Precalc_Figure_05_04_212.jpg\" alt=\"A right triangle with corners labeled A, B, and C. Sides labeled b, c, and 16.5. Angle of 81 degrees also labeled.\" \/><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165134357798\">\n<div id=\"fs-id1165134357799\">\n<p id=\"fs-id1165137573269\">[latex]b=15,\\text{ }\\measuredangle B={15}^{\\circ }[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137705539\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137705539\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137705539\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137705540\">[latex]a=55.9808,c=57.9555[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135475897\">\n<div id=\"fs-id1165135475898\">\n<p id=\"fs-id1165135475899\">[latex]c=200,\\text{ }\\measuredangle B={5}^{\\circ }[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137810074\">\n<div id=\"fs-id1165137810075\">\n<p id=\"fs-id1165137810076\">[latex]c=50,\\text{ }\\measuredangle B={21}^{\\circ }[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137767696\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137767696\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137767696\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137767697\">[latex]a=46.6790,b=17.9184[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135260744\">\n<div id=\"fs-id1165137806834\">\n<p id=\"fs-id1165137806835\">[latex]a=30,\\text{ }\\measuredangle A={27}^{\\circ }[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135331740\">\n<div id=\"fs-id1165135496307\">\n<p id=\"fs-id1165135496308\">[latex]b=3.5,\\text{ }\\measuredangle A={78}^{\\circ }[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135390953\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135390953\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135390953\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135307915\">[latex]a=16.4662,c=16.8341[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137603593\" class=\"bc-section section\">\n<h4>Extensions<\/h4>\n<div id=\"fs-id1165137427200\">\n<div id=\"fs-id1165137404949\">\n<p id=\"fs-id1165137404950\">Find[latex]\\text{ }x.[\/latex]<\/p>\n<p><span id=\"fs-id1165137738251\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132637\/CNX_Precalc_Figure_05_04_213.jpg\" alt=\"A triangle with angles of 63 degrees and 39 degrees and side x. Bisector in triangle with length of 82.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135173378\">\n<div id=\"fs-id1165135173379\">\n<p id=\"fs-id1165135173380\">Find[latex]\\text{ }x.[\/latex]<\/p>\n<p><span id=\"fs-id1165137768593\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132640\/CNX_Precalc_Figure_05_04_214.jpg\" alt=\"A triangle with angles of 36 degrees and 50 degrees and side x. Bisector in triangle with length of 85.\" \/><\/span><\/p>\n<\/div>\n<div id=\"fs-id1165137679296\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137679296\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137679296\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137679297\">188.3159<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135176637\">\n<div id=\"fs-id1165135176638\">\n<p id=\"fs-id1165135176639\">Find[latex]\\text{ }x.[\/latex]<\/p>\n<p><span id=\"fs-id1165137771945\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132643\/CNX_Precalc_Figure_05_04_215.jpg\" alt=\"A right triangle with side of 115 and angle of 35 degrees. Within right triangle there is another right triangle with angle of 56 degrees. Side length difference between two triangles is x.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137432679\">\n<div id=\"fs-id1165137432680\">\n<p id=\"fs-id1165137432681\">Find[latex]\\text{ }x.[\/latex]<\/p>\n<p><span id=\"fs-id1165137469034\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132646\/CNX_Precalc_Figure_05_04_216.jpg\" alt=\"A right triangle with side of 119 and angle of 26 degrees. Within right triangle there is another right triangle with angle of 70 degrees instead of 26 degrees. Difference in side length between two triangles is x.\" \/><\/span><\/p>\n<\/div>\n<div id=\"fs-id1165135160172\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135160172\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135160172\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137435323\">200.6737<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137435326\">\n<div id=\"fs-id1165137444512\">\n<p id=\"fs-id1165137444513\">A radio tower is located 400 feet from a building. From a window in the building, a person determines that the angle of elevation to the top of the tower is[latex]\\text{ }36\u00b0,[\/latex] and that the angle of depression to the bottom of the tower is[latex]\\text{ }23\u00b0.\\text{ }[\/latex]How tall is the tower?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137433654\">\n<div id=\"fs-id1165135353109\">\n<p id=\"fs-id1165135353110\">A radio tower is located 325 feet from a building. From a window in the building, a person determines that the angle of elevation to the top of the tower is[latex]\\text{ }43\u00b0,[\/latex] and that the angle of depression to the bottom of the tower is[latex]\\text{ }31\u00b0.\\text{ }[\/latex]How tall is the tower?