{"id":1590,"date":"2021-05-20T02:03:17","date_gmt":"2021-05-20T02:03:17","guid":{"rendered":"https:\/\/opentextbc.ca\/businesstechnicalmath\/chapter\/6-2\/"},"modified":"2021-11-17T18:21:37","modified_gmt":"2021-11-17T18:21:37","slug":"6-2","status":"publish","type":"chapter","link":"https:\/\/opentextbc.ca\/businesstechnicalmath\/chapter\/6-2\/","title":{"raw":"5.2  Solve Applications: Sine, Cosine and Tangent Ratios.","rendered":"5.2  Solve Applications: Sine, Cosine and Tangent Ratios."},"content":{"raw":"<div data-type=\"note\">[latexpage]<\/div>\r\n<div data-type=\"note\">\r\n<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Learning Objectives<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nBy the end of this section it is expected that you will be able to\r\n<ul>\r\n \t<li>Find missing side of a right triangle using sine, cosine, or tangent ratios<\/li>\r\n \t<li>Find missing angle of a right triangle using sine, cosine, or tangent ratios<\/li>\r\n \t<li>Solve applications using right angle trigonometry<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\">Now, that we know the fundamentals of algebra and geometry associated with a right triangle, we can start exploring trigonometry. Many real life problems can be represented and solved using right angle trigonometry.<\/div>\r\n<h1 id=\"fs-id1168345273636\" data-type=\"note\">Sine, Cosine, and Tangent Ratios<\/h1>\r\n<div id=\"fs-id1168345363664\" class=\"bc-section section\" data-depth=\"1\">\r\n\r\nWe know that any right triangle has three sides and a right angle. The side opposite to the right angle is called the hypotenuse. The other two angles in a right triangle are acute angles (with a measure less than 90 degrees). One of those angles we call reference angle and we use \u03b8 (theta) to represent it.\r\n\r\nThe hypotenuse is always the longest side of a right triangle. The other two sides are called opposite side and adjacent side. The names of those sides depends on which of the two acute angles is being used as a reference angle.\r\n<div id=\"fs-id129640054\">\r\n\r\n[caption id=\"attachment_9622\" align=\"aligncenter\" width=\"420\"]<img class=\"wp-image-9622\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/Fig-1-300x188-1.png\" alt=\"\" width=\"420\" height=\"263\" \/> Figure 1.[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\nIn the right triangle each side is labeled with a lowercase letter to match the uppercase letter of the opposite vertex.\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nLabel the sides of the triangle and find the hypotenuse, opposite, and adjacent.\r\n\r\n<img class=\"alignnone size-full wp-image-9639\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/Fig-9-2.png\" alt=\"\" width=\"216\" height=\"189\" \/>\r\n\r\n<strong>Solution<\/strong>\r\n\r\nWe labeled\u00a0 the sides with a lowercase letter to match the uppercase letter of the opposite vertex.\r\n\r\nc is hypotenuse\r\n\r\na is opposite\r\n\r\nb is adjacent\r\n\r\n<img class=\"alignnone size-full wp-image-9640\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/Fig-10-1.png\" alt=\"\" width=\"205\" height=\"188\" \/>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nLabel the sides of the triangle and find the hypotenuse, opposite and adjacent.\r\n\r\n<img class=\"alignnone wp-image-9629\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/Fig-6-300x201-1.png\" alt=\"\" width=\"231\" height=\"155\" \/>\r\n<div id=\"fs-id1296401\" data-type=\"solution\"><details><summary>Show answer<\/summary>y is hypotenuse\r\n\r\nz is opposite\r\n\r\nx is adjacent\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<h1 id=\"fs-id1168345363664\" data-depth=\"1\">Trigonometric Ratios<\/h1>\r\nTrigonometric ratios are the ratios of the sides in the right triangle. For any right triangle we can define three basic trigonometric ratios: sine, cosine, and tangent.\r\n\r\nLet us refer to <a href=\"#fs-id129640054\">Figure 1<\/a> and define the three basic trigonometric ratios as:\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Three Basic Trigonometric Ratios<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ul>\r\n \t<li>sine \u03b8 = \\(\\frac{\\text{the length of the opposite side}}{\\text{the length of the\u00a0 hypotenuse side}}\\)<\/li>\r\n \t<li>cosine \u03b8 = \\(\\frac{\\text{the length of the adjacent side}}{\\text{the length of the\u00a0 hypotenuse side}}\\)<\/li>\r\n \t<li>tangent \u03b8 = \\(\\frac{\\text{the length of the opposite side}}{\\text{the length of the\u00a0 adjacent side}}\\)<\/li>\r\n<\/ul>\r\nWhere \u03b8 is the measure of a reference angle measured in degrees.\r\n\r\n<\/div>\r\n<\/div>\r\nVery often we use the abbreviations for sine, cosine, and tangent ratios.\r\n<ul>\r\n \t<li>sin \u03b8 = \\(\\frac{\\text{opp}}{\\text{hyp}}\\)<\/li>\r\n \t<li>cos \u03b8 = \\(\\frac{\\text{adj}}{\\text{hyp}}\\)<\/li>\r\n \t<li>tan \u03b8 = \\(\\frac{\\text{opp}}{\\text{adj}}\\)<\/li>\r\n<\/ul>\r\nSome people remember the definition of the trigonometric ratios as SOH CAH TOA.\r\n\r\nLet's use the \\(\\Delta DEF\\) from Example 1 to find the three ratios.\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFor the given triangle find the sine, cosine and tangent ratio.\r\n\r\n<img class=\"alignnone wp-image-9626\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/Fig-5-300x252-1.png\" alt=\"\" width=\"231\" height=\"194\" \/>\r\n\r\n<strong>Solution<\/strong>\r\n\r\nsin \u03b8 = \\(\\frac{\\text{f}}{\\text{d}}\\)\r\n\r\ncos \u03b8 = \\(\\frac{\\text{e}}{\\text{d}}\\)\r\n\r\ntan \u03b8 = \\(\\frac{\\text{f}}{\\text{e}}\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFor the given triangle find the sine cosine and tangent ratio.\r\n\r\n<img class=\"alignnone wp-image-9629\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/Fig-6-300x201-1.png\" alt=\"\" width=\"231\" height=\"155\" \/>\r\n\r\n<\/div>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1296403\" data-type=\"solution\"><details><summary>Show answer<\/summary>sin \u03b8 = \\(\\frac{\\text{z}}{\\text{y}}\\)\r\n\r\ncos \u03b8 = \\(\\frac{\\text{x}}{\\text{y}}\\)\r\n\r\ntan \u03b8 = \\(\\frac{\\text{z}}{\\text{x}}\\)\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\nIn Example 2, our reference angles can be \\( \\angle E\\) \u00a0or \\( \\angle F\\) . Using the definition of trigonometric ratios, we can write sinE = \\(\\frac{\\text{e}}{\\text{d}}\\) , cosE = \\(\\frac{\\text{f}}{\\text{d}}\\), and tanE = \\(\\frac{\\text{e}}{\\text{f}}\\).\r\n\r\nWhen calculating we will usually round the ratios to four decimal places and at the end our final answer to one decimal place unless stated otherwise.\r\n<div id=\"fs-id1168345363664\" class=\"bc-section section\" data-depth=\"1\">\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 3<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFor the given triangle find the sine, cosine and tangent ratios. If necessary round to four decimal places.\r\n\r\n<img class=\"alignnone wp-image-9642\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/Fig-11-300x194-1.png\" alt=\"\" width=\"252\" height=\"163\" \/>\r\n\r\n<strong>Solution<\/strong>\r\n\r\nWe have two possible reference angles: R an S.\r\n\r\nUsing the definitions, the trigonometric ratios for angle R are:\r\n<ul>\r\n \t<li>sin R= \\(\\frac{4}{5}\\) = 0.8<\/li>\r\n \t<li>cos R= \\(\\frac{3}{5}\\) = 0.6<\/li>\r\n \t<li>tan R=\\(\\frac{4}{3}\\) = 1.3333...<\/li>\r\n<\/ul>\r\nUsing the definitions, the trigonometric ratios for angle S:\r\n<ul>\r\n \t<li>sin S = \\(\\frac{3}{5}\\) = 0.6<\/li>\r\n \t<li>cos S = \\(\\frac{4}{5}\\) = 0.8<\/li>\r\n \t<li>tan S = \\(\\frac{3}{4}\\) = 0.75<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 3<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFor the given triangle find the sine, cosine, and tangent ratios. If necessary round to four decimal places.\r\n\r\n<img class=\"alignnone wp-image-9666\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/Fig-17-300x141-1.png\" alt=\"\" width=\"270\" height=\"127\" \/>\r\n<div id=\"fs-id1296405\" data-type=\"solution\"><details open=\"open\"><summary>Show answer<\/summary>\r\n<ul>\r\n \t<li>sin F = \\(\\frac{8}{10}\\) = 0.8<\/li>\r\n \t<li>cos F = \\(\\frac{6}{10}\\) =0.6<\/li>\r\n \t<li>tan F = \\(\\frac{8}{6}\\) = 1.3333...<\/li>\r\n \t<li>sin D =\\(\\frac{6}{10}\\) = 0.6<\/li>\r\n \t<li>cos D =\\(\\frac{8}{10}\\) = 0.8<\/li>\r\n \t<li>tan D =\\(\\frac{6}{8}\\) = 0.75<\/li>\r\n<\/ul>\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\nNow, let us use a scientific calculator to find the trigonometric ratios.\u00a0 Can you find the sin, cos, and tan buttons on your calculator? To find the trigonometric ratios make sure your calculator is in Degree Mode.\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 4<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nUsing a calculator find the trigonometric ratios. If necessary, round to 4 decimal places.\r\n\r\na) sin 30\u00b0\r\n\r\nb) cos 45\u00b0\r\n\r\nc) tan 60\u00b0\r\n\r\n<strong>Solution<\/strong>\r\n\r\nMake sure your calculator is in Degree Mode.\r\n\r\na)\u00a0 Using a calculator find that sin 30\u00b0 = 0.5\r\n\r\nb)\u00a0 Using a calculator find that cos 45\u00b0 = 0.7071 Rounded to 4 decimal places.\r\n\r\nc)\u00a0 Using a calculator find that tan 60\u00b0 = 1.7321 Rounded to 4 decimal places.\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 4<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the trigonometric ratios. If necessary, round to 4 decimal places.\r\n\r\na)\u00a0 sin 60\u00b0\r\n\r\nb)\u00a0 cos 30\u00b0\r\n\r\nc)\u00a0 tan 45\u00b0\r\n<div id=\"fs-id1296407\" data-type=\"solution\"><details open=\"open\"><summary>Show answer<\/summary>a)\u00a0 \u00a0sin 60\u00b0 = 0.8660\r\n\r\nb)\u00a0 \u00a0cos 30\u00b0 = 0.8660\r\n\r\nc)\u00a0 \u00a0tan 45\u00b0 = 1\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<h1>Finding Missing Sides of a Right Triangle<\/h1>\r\nIn this section you will be using trigonometric ratios to solve right triangle problems. We will adapt our problem solving strategy for trigonometry applications. In addition, since those problems will involve the right triangle, it is helpful to draw it (if the drawing is not given) and label it with the given information.We will include this in the first step of the problem solving strategy for trigonometry applications.\r\n<div id=\"fs-id1168345512560\" class=\"howto\" data-type=\"note\">\r\n<div data-type=\"title\">\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">HOW TO: Solve Trigonometry Applications<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ol id=\"fs-id1166426163525\" class=\"stepwise\" type=\"1\">\r\n \t<li><strong data-effect=\"bold\">Read<\/strong> the problem and make sure all the words and ideas are understood. Draw the right triangle and label the given parts.<\/li>\r\n \t<li><strong data-effect=\"bold\">Identify<\/strong> what we are looking for.<\/li>\r\n \t<li><strong data-effect=\"bold\">Label<\/strong> what we are looking for by choosing a variable to represent it.<\/li>\r\n \t<li><strong data-effect=\"bold\">Find <\/strong>the required\u00a0trigonometric ratio.<\/li>\r\n \t<li><strong data-effect=\"bold\">Solve<\/strong> the ratio using good algebra techniques.<\/li>\r\n \t<li><strong data-effect=\"bold\">Check<\/strong> the answer by substituting it back into the ratio in step 4 and by making sure it makes sense in the context of the problem.<\/li>\r\n \t<li><strong data-effect=\"bold\">Answer<\/strong> the question with a complete sentence.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\nIn the next few examples, having given the measure of one acute angle and the length of one side of the right triangle, we will solve the right triangle for the missing sides.\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 5<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n\u00a0Find the missing sides. Round\u00a0 your final answer to two decimal places\r\n\r\n<img class=\"alignnone wp-image-9654\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/Fig-15-300x232-1.png\" alt=\"\" width=\"261\" height=\"202\" \/>\r\n\r\n<strong>Solution<\/strong>\r\n<table style=\"border-collapse: collapse; width: 100%; height: 194px;\" border=\"0\">\r\n<tbody>\r\n<tr style=\"height: 46px;\">\r\n<td style=\"width: 33.5227%; height: 46px;\">1. <strong>Read<\/strong> the problem and make sure all the words and ideas are understood. Draw the right triangle and label\u00a0 the given parts.<\/td>\r\n<td style=\"width: 66.6666%; height: 46px;\" colspan=\"2\">A drawing is given. Angle Y is our reference angle, y is opposite side, z is adjacent side, and x=14 is the hypotenuse.<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 33.5227%; height: 14px;\">2. <strong>Identify<\/strong> what we are looking for.<\/td>\r\n<td style=\"width: 33.3333%; height: 14px;\">a) the opposite side<\/td>\r\n<td style=\"width: 33.3333%; height: 14px;\">b) adjacent side<\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"width: 33.5227%; height: 30px;\">3.<strong>Label<\/strong> what we are looking for by choosing a variable to represent it.<\/td>\r\n<td style=\"width: 33.3333%; height: 30px;\">y=?<\/td>\r\n<td style=\"width: 33.3333%; height: 30px;\">z=?<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 33.5227%; height: 14px;\">4. <strong>Find<\/strong> the required\u00a0trigonometric ratio.<\/td>\r\n<td style=\"width: 33.3333%; height: 14px;\">sin 35\u00b0 =\u00a0 \\(\\frac{y}{14}\\)<\/td>\r\n<td style=\"width: 33.3333%; height: 14px;\">cos 35\u00b0 =\u00a0 \\(\\frac{z}{14}\\)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"width: 33.5227%; height: 30px;\">5. <strong>Solve<\/strong> the ratio using good algebra techniques.<\/td>\r\n<td style=\"width: 33.3333%; height: 30px;\">14 sin 35\u00b0 = y\r\n\r\n8.03 = y<\/td>\r\n<td style=\"width: 33.3333%; height: 30px;\">14 cos 35\u00b0 = z\r\n\r\n11.47 =\u00a0 z<\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"width: 33.5227%; height: 30px;\">6. <strong>Check<\/strong> the answer in the problem and by making sure it makes sense.<\/td>\r\n<td style=\"width: 33.3333%; height: 30px;\">0.57 \\(\\stackrel{?}{=}\\) 8.03 \\(\\div\\) 14\r\n\r\n0.57 = 0.57 \\(\\checkmark\\)<\/td>\r\n<td style=\"width: 33.3333%; height: 30px;\">0.82 \\(\\stackrel{?}{=}\\) 11.47 \\(\\div\\) 14\r\n\r\n0.82 = 0.82 \\(\\checkmark\\)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"width: 33.5227%; height: 30px;\">7. <strong>Answer<\/strong> the question with a complete sentence.<\/td>\r\n<td style=\"width: 33.3333%; height: 30px;\">The opposite side is 8.03<\/td>\r\n<td style=\"width: 33.3333%; height: 30px;\">The adjacent side is 11.47<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 5<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n\u00a0Find the missing sides. Round\u00a0 your final answer to one decimal place.