<\/p>\n<\/div>\n<div id=\"fs-id1165135512511\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135512511\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135512511\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137677809\">498.3471 ft<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137677812\">\n<div id=\"fs-id1165135368447\">\n<p id=\"fs-id1165135368448\">A 200-foot tall monument is located in the distance. From a window in a building, a person determines that the angle of elevation to the top of the monument is[latex]\\text{ }15\u00b0,[\/latex] and that the angle of depression to the bottom of the tower is[latex]\\text{ }2\u00b0.\\text{ }[\/latex]How far is the person from the monument?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137648500\">\n<div id=\"fs-id1165137694171\">\n<p id=\"fs-id1165137694172\">A 400-foot tall monument is located in the distance. From a window in a building, a person determines that the angle of elevation to the top of the monument is[latex]\\text{ }18\u00b0,[\/latex] and that the angle of depression to the bottom of the monument is[latex]\\text{ }3\u00b0.\\text{ }[\/latex]How far is the person from the monument?<\/p>\n<\/div>\n<div id=\"fs-id1165137645701\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137645701\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137645701\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135181440\">1060.09 ft<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135181443\">\n<div id=\"fs-id1165137414694\">\n<p id=\"fs-id1165137414695\">There is an antenna on the top of a building. From a location 300 feet from the base of the building, the angle of elevation to the top of the building is measured to be[latex]\\text{ }40\u00b0.\\text{ }[\/latex]From the same location, the angle of elevation to the top of the antenna is measured to be[latex]\\text{ }43\u00b0.\\text{ }[\/latex]Find the height of the antenna.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137436702\">\n<div id=\"fs-id1165137849558\">\n<p id=\"fs-id1165137849560\">There is lightning rod on the top of a building. From a location 500 feet from the base of the building, the angle of elevation to the top of the building is measured to be[latex]\\text{ }36\u00b0.\\text{ }[\/latex]From the same location, the angle of elevation to the top of the lightning rod is measured to be[latex]\\text{ }38\u00b0.\\text{ }[\/latex]Find the height of the lightning rod.<\/p>\n<\/div>\n<div id=\"fs-id1165137767139\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137767139\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137767139\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135189997\">27.372 ft<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137730425\" class=\"bc-section section\">\n<h4>Real-World Applications<\/h4>\n<div id=\"fs-id1165137451128\">\n<div id=\"fs-id1165137451129\">\n<p id=\"fs-id1165137451130\">A 33-ft ladder leans against a building so that the angle between the ground and the ladder is[latex]\\text{ }80\u00b0.\\text{ }[\/latex]How high does the ladder reach up the side of the building?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137437700\">\n<div id=\"fs-id1165137437702\">\n<p id=\"fs-id1165137437703\">A 23-ft ladder leans against a building so that the angle between the ground and the ladder is[latex]\\text{ }80\u00b0.\\text{ }[\/latex]How high does the ladder reach up the side of the building?<\/p>\n<\/div>\n<div id=\"fs-id1165137437289\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137437289\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137437289\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137437290\">22.6506 ft<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137452345\">\n<div id=\"fs-id1165137452346\">\n<p id=\"fs-id1165137935681\">The angle of elevation to the top of a building in New York is found to be 9 degrees from the ground at a distance of 1 mile from the base of the building. Using this information, find the height of the building.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137665678\">\n<div>\n<p id=\"fs-id1165137665680\">The angle of elevation to the top of a building in Seattle is found to be 2 degrees from the ground at a distance of 2 miles from the base of the building. Using this information, find the height of the building.<\/p>\n<\/div>\n<div id=\"fs-id1165135250591\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135250591\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135250591\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135250592\">368.7633 ft<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135690138\">\n<div id=\"fs-id1165135690140\">\n<p id=\"fs-id1165135690141\">Assuming that a 370-foot tall giant redwood grows vertically, if I walk a certain distance from the tree and measure the angle of elevation to the top of the tree to be[latex]\\text{ }60\u00b0,[\/latex] how far from the base of the tree am I?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137627082\" class=\"review-exercises\">\n<h3>Review Exercises<\/h3>\n<div id=\"fs-id1165135160073\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/6a416d21-7302-4df4-a53d-06ac8a166e31\">Angles<\/a><\/h4>\n<p id=\"fs-id1165135186396\">For the following exercises, convert the angle measures to degrees.<\/p>\n<div id=\"fs-id1165137604792\">\n<div id=\"fs-id1165137604793\">\n<p id=\"fs-id1165137604794\">[latex]\\frac{\\pi }{4}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135173589\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135173589\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135173589\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135173590\">[latex]45\u00b0[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134223337\">\n<div id=\"fs-id1165134223338\">\n<p id=\"fs-id1165134223340\">[latex]-\\frac{5\\pi }{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137445371\">For the following exercises, convert the angle measures to radians.<\/p>\n<div id=\"fs-id1165137923488\">\n<div id=\"fs-id1165137923489\">\n<p id=\"fs-id1165137923490\">-210\u00b0<\/p>\n<\/div>\n<div id=\"fs-id1165135192319\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135192319\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135192319\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135192320\">[latex]-\\frac{7\\pi }{6}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137626943\">\n<div id=\"fs-id1165137812372\">\n<p id=\"fs-id1165137812373\">180\u00b0<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137544293\">\n<div id=\"fs-id1165137544294\">\n<p id=\"fs-id1165137652804\">Find the length of an arc in a circle of radius 7 meters subtended by the central angle of 85\u00b0.<\/p>\n<\/div>\n<div id=\"fs-id1165135527005\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135527005\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135527005\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135527006\">10.385 meters<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137833897\">\n<div id=\"fs-id1165137833898\">\n<p id=\"fs-id1165137901222\">Find the area of the sector of a circle with diameter 32 feet and an angle of[latex]\\text{ }\\frac{3\\pi }{5}\\text{ }[\/latex]radians.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135160645\">For the following exercises, find the angle between 0\u00b0 and 360\u00b0 that is coterminal with the given angle.<\/p>\n<div id=\"fs-id1165137506838\">\n<div id=\"fs-id1165137506839\">\n<p id=\"fs-id1165137506840\">[latex]420\u00b0[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137735631\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137735631\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137735631\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137735632\">[latex]60\u00b0[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137534948\">\n<div id=\"fs-id1165137534949\">\n<p id=\"fs-id1165134069177\">[latex]-80\u00b0[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135149019\">For the following exercises, find the angle between 0 and[latex]\\text{ }2\\pi \\text{ }[\/latex]in radians that is coterminal with the given angle.<\/p>\n<div id=\"fs-id1165137605274\">\n<div id=\"fs-id1165134342624\">\n<p id=\"fs-id1165134342625\">[latex]-\\text{ }\\frac{20\\pi }{11}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137444823\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137444823\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137444823\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137444824\">[latex]\\frac{2\\pi }{11}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137727049\">\n<div id=\"fs-id1165137727050\">\n<p id=\"fs-id1165137727051\">[latex]\\frac{14\\pi }{5}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137415610\">For the following exercises, draw the angle provided in standard position on the Cartesian plane.<\/p>\n<div id=\"fs-id1165137415613\">\n<div id=\"fs-id1165137460755\">\n<p id=\"fs-id1165137460756\">-210\u00b0<\/p>\n<\/div>\n<div id=\"fs-id1165135209568\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135209568\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135209568\" class=\"hidden-answer\" style=\"display: none\"><span id=\"fs-id1165135403289\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132649\/CNX_Precalc_Figure_05_04_217.jpg\" alt=\"A graph of a circle with a negative angle inscribed.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137771820\">\n<div id=\"fs-id1165137771822\">\n<p id=\"fs-id1165135497744\">75\u00b0<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137611468\">\n<div id=\"fs-id1165137611469\">\n<p id=\"fs-id1165137611470\">[latex]\\frac{5\\pi }{4}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137785040\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137785040\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137785040\" class=\"hidden-answer\" style=\"display: none\"><span id=\"fs-id1165135169236\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132651\/CNX_Precalc_Figure_05_04_219.jpg\" alt=\"A graph of a circle with an angle inscribed.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137451538\">\n<div id=\"fs-id1165135299863\">\n<p id=\"fs-id1165135299864\">[latex]-\\frac{\\pi }{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137842349\">\n<div id=\"fs-id1165137842350\">\n<p id=\"fs-id1165137842352\">Find the linear speed of a point on the equator of the earth if the earth has a radius of 3,960 miles and the earth rotates on its axis every 24 hours. Express answer in miles per hour.