\r\n\r\n<img class=\"alignnone wp-image-9665\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/Fig-16.png\" alt=\"\" width=\"209\" height=\"204\" \/>\r\n<div id=\"fs-id1296408\" data-type=\"solution\"><details open=\"open\"><summary>Show answer<\/summary>a = 20.2\r\n\r\nb = 16.4\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 6<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the hypotenuse. Round your final answer to one decimal place.\r\n\r\n<img class=\"alignnone wp-image-9709\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/Fig-25-300x179-1.png\" alt=\"\" width=\"283\" height=\"169\" \/>\r\n\r\n<strong>Solution<\/strong>\r\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%; height: 221px;\" border=\"0\">\r\n<tbody>\r\n<tr style=\"height: 45px;\">\r\n<td>1. <strong>Read <\/strong>the problem and make sure all the words and ideas are understood. Draw the right triangle and label\u00a0 the given parts.<\/td>\r\n<td>A drawing is given. Angle S is our reference angle,\u00a0 s is opposite side, r = 4 is the adjacent side, and p is the hypotenuse<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td>2. <strong>Identify <\/strong>what we are looking for.<\/td>\r\n<td>\u00a0the hypotenuse<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td>3.<strong>Label <\/strong>what we are looking for by choosing a variable to represent it.<\/td>\r\n<td>\u00a0p=?<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td>4. <strong>Find<\/strong> the required\u00a0trigonometric ratio.<\/td>\r\n<td>\u00a0cos 32\u00b0 =\u00a0 \\(\\frac{4}{p}\\)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 60px;\">\r\n<td>5. <strong>Solve <\/strong>the ratio using good algebra techniques.<\/td>\r\n<td>0.8480 = \\(\\frac{4}{p}\\)\r\n\r\np = 4.7170\r\n\r\nRounding the ratios to 4 decimal places<\/td>\r\n<\/tr>\r\n<tr style=\"height: 60px;\">\r\n<td>6. <strong>Check <\/strong>the answer in the problem and by making sure it makes sense.<\/td>\r\n<td>0.8480 \\(\\stackrel{?}{=}\\) \\(\\frac{4}{4.7170}\\)\r\n\r\n0.8480 = 0.8480 \\(\\checkmark\\)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td>7. <strong>Answer <\/strong>the question with a complete sentence.<\/td>\r\n<td>The hypotenuse is 4.7\r\n\r\nRound my final answer to one decimal place.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 6<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the hypotenuse. Round your final answer to one decimal place.\r\n\r\n<img class=\"alignnone size-medium wp-image-9710\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/Fig-27-300x174-1.png\" alt=\"\" width=\"300\" height=\"174\" \/>\r\n<div id=\"fs-id1296409\" data-type=\"solution\"><details open=\"open\"><summary>Show answer<\/summary>p = 22.7\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<h1>Finding Missing Angles of a Right Triangle<\/h1>\r\nSometimes we have a right triangle with only the sides given. How can we find the missing angles? To find the missing angles, we use the inverse of the trigonometric ratios. The inverse buttons sin<sup>-1<\/sup>, cos<sup>-1<\/sup>, and tan<sup>-1<\/sup> are on your scientific calculator.\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 7<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the angles. Round your final answer to one decimal place.\r\n\r\na)\u00a0 sin A = 0.5\r\n\r\nb)\u00a0 cos B = 0.9735\r\n\r\nc)\u00a0 tan C = 2.89358\r\n\r\n<strong>Solution<\/strong>\r\n\r\nUse your calculator and press the 2nd FUNCTION key and then press the SIN, COS, or TAN key\r\n\r\na)\u00a0 A = sin<sup>-1<\/sup>0.5\r\n\r\n\\(\\angle A\\) = 30\u00b0\r\n\r\nb)\u00a0 B = cos<sup>-1<\/sup>0.9735\r\n\r\n\\(\\angle B\\)\u00a0 = 13.2\u00b0\u00a0 \u00a0 \u00a0Rounded to one decimal place\r\n\r\nc) C = tan<sup>-1<\/sup>2.89358\r\n\r\n\\(\\angle C\\) = 70.9\u00b0\u00a0 \u00a0 Rounded to one decimal place\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 7<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the angles. Round your final answer to one decimal place.\r\n\r\na)\u00a0 sin X = 1\r\n\r\nb)\u00a0 cos Y = 0.375\r\n\r\nc)\u00a0 tan Z = 1.676767\r\n<div id=\"fs-id1296411\" data-type=\"solution\"><details open=\"open\"><summary>Show answer<\/summary>a)\u00a0 \\(\\angle X\\)\u00a0 = 90\u00b0\r\n\r\nb)\u00a0 \\(\\angle Y\\) \u00a0= 68\u00b0\r\n\r\nc)\u00a0 \\(\\angle Z\\) \u00a0= 59.2\u00b0\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\nIn the example below we have a right triangle with two sides given. Our acute angles are missing. Let us see what the steps are to find the missing angles.\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 8<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the missing \\(\\angle T\\)\u00a0. Round your final answer to one decimal place.\r\n\r\n<img class=\"alignnone wp-image-9684\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/Fig-19-2-300x216-1.png\" alt=\"\" width=\"269\" height=\"194\" \/>\r\n\r\n<strong>Solution<\/strong>\r\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\r\n<tbody>\r\n<tr>\r\n<td>1. <strong>Read<\/strong> the problem and make sure all the words and ideas are understood. Draw the right triangle and label\u00a0 the given parts.<\/td>\r\n<td>A drawing is given. Angle T is our reference angle,\u00a0 t = 7 is the opposite side,\u00a0 s is adjacent side, and r =11 is the hypotenuse<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2. <strong>Identify<\/strong> what we are looking for.<\/td>\r\n<td>\u00a0 angle T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3.<strong>Label<\/strong> what we are looking for by choosing a variable to represent it.<\/td>\r\n<td>\\(\\angle T\\)\u00a0 =?<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>4. <strong>Find<\/strong> the required\u00a0trigonometric ratio.<\/td>\r\n<td>\u00a0 sin T =\u00a0 \\(\\frac{7}{11}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>5. <strong>Solve<\/strong> the ratio using good algebra techniques.<\/td>\r\n<td>sin T = 0.6364\r\n\r\nT = sin<sup>-1<\/sup>0.6364\r\n\r\n\\(\\angle T\\)\u00a0 = 39.5239\u00b0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>6. <strong>Check<\/strong> the answer in the problem and by making sure it makes sense.<\/td>\r\n<td>sin 39.5239\u00b0\u00a0 \\(\\stackrel{?}{=}\\) 0.6364\r\n\r\n0.6364 = 0.6364 \\(\\checkmark\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>7. <strong>Answer<\/strong> the question with a complete sentence.<\/td>\r\n<td>The missing angle T is\u00a0 39.5\u00b0.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 8<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the missing angle X. Round your final answer to one decimal place.\r\n\r\n<img class=\"alignnone wp-image-9689\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/Fig-21-300x178-1.png\" alt=\"\" width=\"276\" height=\"164\" \/>\r\n<div id=\"fs-id1296413\" data-type=\"solution\"><details open=\"open\"><summary>Show answer<\/summary>20.1\u00b0\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 9<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the missing angle A. Round your final answer to one decimal place.\r\n\r\n<img class=\"alignnone wp-image-9688\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/Fig-20.png\" alt=\"\" width=\"208\" height=\"255\" \/>\r\n\r\n<strong>Solution<\/strong>\r\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\r\n<tbody>\r\n<tr>\r\n<td>1. <strong>Read <\/strong>the problem and make sure all the words and ideas are understood. Draw the right triangle and label\u00a0 the given parts.<\/td>\r\n<td>A drawing is given. Angle A is our reference angle,\u00a0 a = 9 is the opposite side,\u00a0 c = 5\u00a0 is the adjacent side, and b is the hypotenuse<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2. <strong>Identify <\/strong>what we are looking for.<\/td>\r\n<td>\u00a0 angle A<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3.<strong>Label <\/strong>what we are looking for by choosing a variable to represent it.<\/td>\r\n<td>\\(\\angle A\\)\u00a0 =?<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>4. <strong>Find<\/strong> the required\u00a0trigonometric ratio.<\/td>\r\n<td>\u00a0\u00a0tan A =\u00a0 \\(\\frac{9}{5}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>5. <strong>Solve <\/strong>the ratio using good algebra techniques.<\/td>\r\n<td>\u00a0tan A = 1.8\r\n\r\nA = tan<sup>-1<\/sup> 1.8\r\n\r\n\\(\\angle A\\)\u00a0 = 60.9\u00b0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>6. <strong>Check<\/strong> the answer in the problem and by making sure it makes sense.<\/td>\r\n<td>\u00a0tan 60.9\u00b0\u00a0 \\(\\stackrel{?}{=}\\) 1.8\r\n\r\n1.8 = 1.8 \\(\\checkmark\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>7. <strong>Answer <\/strong>the question with a complete sentence.<\/td>\r\n<td>The missing angle A is\u00a0 60.9\u00b0.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 9<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the missing angle C. Round your final answer to one decimal place.\r\n\r\n<img class=\"alignnone wp-image-9688\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/Fig-20.png\" alt=\"\" width=\"182\" height=\"224\" \/>\r\n<div id=\"fs-id1296415\" data-type=\"solution\"><details open=\"open\"><summary>Show answer<\/summary>29.1\u00b0\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<h1>Solving a Right Triangle<\/h1>\r\nFrom the section before we know that any triangle has three sides and three interior angles. In a right triangle, when all six parts of the triangle are known, we say that the right triangle is solved.\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 10<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSolve the right triangle. Round your final answer to one decimal place.\r\n\r\n<img class=\"alignnone wp-image-9715\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/Fig-28-300x272-1.png\" alt=\"\" width=\"259\" height=\"235\" \/>\r\n\r\n<strong>Solution<\/strong>\r\n\r\nSince the sum of angles in any triangle is 180\u00b0, the measure of angle B can be easy calculated.\r\n\r\n\\(\\angle B\\)\u00a0 =\u00a0 180\u00b0 \u2212 90\u00b0 \u2212 42\u00b0\r\n\r\n\\(\\angle B\\)\u00a0 = 48\u00b0\r\n<table style=\"border-collapse: collapse; width: 100%; height: 221px;\" border=\"0\">\r\n<tbody>\r\n<tr style=\"height: 46px;\">\r\n<td style=\"width: 33.5227%; height: 46px;\">1. <strong>Read<\/strong> the problem and make sure all the words and ideas are understood. Draw the right triangle and label the given parts.<\/td>\r\n<td style=\"width: 33.3333%; height: 46px;\" colspan=\"2\">A drawing is given. Angle A is our reference angle,\u00a0 a = 8 is the opposite side, b is the adjacent side, and c is the hypotenuse.<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 33.5227%; height: 14px;\">2. <strong>Identify<\/strong> what we are looking for.<\/td>\r\n<td style=\"width: 33.3333%; height: 14px;\">\u00a0a)\u00a0 \u00a0adjacent side<\/td>\r\n<td style=\"width: 33.3333%; height: 14px;\">b) hypotenuse<\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"width: 33.5227%; height: 30px;\">3.<strong>Label<\/strong> what we are looking for by choosing a variable to represent it.<\/td>\r\n<td style=\"width: 33.3333%; height: 30px;\">b = ?<\/td>\r\n<td style=\"width: 33.3333%; height: 30px;\">c = ?<\/td>\r\n<\/tr>\r\n<tr style=\"height: 41px;\">\r\n<td style=\"width: 33.5227%; height: 41px;\">4. <strong>Find<\/strong> the required\u00a0trigonometric ratio.<\/td>\r\n<td style=\"width: 33.3333%; height: 41px;\">tan 42\u00b0 =\u00a0 \\(\\frac{8}{b}\\)<\/td>\r\n<td style=\"width: 33.3333%; height: 41px;\">sin 42\u00b0 =\u00a0 \\(\\frac{8}{c}\\)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"width: 33.5227%; height: 30px;\">5. <strong>Solve<\/strong> the ratio using good algebra techniques.<\/td>\r\n<td style=\"width: 33.3333%; height: 30px;\">0.9004 = \\(\\frac{8}{b}\\)\r\n\r\n0.9004 b = 8\r\n\r\nb = 8.8849<\/td>\r\n<td style=\"width: 33.3333%; height: 30px;\">0.6691 = \\(\\frac{8}{c}\\)\r\n\r\n0.6691 c = 8\r\n\r\nc = 11.9563<\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"width: 33.5227%; height: 30px;\">6. <strong>Check<\/strong> the answer in the problem and by making sure it makes sense.<\/td>\r\n<td style=\"width: 33.3333%; height: 30px;\">tan 42 \u00b0\u00a0 \\(\\stackrel{?}{=}\\) \\(\\frac{8}{8.8849}\\)\r\n\r\n0.9 = 0.9\u00a0 \\(\\checkmark\\)<\/td>\r\n<td style=\"width: 33.3333%; height: 30px;\">sin 42\u00b0 \\(\\stackrel{?}{=}\\) \u00a0\\(\\frac{8}{11.9563}\\)\r\n\r\n0.6691 = 0.6691 \\(\\checkmark\\)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"width: 33.5227%; height: 30px;\">7. <strong>Answer<\/strong> the question with a complete sentence.<\/td>\r\n<td style=\"width: 33.3333%; height: 30px;\">The adjacent side is 8.9.\r\n\r\nRounded to one decimal place.<\/td>\r\n<td style=\"width: 33.3333%; height: 30px;\">The hypotenuse is 12<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe solved the right triangle\r\n\r\n\\(\\angle A\\) = 42\u00b0\r\n\r\n\\(\\angle B\\) = 48\u00b0\r\n\r\n\\( \\angle C\\) = 90\u00b0\r\n\r\na = 8\r\n\r\nb = 8.9\r\n\r\nc = 12\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 10<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSolve the right triangle. Round your final answer to one decimal place.\r\n\r\n<img class=\"alignnone wp-image-9716\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/Fig-29-300x252-1.png\" alt=\"\" width=\"258\" height=\"217\" \/>\r\n\r\n\\(\\angle A\\)= 21\u00b0\r\n\r\n\\(\\angle B\\) = 69\u00b0\r\n\r\n\\(\\angle C\\) = 90\u00b0\r\n<div id=\"fs-id1296417\" data-type=\"solution\"><details open=\"open\"><summary>Show answer<\/summary>a = 6\r\n\r\nb = 15.6\r\n\r\nc = 16.7\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 11<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSolve the right triangle. Round to two decimal places.\r\n\r\n<img class=\"alignnone size-medium wp-image-9721\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/Fig-31-300x152-1.png\" alt=\"\" width=\"300\" height=\"152\" \/>\r\n\r\n<strong>Solution<\/strong>\r\n<table style=\"border-collapse: collapse; width: 100%; height: 254px;\" border=\"0\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 48.5227%;\">1. <strong>Read <\/strong>the problem and make sure all the words and ideas are understood. Draw the right triangle and label\u00a0 the given parts.<\/td>\r\n<td style=\"width: 66.6666%; height: 14px;\" colspan=\"2\">A drawing is given. Let angle D be our reference angle,\u00a0 d = 4 is the opposite side,\u00a0 f is the adjacent side, and e = 9 is the hypotenuse<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 48.5227%;\">2. <strong>Identify <\/strong>what we are looking for.