<\/p>\n<\/div>\n<div id=\"fs-id1165135194111\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135194111\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135194111\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137475894\">1036.73 miles per hour<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137475897\">\n<div id=\"fs-id1165137745166\">\n<p id=\"fs-id1165137745167\">A car wheel with a diameter of 18 inches spins at the rate of 10 revolutions per second. What is the car&#8217;s speed in miles per hour?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137598589\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/17a6d7f5-c90c-4047-abac-0376d582549b\">Unit Circle: Sine and Cosine Functions<\/a><\/h4>\n<div id=\"fs-id1165137804818\">\n<div id=\"fs-id1165137470140\">\n<p id=\"fs-id1165137470141\">Find the exact value of[latex]\\text{ }\\mathrm{sin}\\text{ }\\frac{\\pi }{3}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137642361\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137642361\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137642361\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137642362\">[latex]\\frac{\\sqrt{3}}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137647355\">\n<div id=\"fs-id1165137647356\">\n<p id=\"fs-id1165135708042\">Find the exact value of[latex]\\text{ }\\mathrm{cos}\\text{ }\\frac{\\pi }{4}.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137566636\">\n<div id=\"fs-id1165137566637\">\n<p id=\"fs-id1165137566638\">Find the exact value of[latex]\\text{ }\\mathrm{cos}\\text{ }\\pi .[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135173474\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135173474\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135173474\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135173475\">\u20131<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133281394\">\n<div id=\"fs-id1165133281395\">\n<p id=\"fs-id1165133281396\">State the reference angle for[latex]\\text{ }300\u00b0.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137465126\">\n<div id=\"fs-id1165137465127\">\n<p id=\"fs-id1165137465128\">State the reference angle for[latex]\\text{ }\\frac{3\\pi }{4}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137756068\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137756068\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137756068\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137723588\">[latex]\\frac{\\pi }{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137426933\">\n<div id=\"fs-id1165137426934\">\n<p id=\"fs-id1165137426935\">Compute cosine of[latex]\\text{ }330\u00b0.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137602620\">\n<div id=\"fs-id1165137602621\">\n<p id=\"fs-id1165137602622\">Compute sine of[latex]\\text{ }\\frac{5\\pi }{4}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137748992\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137748992\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137748992\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137506605\">[latex]-\\frac{\\sqrt{2}}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137573390\">\n<div id=\"fs-id1165137573391\">\n<p id=\"fs-id1165137573392\">State the domain of the sine and cosine functions.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137574969\">\n<div id=\"fs-id1165137574970\">\n<p id=\"fs-id1165137748521\">State the range of the sine and cosine functions.<\/p>\n<\/div>\n<div id=\"fs-id1165137748524\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137748524\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137748524\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137583678\">[latex]\\left[\u20131,1\\right][\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137668652\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/df1a76e7-7b9c-4f58-8a81-afd48cedcf0b\">The Other Trigonometric Functions<\/a><\/h4>\n<p id=\"fs-id1165137657202\">For the following exercises, find the exact value of the given expression.<\/p>\n<div id=\"fs-id1165137920680\">\n<div id=\"fs-id1165137920681\">\n<p id=\"fs-id1165134284487\">[latex]\\mathrm{cos}\\text{ }\\frac{\\pi }{6}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137871054\">\n<div id=\"fs-id1165135512426\">\n<p id=\"fs-id1165135512427\">[latex]\\mathrm{tan}\\text{ }\\frac{\\pi }{4}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137676096\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137676096\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137676096\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137676098\">1<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137596056\">\n<div id=\"fs-id1165137596057\">\n<p id=\"fs-id1165137596058\">[latex]\\mathrm{csc}\\text{ }\\frac{\\pi }{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137656500\">\n<div id=\"fs-id1165137656501\">\n<p id=\"fs-id1165137656502\">[latex]\\mathrm{sec}\\text{ }\\frac{\\pi }{4}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137730531\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137730531\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137730531\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137730532\">[latex]\\sqrt{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165134381589\">For the following exercises, use reference angles to evaluate the given expression.