<\/td>\r\n<td style=\"width: 33.3333%; height: 14px;\">a) angle D<\/td>\r\n<td style=\"width: 33.3333%; height: 14px;\">b) adjacent<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 48.5227%;\">3.<strong>Label <\/strong>what we are looking for by choosing a variable to represent it.<\/td>\r\n<td style=\"width: 33.3333%; height: 14px;\">\\(\\angle D\\)\u00a0 =?<\/td>\r\n<td style=\"width: 33.3333%; height: 14px;\">\u00a0f = ?<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 48.5227%;\">4. <strong>Find<\/strong> the required\u00a0trigonometric ratio.<\/td>\r\n<td style=\"width: 33.3333%; height: 14px;\">sin D =\u00a0 \\(\\frac{4}{9}\\)<\/td>\r\n<td style=\"width: 33.3333%; height: 14px;\">4<sup>2<\/sup>\u00a0 +\u00a0 f<sup>2<\/sup> = 9<sup>2<\/sup><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 48.5227%;\">5. <strong>Solve <\/strong>the ratio using good algebra techniques.<\/td>\r\n<td style=\"width: 33.3333%; height: 92px;\">sin D = 0.4444\r\n\r\nD = sin<sup>-1<\/sup>0.4444\r\n\r\n\\(\\angle D\\)\u00a0 = 26.3850\u00b0<\/td>\r\n<td style=\"width: 33.3333%; height: 92px;\">16 + f<sup>2<\/sup> = 81\r\n\r\nf<sup>2<\/sup> = 81 - 16\r\n\r\nf<sup>2<\/sup> = 65\r\n\r\nf = square root of 65\r\n\r\nf = 8.06<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 48.5227%;\">6. <strong>Check <\/strong>the answer in the problem and by making sure it makes sense.<\/td>\r\n<td style=\"width: 33.3333%; height: 76px;\">sin 26.3850\u00b0 \\(\\stackrel{?}{=}\\) \u00a0\\(\\frac{4}{9}\\)\r\n\r\n0.4444 =0.4444\u00a0 \\(\\checkmark\\)<\/td>\r\n<td style=\"width: 33.3333%; height: 76px;\">4<sup>2<\/sup>\u00a0 +\u00a0 8.06<sup>2<\/sup> \\(\\stackrel{?}{=}\\) 9<sup>2<\/sup>\r\n\r\n81 = 81 \\(\\checkmark\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 48.5227%;\">7. <strong>Answer <\/strong>the question with a complete sentence.<\/td>\r\n<td style=\"width: 33.3333%; height: 30px;\">The missing angle D is 26.39\u00b0.<\/td>\r\n<td style=\"width: 33.3333%; height: 30px;\">The adjacent side is 8.06\u00a0 \u00a0Rounded to two decimal places<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe missing angle F = 180\u00b0 - 90\u00b0 - 26.39\u00b0 = 63.64\u00b0\r\n\r\nWe solved the right triangle\r\n\r\n\\(\\angle D\\) = 26.39\u00b0\r\n\r\n\\(\\angle E\\) = 90\u00b0\r\n\r\n\\(\\angle F\\) = 63.61\u00b0\r\n\r\nd = 4\r\n\r\ne = 9\r\n\r\nf = 8.06\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 11<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSolve the right triangle. Round to one decimal place.\r\n\r\n<img class=\"alignnone size-medium wp-image-9722\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/Fig-32-300x147-1.png\" alt=\"\" width=\"300\" height=\"147\" \/>\r\n\r\n\\(\\angle D\\) = 29.3\u00b0\r\n\r\n\\(\\angle E\\) = 90\u00b0\r\n\r\n\\(\\angle F\\) = 60.7\u00b0\r\n<div id=\"fs-id1296419\" data-type=\"solution\"><details open=\"open\"><summary>Show answer<\/summary>d = 29.4\r\n\r\ne = 18.4\r\n\r\nf = 60.6\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<h1 data-type=\"title\">Solve Applications Using Trigonometric Ratios<\/h1>\r\nIn the previous examples we were able to find missing sides and missing angles of a right triangle. Now, let's use the trigonometric ratios to solve real-life\u00a0 problems.\r\n\r\nMany applications of trigonometric ratios involve understanding of an angle of elevation or angle of depression.\r\n\r\nThe angle of elevation is an angle between the horizontal line (ground) and the observer's line of sight.\r\n\r\n<img class=\"wp-image-11452 aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/9.2_angleOfElevation.png\" alt=\"\" width=\"463\" height=\"356\" \/>\r\n\r\n<\/div>\r\nThe angle of depression is the angle between horizontal line (that is parallel to the ground) and the observer's line of sight.\r\n\r\n<img class=\"wp-image-11451 aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/9.2_angleOfDepression.png\" alt=\"\" width=\"438\" height=\"399\" \/>\r\n<div id=\"fs-id1168345363664\" class=\"bc-section section\" data-depth=\"1\">\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 12<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nJames is standing 31 metres away from the base of the Harbour Centre in Vancouver. He looks up to the top of the building at a 78\u00b0 angle. How tall is the Harbour Centre?\r\n\r\n<strong>Solution<\/strong>\r\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\r\n<tbody>\r\n<tr>\r\n<td>1. <strong>Read <\/strong>the problem and make sure all the words and ideas are understood. Draw the right triangle and label\u00a0 the given parts.<\/td>\r\n<td><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/9.2_Example12-James-1.png\" alt=\"\" width=\"271\" height=\"342\" \/>\r\n\r\nAngle X is our reference angle,\u00a0 x is opposite side, y = 31 m is the adjacent side, and z is the hypotenuse.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2. <strong>Identify <\/strong>what we are looking for.<\/td>\r\n<td>\u00a0The opposite side<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3.<strong>Label <\/strong>what we are looking for by choosing a variable to represent it.<\/td>\r\n<td>\u00a0x=?<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>4. <strong>Find<\/strong> the required\u00a0trigonometric ratio.<\/td>\r\n<td>\u00a0tan 78\u00b0 =\u00a0 \\(\\frac{x}{31}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>5. <strong>Solve <\/strong>the ratio using good algebra techniques.<\/td>\r\n<td>4.7046 = \\(\\frac{x}{31}\\)\r\n\r\nx = 145.8426<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>6. <strong>Check <\/strong>the answer in the problem and by making sure it makes sense.<\/td>\r\n<td>4.7046 \\(\\stackrel{?}{=}\\) \\(\\frac{145.8426}{31}\\)\r\n\r\n4.7046 = 4.7046 \\(\\checkmark\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>7. <strong>Answer <\/strong>the question with a complete sentence.<\/td>\r\n<td>\u00a0The Harbour Centre is 145.8426 metres or rounded to 146 metres.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 12<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nMarta is standing 23 metres away from the base of the tallest apartment building in Prince George and looks at the top of the building at a 62\u00b0 angle. How tall is the building?\r\n<div id=\"fs-id1296422\" data-type=\"solution\"><details open=\"open\"><summary>Show answer<\/summary>43.3 metres\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 13<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nThomas is standing at the top of the building that is 45 metres high and looks at her friend that is standing on the ground, 22 metres from the base of the building. What is the angle of depression?\r\n\r\n<strong>Solution<\/strong>\r\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\r\n<tbody>\r\n<tr>\r\n<td>1. <strong>Read<\/strong> the problem and make sure all the words and ideas are understood. Draw the right triangle and label\u00a0 the given parts.<\/td>\r\n<td>&nbsp;\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/9.2_Thomas.png\" alt=\"\" width=\"374\" height=\"306\" \/>\r\n\r\nAngle Y is our reference angle,\u00a0 y = 45 m is the opposite side,\u00a0 z = 22 m\u00a0 is the adjacent side, and x is the hypotenuse<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2. <strong>Identify<\/strong> what we are looking for.<\/td>\r\n<td>\u00a0 angle Y<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3.<strong>Label<\/strong> what we are looking for by choosing a variable to represent it.<\/td>\r\n<td>\\(\\angle Y\\)\u00a0 =?<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>4. <strong>Find<\/strong> the required\u00a0trigonometric ratio.<\/td>\r\n<td>\u00a0\u00a0tan Y =\u00a0 \\(\\frac{45}{22}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>5. <strong>Solve<\/strong> the ratio using good algebra techniques.<\/td>\r\n<td>\u00a0tan Y = 2.0455\r\n\r\nY = tan <sup>-<\/sup>\u00b92.0455\r\n\r\n\\(\\angle Y\\)\u00a0 = 63.9470\u00b0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>6. <strong>Check<\/strong> the answer in the problem and by making sure it makes sense.<\/td>\r\n<td>\u00a0tan 63.9470\u00b0\u00a0 \\(\\stackrel{?}{=}\\)\u00a0 2.0455\r\n\r\n2.0455\u00a0 = 2.0455\u00a0 \\(\\checkmark\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>7. <strong>Answer<\/strong> the question with a complete sentence.<\/td>\r\n<td>The angle \u00a0of depression is\u00a0 63.9470\u00b0 or\u00a0 64\u00b0 rounded to one decimal place.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 13<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nHemanth is standing on the top of a cliff 250 feet above the ground and looks at his friend that is standing on the ground, 40 feet from the base of the cliff. What is the angle of depression?\r\n<div id=\"fs-id1296423\" data-type=\"solution\"><details open=\"open\"><summary>Show answer<\/summary>80.9\u00b0\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<h1>Key Concepts<\/h1>\r\n<ul id=\"fs-id1168345448170\" data-bullet-style=\"bullet\">\r\n \t<li>Three Basic Trigonometric Ratios: (Where \u03b8 is the measure of a reference angle measured in degrees.)\r\n<ul>\r\n \t<li>sine \u03b8 = \\(\\frac{\\text{the length of the opposite side}}{\\text{the length of the\u00a0 hypotenuse side}}\\)<\/li>\r\n \t<li>cosine \u03b8 = \\(\\frac{\\text{the length of the adjacent side}}{\\text{the length of the\u00a0 hypotenuse side}}\\)<\/li>\r\n \t<li>tangent \u03b8 = \\(\\frac{\\text{the length of the opposite side}}{\\text{the length of the\u00a0 adjacent side}}\\)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong data-effect=\"bold\">Problem-Solving Strategy for\u00a0Trigonometry Applications<\/strong>\r\n<ol id=\"fs-id1166426163525\" class=\"stepwise\" type=\"1\">\r\n \t<li><strong data-effect=\"bold\">Read<\/strong> the problem and make sure all the words and ideas are understood. Draw the right triangle and label\u00a0 the given parts.<\/li>\r\n \t<li><strong data-effect=\"bold\">Identify<\/strong> what we are looking for.<\/li>\r\n \t<li><strong data-effect=\"bold\">Label<\/strong> what we are looking for by choosing a variable to represent it.<\/li>\r\n \t<li><strong data-effect=\"bold\">Find <\/strong>the required\u00a0trigonometric ratio.<\/li>\r\n \t<li><strong data-effect=\"bold\">Solve<\/strong> the ratio using good algebra techniques.<\/li>\r\n \t<li><strong data-effect=\"bold\">Check<\/strong> the answer by substituting it back into the ratio solved in step 5 and by making sure it makes sense in the context of the problem.<\/li>\r\n \t<li><strong data-effect=\"bold\">Answer<\/strong> the question with a complete sentence.<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ul>\r\n<h1>5.2 Exercise Set<\/h1>\r\nLabel the sides of the triangle.\r\n<table style=\"border-collapse: collapse; width: 100%; height: 299px;\" border=\"0\">\r\n<tbody>\r\n<tr style=\"height: 230px;\">\r\n<td>1\r\n\r\n<img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/P16-300x151-1.png\" alt=\"\" width=\"249\" height=\"126\" \/><\/td>\r\n<td>2.\r\n\r\n<img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/P14-1-300x142-1.png\" alt=\"\" width=\"273\" height=\"129\" \/><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<ol start=\"3\">\r\n \t<li>If the reference angle in Question 1 is B, Find the adjacent ?<\/li>\r\n \t<li>If the reference angle in Question 2 is Z, find the opposite?<\/li>\r\n<\/ol>\r\nLabel the sides of the triangle and find the hypotenuse, opposite and adjacent.\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 50%;\">5.\r\n\r\n<img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/P-1-300x128-1.png\" alt=\"\" width=\"300\" height=\"128\" \/><\/td>\r\n<td style=\"width: 50%;\">\u00a06.\r\n\r\n<img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/P3-300x144-1.png\" alt=\"\" width=\"300\" height=\"144\" \/><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nUse your calculator to find the given ratios. Round to four decimal places if necessary:\r\n<ol class=\"twocolumn\" start=\"7\">\r\n \t<li>\\(\\sin {47}^{\\circ}\\)<\/li>\r\n \t<li>\\(\\cos {82}^{\\circ}\\)<\/li>\r\n \t<li>\\(\\tan {12}^{\\circ}\\)<\/li>\r\n \t<li>\\(\\sin {30}^{\\circ}\\)<\/li>\r\n<\/ol>\r\nFor the given triangles, find the sine, cosine and tangent of the \u03b8.\r\n<table style=\"border-collapse: collapse; width: 100%; height: 148px;\" border=\"0\">\r\n<tbody>\r\n<tr style=\"height: 131px;\">\r\n<td>11.\r\n\r\n<img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/P-1-300x128-1.png\" alt=\"\" width=\"300\" height=\"128\" \/><\/td>\r\n<td>12.\r\n\r\n<img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/P3-300x144-1.png\" alt=\"\" width=\"300\" height=\"144\" \/><\/td>\r\n<\/tr>\r\n<tr style=\"height: 17px;\">\r\n<td>13.\r\n\r\n<img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/P10-300x252-1.png\" alt=\"\" width=\"249\" height=\"210\" \/><\/td>\r\n<td>14.\r\n\r\n<img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/P21.png\" alt=\"\" width=\"169\" height=\"227\" \/><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nFor the given triangles, find the missing side. Round it to one decimal place.\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\r\n<tbody>\r\n<tr>\r\n<td>15. Find the hypotenuse.\r\n\r\n<img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/P6.png\" alt=\"\" width=\"195\" height=\"269\" \/><\/td>\r\n<td>16. Find b if a = 6.\r\n\r\n<img class=\"alignnone size-medium wp-image-1906\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/05\/Screenshot-2021-11-17-at-10.20.36-AM-242x300.png\" alt=\"\" width=\"242\" height=\"300\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>17. Find the opposite. <img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/P20-300x146-1.png\" alt=\"\" width=\"300\" height=\"146\" \/><\/td>\r\n<td>18. Find the adjacent. <img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/P19-300x151-1.png\" alt=\"\" width=\"300\" height=\"151\" \/><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nFor the given triangles, find the missing sides. Round it to one decimal place.\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\r\n<tbody>\r\n<tr>\r\n<td>19.\r\n\r\n<img class=\"\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/P8-300x251-1.png\" alt=\"\" width=\"244\" height=\"204\" \/><\/td>\r\n<td>20.\r\n\r\n<img class=\"\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/P9-300x244-1.png\" alt=\"\" width=\"259\" height=\"211\" \/><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSolve the triangles. Round to one decimal place.\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\r\n<tbody>\r\n<tr style=\"height: 259px;\">\r\n<td style=\"width: 49.9086%;\">21.\r\n\r\n<img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/P22-300x147-1.png\" alt=\"\" width=\"280\" height=\"137\" \/><\/td>\r\n<td style=\"width: 49.9086%;\">\u00a022.