<\/p>\n<div id=\"fs-id1165137862404\">\n<div id=\"fs-id1165137862406\">\n<p id=\"fs-id1165137862407\">[latex]\\mathrm{sec}\\text{ }\\frac{11\\pi }{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137588722\">\n<div id=\"fs-id1165137588723\">\n<p id=\"fs-id1165137836532\">[latex]\\mathrm{sec}\\text{ }315\u00b0[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137549719\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137549719\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137549719\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137637844\">[latex]\\sqrt{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135205897\">\n<div id=\"fs-id1165135205898\">\n<p id=\"fs-id1165137580092\">If[latex]\\text{ }\\mathrm{sec}\\left(t\\right)=-2.5\\text{ }[\/latex], what is the[latex]\\text{ }\\text{sec}\\left(-t\\right)?[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165132943553\">\n<div id=\"fs-id1165132943554\">\n<p id=\"fs-id1165137768826\">If[latex]\\text{ }\\text{tan}\\left(t\\right)=-0.6,[\/latex] what is the[latex]\\text{ }\\text{tan}\\left(-t\\right)?[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137780740\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137780740\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137780740\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137780741\">0.6<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135378773\">\n<div id=\"fs-id1165135378774\">\n<p id=\"fs-id1165135378775\">If[latex]\\text{ }\\text{tan}\\left(t\\right)=\\frac{1}{3},[\/latex] find[latex]\\text{ }\\text{tan}\\left(t-\\pi \\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133356011\">\n<div id=\"fs-id1165133356012\">\n<p id=\"fs-id1165133356013\">If[latex]\\text{ }\\text{cos}\\left(t\\right)=\\frac{\\sqrt{2}}{2},[\/latex] find[latex]\\text{ }\\text{sin}\\left(t+2\\pi \\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137574568\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137574568\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137574568\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137574570\">[latex]\\frac{\\sqrt{2}}{2}\\text{ }[\/latex]or[latex]\\text{ }-\\frac{\\sqrt{2}}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137758484\">\n<div id=\"fs-id1165137758485\">\n<p id=\"fs-id1165137871876\">Which trigonometric functions are even?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137871879\">\n<div id=\"fs-id1165137572480\">\n<p id=\"fs-id1165137572482\">Which trigonometric functions are odd?<\/p>\n<\/div>\n<div id=\"fs-id1165137471478\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137471478\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137471478\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137471479\">sine, cosecant, tangent, cotangent<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137911687\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/2d2ba0f3-4818-4665-9f37-7ec8c6d36e52\">Right Triangle Trigonometry<\/a><\/h4>\n<p id=\"fs-id1165137425556\">For the following exercises, use side lengths to evaluate.<\/p>\n<div id=\"fs-id1165137571591\">\n<div id=\"fs-id1165137571592\">\n<p id=\"fs-id1165137571593\">[latex]\\mathrm{cos}\\text{ }\\frac{\\pi }{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137803944\">\n<div id=\"fs-id1165137803945\">\n<p id=\"fs-id1165137658119\">[latex]\\mathrm{cot}\\text{ }\\frac{\\pi }{3}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137663617\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137663617\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137663617\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137665107\">[latex]\\frac{\\sqrt{3}}{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133047532\">\n<div id=\"fs-id1165135693727\">\n<p>[latex]\\mathrm{tan}\\text{ }\\frac{\\pi }{6}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137898078\">\n<div id=\"fs-id1165137898079\">\n<p id=\"fs-id1165137898080\">[latex]\\mathrm{cos}\\left(\\frac{\\pi }{2}\\right)=\\mathrm{sin}\\left(\\text{__\u00b0}\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137605493\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137605493\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137605493\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135193114\">0<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135193118\">\n<div id=\"fs-id1165137653909\">\n<p id=\"fs-id1165137653910\">[latex]\\mathrm{csc}\\left(18\\text{\u00b0}\\right)=\\mathrm{sec}\\left(\\text{__\u00b0}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137724964\">For the following exercises, use the given information to find the lengths of the other two sides of the right triangle.<\/p>\n<div id=\"fs-id1165135528966\">\n<div id=\"fs-id1165135528967\">\n<p id=\"fs-id1165135560804\">[latex]\\mathrm{cos}\\text{ }B=\\frac{3}{5},a=6[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137433002\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137433002\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137433002\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137433003\">[latex]b=8,c=10[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137410615\">\n<div id=\"fs-id1165137410616\">\n<p id=\"fs-id1165137410617\">[latex]\\mathrm{tan}\\text{ }A=\\frac{5}{9},b=6[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137644993\">For the following exercises, use <a class=\"autogenerated-content\" href=\"#Figure_05_04_221\">(Figure)<\/a> to evaluate each trigonometric function.<\/p>\n<div id=\"Figure_05_04_221\" class=\"small\"><span id=\"fs-id1165137532849\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132653\/CNX_Precalc_Figure_05_04_221.jpg\" alt=\"A right triangle with side lengths of 11 and 6. Corners A and B are also labeled.\" \/><\/span><\/div>\n<div id=\"fs-id1165135424671\">\n<div id=\"fs-id1165135424672\">\n<p>[latex]\\mathrm{sin}\\text{ }A[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137734516\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137734516\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137734516\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165134122790\">[latex]\\frac{11\\sqrt{157}}{157}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137558054\">\n<div id=\"fs-id1165137442090\">\n<p id=\"fs-id1165137442091\">[latex]\\mathrm{tan}\\text{ }B[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135192508\">For the following exercises, solve for the unknown sides of the given triangle.<\/p>\n<div id=\"fs-id1165135263670\">\n<div id=\"fs-id1165135263671\"><span id=\"fs-id1165137834282\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132656\/CNX_Precalc_Figure_05_04_222n.jpg\" alt=\"A right triangle with corners labeled A, B, and C. Hyptenuse has length of 4 times square root of 2. Other angles measure 45 degrees.\" \/><\/span><\/div>\n<div id=\"fs-id1165137575901\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137575901\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137575901\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137575902\">[latex]a=4, b=4[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135436551\">\n<div id=\"fs-id1165135436552\"><span id=\"fs-id1165137406146\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132658\/CNX_Precalc_Figure_05_04_223n.jpg\" alt=\"A right triangle with hypotenuse with length 5, and an angle of 30 degrees.\" \/><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165133255048\">\n<div id=\"fs-id1165133255050\">\n<p id=\"fs-id1165133255051\">A 15-ft ladder leans against a building so that the angle between the ground and the ladder is[latex]\\text{ }70\u00b0.\\text{ }[\/latex]How high does the ladder reach up the side of the building?<\/p>\n<\/div>\n<div id=\"fs-id1165135332736\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135332736\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135332736\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137767865\">14.0954 ft<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137767869\">\n<div id=\"fs-id1165135536497\">\n<p id=\"fs-id1165135536498\">The angle of elevation to the top of a building in Baltimore is found to be 4 degrees from the ground at a distance of 1 mile from the base of the building. Using this information, find the height of the building.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137668291\" class=\"practice-test\">\n<h3>Practice Test<\/h3>\n<div id=\"fs-id1165137668059\">\n<div id=\"fs-id1165137735263\">\n<p id=\"fs-id1165137735264\">Convert[latex]\\text{ }\\frac{5\\pi }{6}\\text{ }[\/latex]radians to degrees.<\/p>\n<\/div>\n<div id=\"fs-id1165137560606\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137560606\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137560606\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137560607\">[latex]150\u00b0[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135694590\">\n<div id=\"fs-id1165137767383\">\n<p id=\"fs-id1165137767384\">Convert[latex]\\text{ }-620\u00b0\\text{ }[\/latex]to radians.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137452465\">\n<div id=\"fs-id1165137452466\">\n<p id=\"fs-id1165137452467\">Find the length of a circular arc with a radius 12 centimeters subtended by the central angle of[latex]\\text{ }30\u00b0.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137862788\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137862788\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137862788\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137862790\">6.283 centimeters<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137401698\">\n<div id=\"fs-id1165137401699\">\n<p id=\"fs-id1165137401700\">Find the area of the sector with radius of 8 feet and an angle of[latex]\\text{ }\\frac{5\\pi }{4}\\text{ }[\/latex]radians.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135408494\">\n<div id=\"fs-id1165135183090\">\n<p id=\"fs-id1165135183091\">Find the angle between[latex]\\text{ }0\u00b0\\text{ }[\/latex]and[latex]\\text{ }\\text{360\u00b0}\\text{ }[\/latex]that is coterminal with[latex]\\text{ }375\u00b0.