\r\n\r\n<img class=\"alignnone wp-image-1904 \" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/05\/Screenshot-2021-11-17-at-8.39.55-AM-e1637167343689-239x300.png\" alt=\"\" width=\"212\" height=\"266\" \/><\/td>\r\n<\/tr>\r\n<tr style=\"height: 239px;\">\r\n<td style=\"width: 49.9086%;\">23.\r\n\r\n<img class=\"\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/P7-300x223-1.png\" alt=\"\" width=\"273\" height=\"203\" \/><\/td>\r\n<td style=\"width: 49.9086%;\">24.\r\n\r\n<img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/P15-1-300x169-1.png\" alt=\"\" width=\"273\" height=\"154\" \/><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<ol start=\"25\">\r\n \t<li>Kim stands 75 metres from the bottom of a tree and looks up at the top of the tree at a 48\u00b0 angle. How tall is the tree?<\/li>\r\n \t<li>A tree makes a shadow that is 6 metres long when the angle of elevation to the sun is 52\u00b0. How tall is the tree?<\/li>\r\n \t<li>A ladder that is 15 feet is leaning against a house and makes a 45\u00b0 angle with the ground. How far is the base of the ladder from the house?<\/li>\r\n \t<li>Roxanne is flying a kite and has let out 100 feet of string. The angle of elevation with the ground is 38\u00b0. How high is her kite above the ground?<\/li>\r\n \t<li>Marta is flying a kite and has let out 28 metres of string. If the kite is 10 metres above the ground, what is the angle of elevation?<\/li>\r\n \t<li>An airplane takes off from the ground at the angle of 25\u00b0. If the airplane traveled 200 kilometres, how high above the ground is it?<\/li>\r\n<\/ol>\r\n<h1>Answers:<\/h1>\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\r\n<tbody>\r\n<tr style=\"height: 238px;\">\r\n<td style=\"width: 43.6929%;\">1.\r\n\r\n<img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/Qn1-answers.png\" alt=\"\" width=\"285\" height=\"130\" \/><\/td>\r\n<td style=\"width: 14.4424%;\">3. c<\/td>\r\n<td style=\"width: 41.6819%;\">5.\r\n\r\n<img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/Q-5-ans.png\" alt=\"\" width=\"328\" height=\"150\" \/>\r\n\r\ng is opposite , f is adjacent, and e is hypotenuse<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16px;\">\r\n<td style=\"width: 43.6929%;\">7. 0.7314<\/td>\r\n<td style=\"width: 14.4424%;\">9. 0.2126<\/td>\r\n<td style=\"width: 41.6819%;\">11.\r\n\r\n<img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/Q-5-ans.png\" alt=\"\" width=\"328\" height=\"150\" \/>\r\n\r\nsin \u03b8\u00a0 = \\(\\frac{g}{e}\\), cos \u03b8\u00a0 = \\(\\frac{f}{e}\\), tan \u03b8\u00a0 = \\(\\frac{g}{f}\\)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<ol>\r\n \t<li value=\"13\">sin \u03b8\u00a0 = \\(\\frac{s}{r}\\), cos \u03b8\u00a0 = \\(\\frac{t}{r}\\), tan \u03b8\u00a0 = \\(\\frac{s}{t}\\)<\/li>\r\n \t<li value=\"15\">b = 19.8<\/li>\r\n \t<li value=\"17\">c = 12<\/li>\r\n \t<li value=\"19\">y = 19.3, z = 8.2<\/li>\r\n \t<li value=\"21\">\\(\\angle B\\) = 61\u00b0\\(\\angle C\\) = 29\u00b0\\(\\angle D\\) = 90\u00b0b = 38.5c = 21.3d = 44<\/li>\r\n \t<li value=\"23\">\\( \\angle T\\) = 36.9\u00b0\\(\\angle R\\) = 90\u00b0\\(\\angle S\\) = 53.1\u00b0t = 15r = 25s = 20<\/li>\r\n \t<li value=\"25\">83.3 m<\/li>\r\n \t<li value=\"27\">10.6 ft<\/li>\r\n \t<li value=\"29\">20.9\u00b0<\/li>\r\n<\/ol>\r\n<h1><strong>Attribution:<\/strong><\/h1>\r\nThis chapter has been adapted from \u201cSolve Applications: Sine, Cosine and Tangent Ratios\u201d in <a href=\"https:\/\/opentextbc.ca\/introalgebra\/\">Introductory Algebra<\/a>\u00a0by Izabela Mazur, which is under a <a href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY 4.0 Licence<\/a>. See the Adaptation Statement for more information.","rendered":"<div data-type=\"note\"><\/div>\n<div data-type=\"note\">\n<div class=\"textbox textbox--learning-objectives\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Learning Objectives<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>By the end of this section it is expected that you will be able to<\/p>\n<ul>\n<li>Find missing side of a right triangle using sine, cosine, or tangent ratios<\/li>\n<li>Find missing angle of a right triangle using sine, cosine, or tangent ratios<\/li>\n<li>Solve applications using right angle trigonometry<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\">Now, that we know the fundamentals of algebra and geometry associated with a right triangle, we can start exploring trigonometry. Many real life problems can be represented and solved using right angle trigonometry.<\/div>\n<h1 id=\"fs-id1168345273636\" data-type=\"note\">Sine, Cosine, and Tangent Ratios<\/h1>\n<div id=\"fs-id1168345363664\" class=\"bc-section section\" data-depth=\"1\">\n<p>We know that any right triangle has three sides and a right angle. The side opposite to the right angle is called the hypotenuse. The other two angles in a right triangle are acute angles (with a measure less than 90 degrees). One of those angles we call reference angle and we use \u03b8 (theta) to represent it.<\/p>\n<p>The hypotenuse is always the longest side of a right triangle. The other two sides are called opposite side and adjacent side. The names of those sides depends on which of the two acute angles is being used as a reference angle.<\/p>\n<div id=\"fs-id129640054\">\n<figure id=\"attachment_9622\" aria-describedby=\"caption-attachment-9622\" style=\"width: 420px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-9622\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/Fig-1-300x188-1.png\" alt=\"\" width=\"420\" height=\"263\" \/><figcaption id=\"caption-attachment-9622\" class=\"wp-caption-text\">Figure 1.<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<p>In the right triangle each side is labeled with a lowercase letter to match the uppercase letter of the opposite vertex.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Label the sides of the triangle and find the hypotenuse, opposite, and adjacent.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-9639\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/Fig-9-2.png\" alt=\"\" width=\"216\" height=\"189\" \/><\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>We labeled\u00a0 the sides with a lowercase letter to match the uppercase letter of the opposite vertex.<\/p>\n<p>c is hypotenuse<\/p>\n<p>a is opposite<\/p>\n<p>b is adjacent<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-9640\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/Fig-10-1.png\" alt=\"\" width=\"205\" height=\"188\" \/><\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Label the sides of the triangle and find the hypotenuse, opposite and adjacent.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-9629\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/Fig-6-300x201-1.png\" alt=\"\" width=\"231\" height=\"155\" \/><\/p>\n<div id=\"fs-id1296401\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p>y is hypotenuse<\/p>\n<p>z is opposite<\/p>\n<p>x is adjacent<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<h1 id=\"fs-id1168345363664\" data-depth=\"1\">Trigonometric Ratios<\/h1>\n<p>Trigonometric ratios are the ratios of the sides in the right triangle. For any right triangle we can define three basic trigonometric ratios: sine, cosine, and tangent.<\/p>\n<p>Let us refer to <a href=\"#fs-id129640054\">Figure 1<\/a> and define the three basic trigonometric ratios as:<\/p>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Three Basic Trigonometric Ratios<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ul>\n<li>sine \u03b8 = <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-264a6a31a1d2f7f3d93f6495d99aac3d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#108;&#101;&#110;&#103;&#116;&#104;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#111;&#112;&#112;&#111;&#115;&#105;&#116;&#101;&#32;&#115;&#105;&#100;&#101;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#108;&#101;&#110;&#103;&#116;&#104;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#32;&#104;&#121;&#112;&#111;&#116;&#101;&#110;&#117;&#115;&#101;&#32;&#115;&#105;&#100;&#101;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"201\" style=\"vertical-align: -9px;\" \/><\/li>\n<li>cosine \u03b8 = <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-857da9e2cb11012174ba178d506c1288_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#108;&#101;&#110;&#103;&#116;&#104;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#97;&#100;&#106;&#97;&#99;&#101;&#110;&#116;&#32;&#115;&#105;&#100;&#101;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#108;&#101;&#110;&#103;&#116;&#104;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#32;&#104;&#121;&#112;&#111;&#116;&#101;&#110;&#117;&#115;&#101;&#32;&#115;&#105;&#100;&#101;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"201\" style=\"vertical-align: -9px;\" \/><\/li>\n<li>tangent \u03b8 = <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-40ec3e42f10b611cf2916f1f8fdab7d7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#108;&#101;&#110;&#103;&#116;&#104;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#111;&#112;&#112;&#111;&#115;&#105;&#116;&#101;&#32;&#115;&#105;&#100;&#101;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#108;&#101;&#110;&#103;&#116;&#104;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#32;&#97;&#100;&#106;&#97;&#99;&#101;&#110;&#116;&#32;&#115;&#105;&#100;&#101;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"184\" style=\"vertical-align: -9px;\" \/><\/li>\n<\/ul>\n<p>Where \u03b8 is the measure of a reference angle measured in degrees.<\/p>\n<\/div>\n<\/div>\n<p>Very often we use the abbreviations for sine, cosine, and tangent ratios.<\/p>\n<ul>\n<li>sin \u03b8 = <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-2dc931ea66f1240a746487bdcd386c4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#111;&#112;&#112;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#104;&#121;&#112;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"23\" style=\"vertical-align: -9px;\" \/><\/li>\n<li>cos \u03b8 = <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-10aac2fba6db8f5738555aec18d9f5ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#100;&#106;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#104;&#121;&#112;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"23\" style=\"vertical-align: -9px;\" \/><\/li>\n<li>tan \u03b8 = <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-4f3a1211bc0bba9e6b5d5f573a456ffb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#111;&#112;&#112;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#100;&#106;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"23\" style=\"vertical-align: -9px;\" \/><\/li>\n<\/ul>\n<p>Some people remember the definition of the trigonometric ratios as SOH CAH TOA.<\/p>\n<p>Let&#8217;s use the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-be0f8a277df93e7bac1f9fad073a9d23_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#68;&#69;&#70;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"58\" style=\"vertical-align: 0px;\" \/> from Example 1 to find the three ratios.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>For the given triangle find the sine, cosine and tangent ratio.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-9626\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/Fig-5-300x252-1.png\" alt=\"\" width=\"231\" height=\"194\" \/><\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>sin \u03b8 = <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-cef6ed083da81d1d1e16d3c9a20f944a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#102;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#100;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"8\" style=\"vertical-align: -6px;\" \/><\/p>\n<p>cos \u03b8 = <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ee62d1ed1fb46979dab93d81653b5d13_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#101;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#100;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"8\" style=\"vertical-align: -6px;\" \/><\/p>\n<p>tan \u03b8 = <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-23157696d1a9a1a5b0210db25061abaf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#102;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#101;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>For the given triangle find the sine cosine and tangent ratio.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-9629\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/Fig-6-300x201-1.png\" alt=\"\" width=\"231\" height=\"155\" \/><\/p>\n<\/div>\n<div class=\"textbox__content\">\n<div id=\"fs-id1296403\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p>sin \u03b8 = <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-faaa442f82eb64308e384d4dec24dc90_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#122;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#121;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"8\" style=\"vertical-align: -9px;\" \/><\/p>\n<p>cos \u03b8 = <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-f8e0bff2f527dade524f2121836dbb97_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#121;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"8\" style=\"vertical-align: -9px;\" \/><\/p>\n<p>tan \u03b8 = <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d1b6b991250864e9248a3b8c15c75c7b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#122;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"8\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<p>In Example 2, our reference angles can be <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0e1bd510bc2435d28df8b8ff340d980b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#69;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: 0px;\" \/> \u00a0or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-225d2eda571002518f953558c776be31_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#70;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: 0px;\" \/> . Using the definition of trigonometric ratios, we can write sinE = <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ee62d1ed1fb46979dab93d81653b5d13_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#101;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#100;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"8\" style=\"vertical-align: -6px;\" \/> , cosE = <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-cef6ed083da81d1d1e16d3c9a20f944a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#102;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#100;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"8\" style=\"vertical-align: -6px;\" \/>, and tanE = <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-aca49bff5657630dbdc5e3b4154cee57_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#101;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#102;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"7\" style=\"vertical-align: -6px;\" \/>.<\/p>\n<p>When calculating we will usually round the ratios to four decimal places and at the end our final answer to one decimal place unless stated otherwise.<\/p>\n<div id=\"fs-id1168345363664\" class=\"bc-section section\" data-depth=\"1\">\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 3<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>For the given triangle find the sine, cosine and tangent ratios. If necessary round to four decimal places.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-9642\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/Fig-11-300x194-1.png\" alt=\"\" width=\"252\" height=\"163\" \/><\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>We have two possible reference angles: R an S.<\/p>\n<p>Using the definitions, the trigonometric ratios for angle R are:<\/p>\n<ul>\n<li>sin R= <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c995a9c919f066bf6863b7f22d9cc88f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> = 0.8<\/li>\n<li>cos R= <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-56fd13ae94393ab95b21e4d3651f1ec2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> = 0.6<\/li>\n<li>tan R=<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0d50443e9b625e846a8f38e891bba33d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> = 1.3333&#8230;<\/li>\n<\/ul>\n<p>Using the definitions, the trigonometric ratios for angle S:<\/p>\n<ul>\n<li>sin S = <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-56fd13ae94393ab95b21e4d3651f1ec2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> = 0.6<\/li>\n<li>cos S = <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c995a9c919f066bf6863b7f22d9cc88f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> = 0.8<\/li>\n<li>tan S = <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-469181e98c8ec4b1bce8f57ea4ba4e31_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> = 0.75<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 3<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>For the given triangle find the sine, cosine, and tangent ratios. If necessary round to four decimal places.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-9666\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/Fig-17-300x141-1.png\" alt=\"\" width=\"270\" height=\"127\" \/><\/p>\n<div id=\"fs-id1296405\" data-type=\"solution\">\n<details open=\"open\">\n<summary>Show answer<\/summary>\n<ul>\n<li>sin F = <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d7808622860c3d766e41b1ea4ac20825_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#49;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -7px;\" \/> = 0.8<\/li>\n<li>cos F = <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-601b4e8a9699caeba2ea4963b029072a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#49;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -7px;\" \/> =0.6<\/li>\n<li>tan F = <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-413512946ef62757d742dc6e83235ad9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> = 1.3333&#8230;<\/li>\n<li>sin D =<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-601b4e8a9699caeba2ea4963b029072a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#49;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -7px;\" \/> = 0.6<\/li>\n<li>cos D =<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d7808622860c3d766e41b1ea4ac20825_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#49;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -7px;\" \/> = 0.8<\/li>\n<li>tan D =<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-8d9511690ce9e3e1d1d8e0fa566bd68b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> = 0.75<\/li>\n<\/ul>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<p>Now, let us use a scientific calculator to find the trigonometric ratios.\u00a0 Can you find the sin, cos, and tan buttons on your calculator? To find the trigonometric ratios make sure your calculator is in Degree Mode.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 4<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Using a calculator find the trigonometric ratios. If necessary, round to 4 decimal places.<\/p>\n<p>a) sin 30\u00b0<\/p>\n<p>b) cos 45\u00b0<\/p>\n<p>c) tan 60\u00b0<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>Make sure your calculator is in Degree Mode.<\/p>\n<p>a)\u00a0 Using a calculator find that sin 30\u00b0 = 0.5<\/p>\n<p>b)\u00a0 Using a calculator find that cos 45\u00b0 = 0.7071 Rounded to 4 decimal places.<\/p>\n<p>c)\u00a0 Using a calculator find that tan 60\u00b0 = 1.7321 Rounded to 4 decimal places.<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 4<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the trigonometric ratios. If necessary, round to 4 decimal places.<\/p>\n<p>a)\u00a0 sin 60\u00b0<\/p>\n<p>b)\u00a0 cos 30\u00b0<\/p>\n<p>c)\u00a0 tan 45\u00b0<\/p>\n<div id=\"fs-id1296407\" data-type=\"solution\">\n<details open=\"open\">\n<summary>Show answer<\/summary>\n<p>a)\u00a0 \u00a0sin 60\u00b0 = 0.8660<\/p>\n<p>b)\u00a0 \u00a0cos 30\u00b0 = 0.8660<\/p>\n<p>c)\u00a0 \u00a0tan 45\u00b0 = 1<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<h1>Finding Missing Sides of a Right Triangle<\/h1>\n<p>In this section you will be using trigonometric ratios to solve right triangle problems. We will adapt our problem solving strategy for trigonometry applications. In addition, since those problems will involve the right triangle, it is helpful to draw it (if the drawing is not given) and label it with the given information.We will include this in the first step of the problem solving strategy for trigonometry applications.<\/p>\n<div id=\"fs-id1168345512560\" class=\"howto\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">HOW TO: Solve Trigonometry Applications<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ol id=\"fs-id1166426163525\" class=\"stepwise\" type=\"1\">\n<li><strong data-effect=\"bold\">Read<\/strong> the problem and make sure all the words and ideas are understood. Draw the right triangle and label the given parts.<\/li>\n<li><strong data-effect=\"bold\">Identify<\/strong> what we are looking for.<\/li>\n<li><strong data-effect=\"bold\">Label<\/strong> what we are looking for by choosing a variable to represent it.<\/li>\n<li><strong data-effect=\"bold\">Find <\/strong>the required\u00a0trigonometric ratio.<\/li>\n<li><strong data-effect=\"bold\">Solve<\/strong> the ratio using good algebra techniques.<\/li>\n<li><strong data-effect=\"bold\">Check<\/strong> the answer by substituting it back into the ratio in step 4 and by making sure it makes sense in the context of the problem.<\/li>\n<li><strong data-effect=\"bold\">Answer<\/strong> the question with a complete sentence.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p>In the next few examples, having given the measure of one acute angle and the length of one side of the right triangle, we will solve the right triangle for the missing sides.<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 5<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>\u00a0Find the missing sides. Round\u00a0 your final answer to two decimal places<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-9654\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/Fig-15-300x232-1.png\" alt=\"\" width=\"261\" height=\"202\" \/><\/p>\n<p><strong>Solution<\/strong><\/p>\n<table style=\"border-collapse: collapse; width: 100%; height: 194px;\">\n<tbody>\n<tr style=\"height: 46px;\">\n<td style=\"width: 33.5227%; height: 46px;\">1. <strong>Read<\/strong> the problem and make sure all the words and ideas are understood. Draw the right triangle and label\u00a0 the given parts.<\/td>\n<td style=\"width: 66.6666%; height: 46px;\" colspan=\"2\">A drawing is given. Angle Y is our reference angle, y is opposite side, z is adjacent side, and x=14 is the hypotenuse.<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 33.5227%; height: 14px;\">2. <strong>Identify<\/strong> what we are looking for.<\/td>\n<td style=\"width: 33.3333%; height: 14px;\">a) the opposite side<\/td>\n<td style=\"width: 33.3333%; height: 14px;\">b) adjacent side<\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"width: 33.5227%; height: 30px;\">3.<strong>Label<\/strong> what we are looking for by choosing a variable to represent it.<\/td>\n<td style=\"width: 33.3333%; height: 30px;\">y=?<\/td>\n<td style=\"width: 33.3333%; height: 30px;\">z=?<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 33.5227%; height: 14px;\">4. <strong>Find<\/strong> the required\u00a0trigonometric ratio.<\/td>\n<td style=\"width: 33.3333%; height: 14px;\">sin 35\u00b0 =\u00a0 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-beb642ff08e42931ca869271bbc6e0c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#121;&#125;&#123;&#49;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"14\" style=\"vertical-align: -6px;\" \/><\/td>\n<td style=\"width: 33.3333%; height: 14px;\">cos 35\u00b0 =\u00a0 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7d6d617aee4720ea92d76f001e664389_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#122;&#125;&#123;&#49;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"14\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"width: 33.5227%; height: 30px;\">5. <strong>Solve<\/strong> the ratio using good algebra techniques.<\/td>\n<td style=\"width: 33.3333%; height: 30px;\">14 sin 35\u00b0 = y<\/p>\n<p>8.03 = y<\/td>\n<td style=\"width: 33.3333%; height: 30px;\">14 cos 35\u00b0 = z<\/p>\n<p>11.47 =\u00a0 z<\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"width: 33.5227%; height: 30px;\">6. <strong>Check<\/strong> the answer in the problem and by making sure it makes sense.<\/td>\n<td style=\"width: 33.3333%; height: 30px;\">0.57 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-291fc22ffe86153c31295cf8317425ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"13\" style=\"vertical-align: 2px;\" \/> 8.03 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ea134507bb0dd8200c74b73e133b84e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#105;&#118;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"13\" style=\"vertical-align: -1px;\" \/> 14<\/p>\n<p>0.57 = 0.57 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-46376855baeb2c7a9b4e7ae04ce9831b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#99;&#104;&#101;&#99;&#107;&#109;&#97;&#114;&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"13\" style=\"vertical-align: -1px;\" \/><\/td>\n<td style=\"width: 33.3333%; height: 30px;\">0.82 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-291fc22ffe86153c31295cf8317425ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"13\" style=\"vertical-align: 2px;\" \/> 11.47 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ea134507bb0dd8200c74b73e133b84e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#105;&#118;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"13\" style=\"vertical-align: -1px;\" \/> 14<\/p>\n<p>0.82 = 0.82 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-46376855baeb2c7a9b4e7ae04ce9831b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#99;&#104;&#101;&#99;&#107;&#109;&#97;&#114;&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"13\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"width: 33.5227%; height: 30px;\">7. <strong>Answer<\/strong> the question with a complete sentence.<\/td>\n<td style=\"width: 33.3333%; height: 30px;\">The opposite side is 8.03<\/td>\n<td style=\"width: 33.3333%; height: 30px;\">The adjacent side is 11.47<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 5<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>\u00a0Find the missing sides. Round\u00a0 your final answer to one decimal place.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-9665\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/Fig-16.png\" alt=\"\" width=\"209\" height=\"204\" \/><\/p>\n<div id=\"fs-id1296408\" data-type=\"solution\">\n<details open=\"open\">\n<summary>Show answer<\/summary>\n<p>a = 20.2<\/p>\n<p>b = 16.4<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 6<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the hypotenuse. Round your final answer to one decimal place.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-9709\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/Fig-25-300x179-1.png\" alt=\"\" width=\"283\" height=\"169\" \/><\/p>\n<p><strong>Solution<\/strong><\/p>\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%; height: 221px;\">\n<tbody>\n<tr style=\"height: 45px;\">\n<td>1. <strong>Read <\/strong>the problem and make sure all the words and ideas are understood. Draw the right triangle and label\u00a0 the given parts.<\/td>\n<td>A drawing is given. Angle S is our reference angle,\u00a0 s is opposite side, r = 4 is the adjacent side, and p is the hypotenuse<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td>2. <strong>Identify <\/strong>what we are looking for.<\/td>\n<td>\u00a0the hypotenuse<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td>3.<strong>Label <\/strong>what we are looking for by choosing a variable to represent it.<\/td>\n<td>\u00a0p=?<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td>4. <strong>Find<\/strong> the required\u00a0trigonometric ratio.<\/td>\n<td>\u00a0cos 32\u00b0 =\u00a0 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-065f241caf9c921ae95ac63f9a848a56_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#112;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"8\" style=\"vertical-align: -9px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 60px;\">\n<td>5. <strong>Solve <\/strong>the ratio using good algebra techniques.<\/td>\n<td>0.8480 = <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-065f241caf9c921ae95ac63f9a848a56_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#112;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"8\" style=\"vertical-align: -9px;\" \/><\/p>\n<p>p = 4.7170<\/p>\n<p>Rounding the ratios to 4 decimal places<\/td>\n<\/tr>\n<tr style=\"height: 60px;\">\n<td>6. <strong>Check <\/strong>the answer in the problem and by making sure it makes sense.<\/td>\n<td>0.8480 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-291fc22ffe86153c31295cf8317425ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"13\" style=\"vertical-align: 2px;\" \/> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-a36a7f5a9c43ee4506e2c518bca89d26_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#52;&#46;&#55;&#49;&#55;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"39\" style=\"vertical-align: -6px;\" \/><\/p>\n<p>0.8480 = 0.8480 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-46376855baeb2c7a9b4e7ae04ce9831b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#99;&#104;&#101;&#99;&#107;&#109;&#97;&#114;&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"13\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td>7. <strong>Answer <\/strong>the question with a complete sentence.