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135161403\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135161403\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135161403\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135161404\">[latex]15\u00b0[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137619343\">\n<div id=\"fs-id1165137768782\">\n<p id=\"fs-id1165137768783\">Find the angle between 0 and[latex]\\text{ }2\\pi \\text{ }[\/latex]in radians that is coterminal with[latex]\\text{ }-\\frac{4\\pi }{7}.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137931243\">\n<div id=\"fs-id1165137931244\">\n<p id=\"fs-id1165137931245\">Draw the angle[latex]\\text{ }315\u00b0\\text{ }[\/latex]in standard position on the Cartesian plane.<\/p>\n<\/div>\n<div id=\"fs-id1165137694225\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137694225\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137694225\" class=\"hidden-answer\" style=\"display: none\"><span id=\"fs-id1165137592067\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132701\/CNX_Precalc_Figure_05_04_224.jpg\" alt=\"A graph of a circle with an angle inscribed.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137851378\">\n<div id=\"fs-id1165137402102\">\n<p id=\"fs-id1165137402103\">Draw the angle[latex]\\text{ }-\\frac{\\pi }{6}\\text{ }[\/latex]in standard position on the Cartesian plane.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137653537\">\n<div id=\"fs-id1165137653538\">\n<p id=\"fs-id1165137653540\">A carnival has a Ferris wheel with a diameter of 80 feet. The time for the Ferris wheel to make one revolution is 75 seconds. What is the linear speed in feet per second of a point on the Ferris wheel? What is the angular speed in radians per second?<\/p>\n<\/div>\n<div id=\"fs-id1165137725827\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137725827\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137725827\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137557535\">3.351 feet per second,[latex]\\text{ }\\frac{2\\pi }{75}\\text{ }[\/latex]radians per second<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135160182\">\n<div id=\"fs-id1165135160183\">\n<p id=\"fs-id1165135160184\">Find the exact value of[latex]\\text{ }\\mathrm{sin}\\text{ }\\frac{\\pi }{6}.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137575005\">\n<div id=\"fs-id1165137575006\">\n<p id=\"fs-id1165135639873\">Compute sine of[latex]\\text{ }240\u00b0.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137408475\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137408475\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137408475\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137408476\">[latex]-\\frac{\\sqrt{3}}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135343004\">\n<div id=\"fs-id1165135593415\">\n<p id=\"fs-id1165135593416\">State the domain of the sine and cosine functions.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135593420\">\n<div id=\"fs-id1165135192267\">\n<p id=\"fs-id1165135192268\">State the range of the sine and cosine functions.<\/p>\n<\/div>\n<div id=\"fs-id1165135192271\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135192271\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135192271\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165134269017\">[latex]\\left[\u20131,1\\right][\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135344124\">\n<div id=\"fs-id1165137605201\">\n<p id=\"fs-id1165137605202\">Find the exact value of[latex]\\text{ }\\mathrm{cot}\\text{ }\\frac{\\pi }{4}.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137722364\">\n<div id=\"fs-id1165137722365\">\n<p id=\"fs-id1165137722366\">Find the exact value of[latex]\\text{ }\\mathrm{tan}\\text{ }\\frac{\\pi }{3}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137823309\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137823309\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137823309\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137823310\">[latex]\\sqrt{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137664264\">\n<div id=\"fs-id1165137664265\">\n<p id=\"fs-id1165137603262\">Use reference angles to evaluate[latex]\\text{ }\\mathrm{csc}\\text{ }\\frac{7\\pi }{4}.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137415501\">\n<div id=\"fs-id1165137415502\">\n<p id=\"fs-id1165137415504\">Use reference angles to evaluate[latex]\\text{ }\\mathrm{tan}\\text{ }210\u00b0.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135405233\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135405233\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135405233\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135405234\">[latex]\\frac{\\sqrt{3}}{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135181505\">\n<div id=\"fs-id1165135181506\">\n<p id=\"fs-id1165135181507\">If[latex]\\text{ }\\text{csc}\\text{ }t=0.68,[\/latex]what is the[latex]\\text{ }\\text{csc}\\left(-t\\right)?