<\/td>\n<td>The hypotenuse is 4.7<\/p>\n<p>Round my final answer to one decimal place.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 6<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the hypotenuse. Round your final answer to one decimal place.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-9710\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/Fig-27-300x174-1.png\" alt=\"\" width=\"300\" height=\"174\" \/><\/p>\n<div id=\"fs-id1296409\" data-type=\"solution\">\n<details open=\"open\">\n<summary>Show answer<\/summary>\n<p>p = 22.7<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<h1>Finding Missing Angles of a Right Triangle<\/h1>\n<p>Sometimes we have a right triangle with only the sides given. How can we find the missing angles? To find the missing angles, we use the inverse of the trigonometric ratios. The inverse buttons sin<sup>-1<\/sup>, cos<sup>-1<\/sup>, and tan<sup>-1<\/sup> are on your scientific calculator.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 7<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the angles. Round your final answer to one decimal place.<\/p>\n<p>a)\u00a0 sin A = 0.5<\/p>\n<p>b)\u00a0 cos B = 0.9735<\/p>\n<p>c)\u00a0 tan C = 2.89358<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>Use your calculator and press the 2nd FUNCTION key and then press the SIN, COS, or TAN key<\/p>\n<p>a)\u00a0 A = sin<sup>-1<\/sup>0.5<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c9644d971857e5b6fb1370bb8de85b92_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"26\" style=\"vertical-align: 0px;\" \/> = 30\u00b0<\/p>\n<p>b)\u00a0 B = cos<sup>-1<\/sup>0.9735<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-6150531321f68bd917536a4352de756f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: 0px;\" \/>\u00a0 = 13.2\u00b0\u00a0 \u00a0 \u00a0Rounded to one decimal place<\/p>\n<p>c) C = tan<sup>-1<\/sup>2.89358<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-199478770c6257562ee854658a4e1f50_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: 0px;\" \/> = 70.9\u00b0\u00a0 \u00a0 Rounded to one decimal place<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 7<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the angles. Round your final answer to one decimal place.<\/p>\n<p>a)\u00a0 sin X = 1<\/p>\n<p>b)\u00a0 cos Y = 0.375<\/p>\n<p>c)\u00a0 tan Z = 1.676767<\/p>\n<div id=\"fs-id1296411\" data-type=\"solution\">\n<details open=\"open\">\n<summary>Show answer<\/summary>\n<p>a)\u00a0 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b84f4cd333212df1ed0b8a1e64bc195c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#88;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"29\" style=\"vertical-align: 0px;\" \/>\u00a0 = 90\u00b0<\/p>\n<p>b)\u00a0 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-a8672f26aeb43591efd785d68a7dbfea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#89;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: 0px;\" \/> \u00a0= 68\u00b0<\/p>\n<p>c)\u00a0 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d5ceb3657ebff886968490f7f61d1ee9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#90;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"26\" style=\"vertical-align: 0px;\" \/> \u00a0= 59.2\u00b0<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<p>In the example below we have a right triangle with two sides given. Our acute angles are missing. Let us see what the steps are to find the missing angles.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 8<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the missing <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-4233306bbe710ee1fc36720ac41656d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#84;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"26\" style=\"vertical-align: 0px;\" \/>\u00a0. Round your final answer to one decimal place.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-9684\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/Fig-19-2-300x216-1.png\" alt=\"\" width=\"269\" height=\"194\" \/><\/p>\n<p><strong>Solution<\/strong><\/p>\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td>1. <strong>Read<\/strong> the problem and make sure all the words and ideas are understood. Draw the right triangle and label\u00a0 the given parts.<\/td>\n<td>A drawing is given. Angle T is our reference angle,\u00a0 t = 7 is the opposite side,\u00a0 s is adjacent side, and r =11 is the hypotenuse<\/td>\n<\/tr>\n<tr>\n<td>2. <strong>Identify<\/strong> what we are looking for.<\/td>\n<td>\u00a0 angle T<\/td>\n<\/tr>\n<tr>\n<td>3.<strong>Label<\/strong> what we are looking for by choosing a variable to represent it.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-4233306bbe710ee1fc36720ac41656d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#84;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"26\" style=\"vertical-align: 0px;\" \/>\u00a0 =?<\/td>\n<\/tr>\n<tr>\n<td>4. <strong>Find<\/strong> the required\u00a0trigonometric ratio.<\/td>\n<td>\u00a0 sin T =\u00a0 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-4ac411931c681fd21df5ea3499b79636_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#49;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>5. <strong>Solve<\/strong> the ratio using good algebra techniques.<\/td>\n<td>sin T = 0.6364<\/p>\n<p>T = sin<sup>-1<\/sup>0.6364<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-4233306bbe710ee1fc36720ac41656d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#84;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"26\" style=\"vertical-align: 0px;\" \/>\u00a0 = 39.5239\u00b0<\/td>\n<\/tr>\n<tr>\n<td>6. <strong>Check<\/strong> the answer in the problem and by making sure it makes sense.<\/td>\n<td>sin 39.5239\u00b0\u00a0 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-291fc22ffe86153c31295cf8317425ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"13\" style=\"vertical-align: 2px;\" \/> 0.6364<\/p>\n<p>0.6364 = 0.6364 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-46376855baeb2c7a9b4e7ae04ce9831b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#99;&#104;&#101;&#99;&#107;&#109;&#97;&#114;&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"13\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>7. <strong>Answer<\/strong> the question with a complete sentence.<\/td>\n<td>The missing angle T is\u00a0 39.5\u00b0.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 8<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the missing angle X. Round your final answer to one decimal place.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-9689\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/Fig-21-300x178-1.png\" alt=\"\" width=\"276\" height=\"164\" \/><\/p>\n<div id=\"fs-id1296413\" data-type=\"solution\">\n<details open=\"open\">\n<summary>Show answer<\/summary>\n<p>20.1\u00b0<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 9<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the missing angle A. Round your final answer to one decimal place.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-9688\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/Fig-20.png\" alt=\"\" width=\"208\" height=\"255\" \/><\/p>\n<p><strong>Solution<\/strong><\/p>\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td>1. <strong>Read <\/strong>the problem and make sure all the words and ideas are understood. Draw the right triangle and label\u00a0 the given parts.<\/td>\n<td>A drawing is given. Angle A is our reference angle,\u00a0 a = 9 is the opposite side,\u00a0 c = 5\u00a0 is the adjacent side, and b is the hypotenuse<\/td>\n<\/tr>\n<tr>\n<td>2. <strong>Identify <\/strong>what we are looking for.<\/td>\n<td>\u00a0 angle A<\/td>\n<\/tr>\n<tr>\n<td>3.<strong>Label <\/strong>what we are looking for by choosing a variable to represent it.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c9644d971857e5b6fb1370bb8de85b92_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"26\" style=\"vertical-align: 0px;\" \/>\u00a0 =?<\/td>\n<\/tr>\n<tr>\n<td>4. <strong>Find<\/strong> the required\u00a0trigonometric ratio.<\/td>\n<td>\u00a0\u00a0tan A =\u00a0 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-577d63c13081877c1b742b6c20f6f99c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>5. <strong>Solve <\/strong>the ratio using good algebra techniques.<\/td>\n<td>\u00a0tan A = 1.8<\/p>\n<p>A = tan<sup>-1<\/sup> 1.8<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c9644d971857e5b6fb1370bb8de85b92_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"26\" style=\"vertical-align: 0px;\" \/>\u00a0 = 60.9\u00b0<\/td>\n<\/tr>\n<tr>\n<td>6. <strong>Check<\/strong> the answer in the problem and by making sure it makes sense.<\/td>\n<td>\u00a0tan 60.9\u00b0\u00a0 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-291fc22ffe86153c31295cf8317425ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"13\" style=\"vertical-align: 2px;\" \/> 1.8<\/p>\n<p>1.8 = 1.8 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-46376855baeb2c7a9b4e7ae04ce9831b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#99;&#104;&#101;&#99;&#107;&#109;&#97;&#114;&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"13\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>7. <strong>Answer <\/strong>the question with a complete sentence.<\/td>\n<td>The missing angle A is\u00a0 60.9\u00b0.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 9<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the missing angle C. Round your final answer to one decimal place.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-9688\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/Fig-20.png\" alt=\"\" width=\"182\" height=\"224\" \/><\/p>\n<div id=\"fs-id1296415\" data-type=\"solution\">\n<details open=\"open\">\n<summary>Show answer<\/summary>\n<p>29.1\u00b0<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<h1>Solving a Right Triangle<\/h1>\n<p>From the section before we know that any triangle has three sides and three interior angles. In a right triangle, when all six parts of the triangle are known, we say that the right triangle is solved.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 10<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Solve the right triangle. Round your final answer to one decimal place.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-9715\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/Fig-28-300x272-1.png\" alt=\"\" width=\"259\" height=\"235\" \/><\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>Since the sum of angles in any triangle is 180\u00b0, the measure of angle B can be easy calculated.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-6150531321f68bd917536a4352de756f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: 0px;\" \/>\u00a0 =\u00a0 180\u00b0 \u2212 90\u00b0 \u2212 42\u00b0<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-6150531321f68bd917536a4352de756f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: 0px;\" \/>\u00a0 = 48\u00b0<\/p>\n<table style=\"border-collapse: collapse; width: 100%; height: 221px;\">\n<tbody>\n<tr style=\"height: 46px;\">\n<td style=\"width: 33.5227%; height: 46px;\">1. <strong>Read<\/strong> the problem and make sure all the words and ideas are understood. Draw the right triangle and label the given parts.<\/td>\n<td style=\"width: 33.3333%; height: 46px;\" colspan=\"2\">A drawing is given. Angle A is our reference angle,\u00a0 a = 8 is the opposite side, b is the adjacent side, and c is the hypotenuse.<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 33.5227%; height: 14px;\">2. <strong>Identify<\/strong> what we are looking for.<\/td>\n<td style=\"width: 33.3333%; height: 14px;\">\u00a0a)\u00a0 \u00a0adjacent side<\/td>\n<td style=\"width: 33.3333%; height: 14px;\">b) hypotenuse<\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"width: 33.5227%; height: 30px;\">3.<strong>Label<\/strong> what we are looking for by choosing a variable to represent it.<\/td>\n<td style=\"width: 33.3333%; height: 30px;\">b = ?<\/td>\n<td style=\"width: 33.3333%; height: 30px;\">c = ?<\/td>\n<\/tr>\n<tr style=\"height: 41px;\">\n<td style=\"width: 33.5227%; height: 41px;\">4. <strong>Find<\/strong> the required\u00a0trigonometric ratio.<\/td>\n<td style=\"width: 33.3333%; height: 41px;\">tan 42\u00b0 =\u00a0 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-101afdd8664cd730254b9db4a409c6d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#98;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><\/td>\n<td style=\"width: 33.3333%; height: 41px;\">sin 42\u00b0 =\u00a0 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-51911989659c0698664d7241dfb9c718_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#99;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"width: 33.5227%; height: 30px;\">5. <strong>Solve<\/strong> the ratio using good algebra techniques.<\/td>\n<td style=\"width: 33.3333%; height: 30px;\">0.9004 = <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-101afdd8664cd730254b9db4a409c6d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#98;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><\/p>\n<p>0.9004 b = 8<\/p>\n<p>b = 8.8849<\/td>\n<td style=\"width: 33.3333%; height: 30px;\">0.6691 = <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-51911989659c0698664d7241dfb9c718_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#99;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><\/p>\n<p>0.6691 c = 8<\/p>\n<p>c = 11.9563<\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"width: 33.5227%; height: 30px;\">6. <strong>Check<\/strong> the answer in the problem and by making sure it makes sense.<\/td>\n<td style=\"width: 33.3333%; height: 30px;\">tan 42 \u00b0\u00a0 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-291fc22ffe86153c31295cf8317425ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"13\" style=\"vertical-align: 2px;\" \/> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-74188c6a42df74233a044cc884d4a808_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#56;&#46;&#56;&#56;&#52;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"39\" style=\"vertical-align: -6px;\" \/><\/p>\n<p>0.9 = 0.9\u00a0 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-46376855baeb2c7a9b4e7ae04ce9831b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#99;&#104;&#101;&#99;&#107;&#109;&#97;&#114;&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"13\" style=\"vertical-align: -1px;\" \/><\/td>\n<td style=\"width: 33.3333%; height: 30px;\">sin 42\u00b0 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-291fc22ffe86153c31295cf8317425ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"13\" style=\"vertical-align: 2px;\" \/> \u00a0<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-cba83ee1c601bf859ac81cf6d79a51f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#49;&#49;&#46;&#57;&#53;&#54;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"46\" style=\"vertical-align: -6px;\" \/><\/p>\n<p>0.6691 = 0.6691 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-46376855baeb2c7a9b4e7ae04ce9831b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#99;&#104;&#101;&#99;&#107;&#109;&#97;&#114;&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"13\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"width: 33.5227%; height: 30px;\">7. <strong>Answer<\/strong> the question with a complete sentence.<\/td>\n<td style=\"width: 33.3333%; height: 30px;\">The adjacent side is 8.9.<\/p>\n<p>Rounded to one decimal place.<\/td>\n<td style=\"width: 33.3333%; height: 30px;\">The hypotenuse is 12<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We solved the right triangle<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c9644d971857e5b6fb1370bb8de85b92_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"26\" style=\"vertical-align: 0px;\" \/> = 42\u00b0<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-6150531321f68bd917536a4352de756f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: 0px;\" \/> = 48\u00b0<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-199478770c6257562ee854658a4e1f50_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: 0px;\" \/> = 90\u00b0<\/p>\n<p>a = 8<\/p>\n<p>b = 8.9<\/p>\n<p>c = 12<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 10<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Solve the right triangle. Round your final answer to one decimal place.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-9716\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/Fig-29-300x252-1.png\" alt=\"\" width=\"258\" height=\"217\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c9644d971857e5b6fb1370bb8de85b92_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"26\" style=\"vertical-align: 0px;\" \/>= 21\u00b0<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-6150531321f68bd917536a4352de756f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: 0px;\" \/> = 69\u00b0<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-199478770c6257562ee854658a4e1f50_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: 0px;\" \/> = 90\u00b0<\/p>\n<div id=\"fs-id1296417\" data-type=\"solution\">\n<details open=\"open\">\n<summary>Show answer<\/summary>\n<p>a = 6<\/p>\n<p>b = 15.6<\/p>\n<p>c = 16.7<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 11<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Solve the right triangle. Round to two decimal places.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-9721\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/Fig-31-300x152-1.png\" alt=\"\" width=\"300\" height=\"152\" \/><\/p>\n<p><strong>Solution<\/strong><\/p>\n<table style=\"border-collapse: collapse; width: 100%; height: 254px;\">\n<tbody>\n<tr>\n<td style=\"width: 48.5227%;\">1. <strong>Read <\/strong>the problem and make sure all the words and ideas are understood. Draw the right triangle and label\u00a0 the given parts.<\/td>\n<td style=\"width: 66.6666%; height: 14px;\" colspan=\"2\">A drawing is given. Let angle D be our reference angle,\u00a0 d = 4 is the opposite side,\u00a0 f is the adjacent side, and e = 9 is the hypotenuse<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 48.5227%;\">2. <strong>Identify <\/strong>what we are looking for.<\/td>\n<td style=\"width: 33.3333%; height: 14px;\">a) angle D<\/td>\n<td style=\"width: 33.3333%; height: 14px;\">b) adjacent<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 48.5227%;\">3.<strong>Label <\/strong>what we are looking for by choosing a variable to represent it.<\/td>\n<td style=\"width: 33.3333%; height: 14px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-f4dcfcb6aad74038f145771931316aa8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#68;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"28\" style=\"vertical-align: 0px;\" \/>\u00a0 =?<\/td>\n<td style=\"width: 33.3333%; height: 14px;\">\u00a0f = ?<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 48.5227%;\">4. <strong>Find<\/strong> the required\u00a0trigonometric ratio.<\/td>\n<td style=\"width: 33.3333%; height: 14px;\">sin D =\u00a0 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-03691bec54827131718c68dfdca523bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><\/td>\n<td style=\"width: 33.3333%; height: 14px;\">4<sup>2<\/sup>\u00a0 +\u00a0 f<sup>2<\/sup> = 9<sup>2<\/sup><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 48.5227%;\">5. <strong>Solve <\/strong>the ratio using good algebra techniques.<\/td>\n<td style=\"width: 33.3333%; height: 92px;\">sin D = 0.4444<\/p>\n<p>D = sin<sup>-1<\/sup>0.4444<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-f4dcfcb6aad74038f145771931316aa8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#68;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"28\" style=\"vertical-align: 0px;\" \/>\u00a0 = 26.3850\u00b0<\/td>\n<td style=\"width: 33.3333%; height: 92px;\">16 + f<sup>2<\/sup> = 81<\/p>\n<p>f<sup>2<\/sup> = 81 &#8211; 16<\/p>\n<p>f<sup>2<\/sup> = 65<\/p>\n<p>f = square root of 65<\/p>\n<p>f = 8.06<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 48.5227%;\">6. <strong>Check <\/strong>the answer in the problem and by making sure it makes sense.<\/td>\n<td style=\"width: 33.3333%; height: 76px;\">sin 26.3850\u00b0 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-291fc22ffe86153c31295cf8317425ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"13\" style=\"vertical-align: 2px;\" \/> \u00a0<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-03691bec54827131718c68dfdca523bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><\/p>\n<p>0.4444 =0.4444\u00a0 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-46376855baeb2c7a9b4e7ae04ce9831b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#99;&#104;&#101;&#99;&#107;&#109;&#97;&#114;&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"13\" style=\"vertical-align: -1px;\" \/><\/td>\n<td style=\"width: 33.3333%; height: 76px;\">4<sup>2<\/sup>\u00a0 +\u00a0 8.06<sup>2<\/sup> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-291fc22ffe86153c31295cf8317425ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"13\" style=\"vertical-align: 2px;\" \/> 9<sup>2<\/sup><\/p>\n<p>81 = 81 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-46376855baeb2c7a9b4e7ae04ce9831b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#99;&#104;&#101;&#99;&#107;&#109;&#97;&#114;&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"13\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 48.5227%;\">7. <strong>Answer <\/strong>the question with a complete sentence.<\/td>\n<td style=\"width: 33.3333%; height: 30px;\">The missing angle D is 26.39\u00b0.<\/td>\n<td style=\"width: 33.3333%; height: 30px;\">The adjacent side is 8.06\u00a0 \u00a0Rounded to two decimal places<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The missing angle F = 180\u00b0 &#8211; 90\u00b0 &#8211; 26.39\u00b0 = 63.64\u00b0<\/p>\n<p>We solved the right triangle<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-f4dcfcb6aad74038f145771931316aa8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#68;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"28\" style=\"vertical-align: 0px;\" \/> = 26.39\u00b0<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0e1bd510bc2435d28df8b8ff340d980b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#69;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: 0px;\" \/> = 90\u00b0<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-225d2eda571002518f953558c776be31_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#70;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: 0px;\" \/> = 63.61\u00b0<\/p>\n<p>d = 4<\/p>\n<p>e = 9<\/p>\n<p>f = 8.06<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 11<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Solve the right triangle. Round to one decimal place.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-9722\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/Fig-32-300x147-1.png\" alt=\"\" width=\"300\" height=\"147\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-f4dcfcb6aad74038f145771931316aa8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#68;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"28\" style=\"vertical-align: 0px;\" \/> = 29.3\u00b0<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0e1bd510bc2435d28df8b8ff340d980b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#69;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: 0px;\" \/> = 90\u00b0<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-225d2eda571002518f953558c776be31_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#70;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: 0px;\" \/> = 60.7\u00b0<\/p>\n<div id=\"fs-id1296419\" data-type=\"solution\">\n<details open=\"open\">\n<summary>Show answer<\/summary>\n<p>d = 29.4<\/p>\n<p>e = 18.4<\/p>\n<p>f = 60.6<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<h1 data-type=\"title\">Solve Applications Using Trigonometric Ratios<\/h1>\n<p>In the previous examples we were able to find missing sides and missing angles of a right triangle. Now, let&#8217;s use the trigonometric ratios to solve real-life\u00a0 problems.<\/p>\n<p>Many applications of trigonometric ratios involve understanding of an angle of elevation or angle of depression.<\/p>\n<p>The angle of elevation is an angle between the horizontal line (ground) and the observer&#8217;s line of sight.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-11452 aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/9.2_angleOfElevation.png\" alt=\"\" width=\"463\" height=\"356\" \/><\/p>\n<\/div>\n<p>The angle of depression is the angle between horizontal line (that is parallel to the ground) and the observer&#8217;s line of sight.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-11451 aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/9.2_angleOfDepression.png\" alt=\"\" width=\"438\" height=\"399\" \/><\/p>\n<div id=\"fs-id1168345363664\" class=\"bc-section section\" data-depth=\"1\">\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 12<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>James is standing 31 metres away from the base of the Harbour Centre in Vancouver. He looks up to the top of the building at a 78\u00b0 angle. How tall is the Harbour Centre?<\/p>\n<p><strong>Solution<\/strong><\/p>\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td>1. <strong>Read <\/strong>the problem and make sure all the words and ideas are understood. Draw the right triangle and label\u00a0 the given parts.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/9.2_Example12-James-1.png\" alt=\"\" width=\"271\" height=\"342\" \/><\/p>\n<p>Angle X is our reference angle,\u00a0 x is opposite side, y = 31 m is the adjacent side, and z is the hypotenuse.<\/td>\n<\/tr>\n<tr>\n<td>2. <strong>Identify <\/strong>what we are looking for.<\/td>\n<td>\u00a0The opposite side<\/td>\n<\/tr>\n<tr>\n<td>3.<strong>Label <\/strong>what we are looking for by choosing a variable to represent it.<\/td>\n<td>\u00a0x=?<\/td>\n<\/tr>\n<tr>\n<td>4. <strong>Find<\/strong> the required\u00a0trigonometric ratio.<\/td>\n<td>\u00a0tan 78\u00b0 =\u00a0 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-f47497c69591ef7242a4b3ed377f1d76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#125;&#123;&#51;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"14\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>5. <strong>Solve <\/strong>the ratio using good algebra techniques.<\/td>\n<td>4.7046 = <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-f47497c69591ef7242a4b3ed377f1d76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#125;&#123;&#51;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"14\" style=\"vertical-align: -6px;\" \/><\/p>\n<p>x = 145.8426<\/td>\n<\/tr>\n<tr>\n<td>6. <strong>Check <\/strong>the answer in the problem and by making sure it makes sense.<\/td>\n<td>4.7046 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-291fc22ffe86153c31295cf8317425ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"13\" style=\"vertical-align: 2px;\" \/> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-1f8cd50f77437261df9939f10b70e790_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#52;&#53;&#46;&#56;&#52;&#50;&#54;&#125;&#123;&#51;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"53\" style=\"vertical-align: -6px;\" \/><\/p>\n<p>4.7046 = 4.7046 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-46376855baeb2c7a9b4e7ae04ce9831b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#99;&#104;&#101;&#99;&#107;&#109;&#97;&#114;&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"13\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>7. <strong>Answer <\/strong>the question with a complete sentence.<\/td>\n<td>\u00a0The Harbour Centre is 145.8426 metres or rounded to 146 metres.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 12<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Marta is standing 23 metres away from the base of the tallest apartment building in Prince George and looks at the top of the building at a 62\u00b0 angle. How tall is the building?<\/p>\n<div id=\"fs-id1296422\" data-type=\"solution\">\n<details open=\"open\">\n<summary>Show answer<\/summary>\n<p>43.3 metres<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 13<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Thomas is standing at the top of the building that is 45 metres high and looks at her friend that is standing on the ground, 22 metres from the base of the building. What is the angle of depression?<\/p>\n<p><strong>Solution<\/strong><\/p>\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td>1. <strong>Read<\/strong> the problem and make sure all the words and ideas are understood. Draw the right triangle and label\u00a0 the given parts.<\/td>\n<td>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/9.2_Thomas.png\" alt=\"\" width=\"374\" height=\"306\" \/><\/p>\n<p>Angle Y is our reference angle,\u00a0 y = 45 m is the opposite side,\u00a0 z = 22 m\u00a0 is the adjacent side, and x is the hypotenuse<\/td>\n<\/tr>\n<tr>\n<td>2. <strong>Identify<\/strong> what we are looking for.