[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137634897\">\n<div id=\"fs-id1165137634898\">\n<p id=\"fs-id1165137634899\">If[latex]\\text{ }\\text{cos}\\text{ }\\text{t}=\\frac{\\sqrt{3}}{2},[\/latex]find[latex]\\text{ }\\text{cos}\\left(t-2\\pi \\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137447886\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137447886\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137447886\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137447887\">[latex]\\frac{\\sqrt{3}}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137730237\">\n<div id=\"fs-id1165137432724\">\n<p id=\"fs-id1165137432725\">Which trigonometric functions are even?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137432728\">\n<div id=\"fs-id1165137891501\">\n<p id=\"fs-id1165137891502\">Find the missing angle:[latex]\\text{ }\\mathrm{cos}\\left(\\frac{\\pi }{6}\\right)=\\mathrm{sin}\\left(___\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137476439\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137476439\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137476439\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137476440\">[latex]\\frac{\\pi }{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165135196915\">\n<p id=\"fs-id1165135196916\">Find the missing sides of the triangle[latex]\\text{ }ABC:\\mathrm{sin}\\text{ }B=\\frac{3}{4},c=12[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137661200\">\n<div id=\"fs-id1165137661201\">\n<p id=\"fs-id1165137661202\">Find the missing sides of the triangle.<\/p>\n<p><span id=\"fs-id1165137410882\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132703\/CNX_Precalc_Figure_05_04_226.jpg\" alt=\"A right triangle with hyptenuse length of 9 and angle measure of 60 degrees.\" \/><\/span><\/p>\n<\/div>\n<div id=\"fs-id1165134113903\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165134113903\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165134113903\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165134113904\">[latex]a=\\frac{9}{2},b=\\frac{9\\sqrt{3}}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165132971678\">\n<div id=\"fs-id1165137480414\">\n<p id=\"fs-id1165137480415\">The angle of elevation to the top of a building in Chicago is found to be 9 degrees from the ground at a distance of 2000 feet from the base of the building. Using this information, find the height of the building.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id1165137446119\">\n<dt>adjacent side<\/dt>\n<dd id=\"fs-id1165137446123\">in a right triangle, the side between a given angle and the right angle<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137465232\">\n<dt>angle of depression<\/dt>\n<dd id=\"fs-id1165135175026\">the angle between the horizontal and the line from the object to the observer\u2019s eye, assuming the object is positioned lower than the observer<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137447602\">\n<dt>angle of elevation<\/dt>\n<dd id=\"fs-id1165135185281\">the angle between the horizontal and the line from the object to the observer\u2019s eye, assuming the object is positioned higher than the observer<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137558543\">\n<dt>opposite side<\/dt>\n<dd id=\"fs-id1165137558547\">in a right triangle, the side most distant from a given angle<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135477488\">\n<dt>hypotenuse<\/dt>\n<dd id=\"fs-id1165137588091\">the side of a right triangle opposite the right angle<\/dd>\n<\/dl>\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-608\">\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>Right Triangle Trigonometry. <strong>Authored by<\/strong>: Douglas Hoffman. <strong>Provided by<\/strong>: Openstax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/cnx.org\/contents\/8si1Yf2B@2.21:LSug80gY@6\/Right-Triangle-Trigonometry\">https:\/\/cnx.org\/contents\/8si1Yf2B@2.21:LSug80gY@6\/Right-Triangle-Trigonometry<\/a>. <strong>Project<\/strong>: Essential Precalcus, Part 2. <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":311,"menu_order":1,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Right Triangle Trigonometry\",\"author\":\"Douglas Hoffman\",\"organization\":\"Openstax\",\"url\":\"https:\/\/cnx.org\/contents\/8si1Yf2B@2.21:LSug80gY@6\/Right-Triangle-Trigonometry\",\"project\":\"Essential Precalcus, Part 2\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":null,"pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-608","chapter","type-chapter","status-web-only","hentry"],"part":609,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/608","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/users\/311"}],"version-history":[{"count":4,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/608\/revisions"}],"predecessor-version":[{"id":2115,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/608\/revisions\/2115"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/parts\/609"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/608\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/media?parent=608"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=608"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/contributor?post=608"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/license?post=608"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}