<\/td>\n<td>\u00a0 angle Y<\/td>\n<\/tr>\n<tr>\n<td>3.<strong>Label<\/strong> what we are looking for by choosing a variable to represent it.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-a8672f26aeb43591efd785d68a7dbfea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#89;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: 0px;\" \/>\u00a0 =?<\/td>\n<\/tr>\n<tr>\n<td>4. <strong>Find<\/strong> the required\u00a0trigonometric ratio.<\/td>\n<td>\u00a0\u00a0tan Y =\u00a0 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-33d34dbc5d8c2f7c61932e5ebba07c72_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#53;&#125;&#123;&#50;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>5. <strong>Solve<\/strong> the ratio using good algebra techniques.<\/td>\n<td>\u00a0tan Y = 2.0455<\/p>\n<p>Y = tan <sup>&#8211;<\/sup>\u00b92.0455<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-a8672f26aeb43591efd785d68a7dbfea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#89;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: 0px;\" \/>\u00a0 = 63.9470\u00b0<\/td>\n<\/tr>\n<tr>\n<td>6. <strong>Check<\/strong> the answer in the problem and by making sure it makes sense.<\/td>\n<td>\u00a0tan 63.9470\u00b0\u00a0 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-291fc22ffe86153c31295cf8317425ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"13\" style=\"vertical-align: 2px;\" \/>\u00a0 2.0455<\/p>\n<p>2.0455\u00a0 = 2.0455\u00a0 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-46376855baeb2c7a9b4e7ae04ce9831b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#99;&#104;&#101;&#99;&#107;&#109;&#97;&#114;&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"13\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>7. <strong>Answer<\/strong> the question with a complete sentence.<\/td>\n<td>The angle \u00a0of depression is\u00a0 63.9470\u00b0 or\u00a0 64\u00b0 rounded to one decimal place.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 13<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Hemanth is standing on the top of a cliff 250 feet above the ground and looks at his friend that is standing on the ground, 40 feet from the base of the cliff. What is the angle of depression?<\/p>\n<div id=\"fs-id1296423\" data-type=\"solution\">\n<details open=\"open\">\n<summary>Show answer<\/summary>\n<p>80.9\u00b0<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<h1>Key Concepts<\/h1>\n<ul id=\"fs-id1168345448170\" data-bullet-style=\"bullet\">\n<li>Three Basic Trigonometric Ratios: (Where \u03b8 is the measure of a reference angle measured in degrees.)\n<ul>\n<li>sine \u03b8 = <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-264a6a31a1d2f7f3d93f6495d99aac3d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#108;&#101;&#110;&#103;&#116;&#104;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#111;&#112;&#112;&#111;&#115;&#105;&#116;&#101;&#32;&#115;&#105;&#100;&#101;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#108;&#101;&#110;&#103;&#116;&#104;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#32;&#104;&#121;&#112;&#111;&#116;&#101;&#110;&#117;&#115;&#101;&#32;&#115;&#105;&#100;&#101;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"201\" style=\"vertical-align: -9px;\" \/><\/li>\n<li>cosine \u03b8 = <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-857da9e2cb11012174ba178d506c1288_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#108;&#101;&#110;&#103;&#116;&#104;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#97;&#100;&#106;&#97;&#99;&#101;&#110;&#116;&#32;&#115;&#105;&#100;&#101;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#108;&#101;&#110;&#103;&#116;&#104;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#32;&#104;&#121;&#112;&#111;&#116;&#101;&#110;&#117;&#115;&#101;&#32;&#115;&#105;&#100;&#101;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"201\" style=\"vertical-align: -9px;\" \/><\/li>\n<li>tangent \u03b8 = <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-40ec3e42f10b611cf2916f1f8fdab7d7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#108;&#101;&#110;&#103;&#116;&#104;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#111;&#112;&#112;&#111;&#115;&#105;&#116;&#101;&#32;&#115;&#105;&#100;&#101;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#108;&#101;&#110;&#103;&#116;&#104;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#32;&#97;&#100;&#106;&#97;&#99;&#101;&#110;&#116;&#32;&#115;&#105;&#100;&#101;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"184\" style=\"vertical-align: -9px;\" \/><\/li>\n<\/ul>\n<\/li>\n<li><strong data-effect=\"bold\">Problem-Solving Strategy for\u00a0Trigonometry Applications<\/strong>\n<ol id=\"fs-id1166426163525\" class=\"stepwise\" type=\"1\">\n<li><strong data-effect=\"bold\">Read<\/strong> the problem and make sure all the words and ideas are understood. Draw the right triangle and label\u00a0 the given parts.<\/li>\n<li><strong data-effect=\"bold\">Identify<\/strong> what we are looking for.<\/li>\n<li><strong data-effect=\"bold\">Label<\/strong> what we are looking for by choosing a variable to represent it.<\/li>\n<li><strong data-effect=\"bold\">Find <\/strong>the required\u00a0trigonometric ratio.<\/li>\n<li><strong data-effect=\"bold\">Solve<\/strong> the ratio using good algebra techniques.<\/li>\n<li><strong data-effect=\"bold\">Check<\/strong> the answer by substituting it back into the ratio solved in step 5 and by making sure it makes sense in the context of the problem.<\/li>\n<li><strong data-effect=\"bold\">Answer<\/strong> the question with a complete sentence.<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<h1>5.2 Exercise Set<\/h1>\n<p>Label the sides of the triangle.<\/p>\n<table style=\"border-collapse: collapse; width: 100%; height: 299px;\">\n<tbody>\n<tr style=\"height: 230px;\">\n<td>1<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/P16-300x151-1.png\" alt=\"\" width=\"249\" height=\"126\" \/><\/td>\n<td>2.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/P14-1-300x142-1.png\" alt=\"\" width=\"273\" height=\"129\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol start=\"3\">\n<li>If the reference angle in Question 1 is B, Find the adjacent ?<\/li>\n<li>If the reference angle in Question 2 is Z, find the opposite?<\/li>\n<\/ol>\n<p>Label the sides of the triangle and find the hypotenuse, opposite and adjacent.<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\">5.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/P-1-300x128-1.png\" alt=\"\" width=\"300\" height=\"128\" \/><\/td>\n<td style=\"width: 50%;\">\u00a06.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/P3-300x144-1.png\" alt=\"\" width=\"300\" height=\"144\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>Use your calculator to find the given ratios. Round to four decimal places if necessary:<\/p>\n<ol class=\"twocolumn\" start=\"7\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-189698f5a13240102a397ba53055faeb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#105;&#110;&#32;&#123;&#52;&#55;&#125;&#94;&#123;&#92;&#99;&#105;&#114;&#99;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"49\" style=\"vertical-align: 0px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-fecb5952e9c750a6679530240a6ac3b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#99;&#111;&#115;&#32;&#123;&#56;&#50;&#125;&#94;&#123;&#92;&#99;&#105;&#114;&#99;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"51\" style=\"vertical-align: 0px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ad047ca4c779ee5d8cee9e91ecb10544_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#97;&#110;&#32;&#123;&#49;&#50;&#125;&#94;&#123;&#92;&#99;&#105;&#114;&#99;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"53\" style=\"vertical-align: -1px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-30a61a190a1c83d3185f826b534ad029_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#105;&#110;&#32;&#123;&#51;&#48;&#125;&#94;&#123;&#92;&#99;&#105;&#114;&#99;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"49\" style=\"vertical-align: 0px;\" \/><\/li>\n<\/ol>\n<p>For the given triangles, find the sine, cosine and tangent of the \u03b8.<\/p>\n<table style=\"border-collapse: collapse; width: 100%; height: 148px;\">\n<tbody>\n<tr style=\"height: 131px;\">\n<td>11.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/P-1-300x128-1.png\" alt=\"\" width=\"300\" height=\"128\" \/><\/td>\n<td>12.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/P3-300x144-1.png\" alt=\"\" width=\"300\" height=\"144\" \/><\/td>\n<\/tr>\n<tr style=\"height: 17px;\">\n<td>13.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/P10-300x252-1.png\" alt=\"\" width=\"249\" height=\"210\" \/><\/td>\n<td>14.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/P21.png\" alt=\"\" width=\"169\" height=\"227\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>For the given triangles, find the missing side. Round it to one decimal place.<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td>15. Find the hypotenuse.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/P6.png\" alt=\"\" width=\"195\" height=\"269\" \/><\/td>\n<td>16. Find b if a = 6.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-1906\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/05\/Screenshot-2021-11-17-at-10.20.36-AM-242x300.png\" alt=\"\" width=\"242\" height=\"300\" srcset=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/05\/Screenshot-2021-11-17-at-10.20.36-AM-242x300.png 242w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/05\/Screenshot-2021-11-17-at-10.20.36-AM-65x81.png 65w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/05\/Screenshot-2021-11-17-at-10.20.36-AM-225x279.png 225w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/05\/Screenshot-2021-11-17-at-10.20.36-AM-350x434.png 350w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/05\/Screenshot-2021-11-17-at-10.20.36-AM.png 526w\" sizes=\"auto, (max-width: 242px) 100vw, 242px\" \/><\/td>\n<\/tr>\n<tr>\n<td>17. Find the opposite. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/P20-300x146-1.png\" alt=\"\" width=\"300\" height=\"146\" \/><\/td>\n<td>18. Find the adjacent. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/P19-300x151-1.png\" alt=\"\" width=\"300\" height=\"151\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>For the given triangles, find the missing sides. Round it to one decimal place.<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td>19.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/P8-300x251-1.png\" alt=\"\" width=\"244\" height=\"204\" \/><\/td>\n<td>20.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/P9-300x244-1.png\" alt=\"\" width=\"259\" height=\"211\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Solve the triangles. Round to one decimal place.<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr style=\"height: 259px;\">\n<td style=\"width: 49.9086%;\">21.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/P22-300x147-1.png\" alt=\"\" width=\"280\" height=\"137\" \/><\/td>\n<td style=\"width: 49.9086%;\">\u00a022.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1904\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/05\/Screenshot-2021-11-17-at-8.39.55-AM-e1637167343689-239x300.png\" alt=\"\" width=\"212\" height=\"266\" srcset=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/05\/Screenshot-2021-11-17-at-8.39.55-AM-e1637167343689-239x300.png 239w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/05\/Screenshot-2021-11-17-at-8.39.55-AM-e1637167343689-65x81.png 65w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/05\/Screenshot-2021-11-17-at-8.39.55-AM-e1637167343689-225x282.png 225w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/05\/Screenshot-2021-11-17-at-8.39.55-AM-e1637167343689-350x439.png 350w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/05\/Screenshot-2021-11-17-at-8.39.55-AM-e1637167343689.png 470w\" sizes=\"auto, (max-width: 212px) 100vw, 212px\" \/><\/td>\n<\/tr>\n<tr style=\"height: 239px;\">\n<td style=\"width: 49.9086%;\">23.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/P7-300x223-1.png\" alt=\"\" width=\"273\" height=\"203\" \/><\/td>\n<td style=\"width: 49.9086%;\">24.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/P15-1-300x169-1.png\" alt=\"\" width=\"273\" height=\"154\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol start=\"25\">\n<li>Kim stands 75 metres from the bottom of a tree and looks up at the top of the tree at a 48\u00b0 angle. How tall is the tree?<\/li>\n<li>A tree makes a shadow that is 6 metres long when the angle of elevation to the sun is 52\u00b0. How tall is the tree?<\/li>\n<li>A ladder that is 15 feet is leaning against a house and makes a 45\u00b0 angle with the ground. How far is the base of the ladder from the house?<\/li>\n<li>Roxanne is flying a kite and has let out 100 feet of string. The angle of elevation with the ground is 38\u00b0. How high is her kite above the ground?<\/li>\n<li>Marta is flying a kite and has let out 28 metres of string. If the kite is 10 metres above the ground, what is the angle of elevation?<\/li>\n<li>An airplane takes off from the ground at the angle of 25\u00b0. If the airplane traveled 200 kilometres, how high above the ground is it?<\/li>\n<\/ol>\n<h1>Answers:<\/h1>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr style=\"height: 238px;\">\n<td style=\"width: 43.6929%;\">1.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/Qn1-answers.png\" alt=\"\" width=\"285\" height=\"130\" \/><\/td>\n<td style=\"width: 14.4424%;\">3. c<\/td>\n<td style=\"width: 41.6819%;\">5.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/Q-5-ans.png\" alt=\"\" width=\"328\" height=\"150\" \/><\/p>\n<p>g is opposite , f is adjacent, and e is hypotenuse<\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"width: 43.6929%;\">7. 0.7314<\/td>\n<td style=\"width: 14.4424%;\">9. 0.2126<\/td>\n<td style=\"width: 41.6819%;\">11.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/05\/Q-5-ans.png\" alt=\"\" width=\"328\" height=\"150\" \/><\/p>\n<p>sin \u03b8\u00a0 = <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ac15e6133be3cc474f298bb54824c76e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#103;&#125;&#123;&#101;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"8\" style=\"vertical-align: -6px;\" \/>, cos \u03b8\u00a0 = <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-1400ee01588fbe02959d2272d497e7f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#102;&#125;&#123;&#101;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"9\" style=\"vertical-align: -6px;\" \/>, tan \u03b8\u00a0 = <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-154ba01c1a1e74386321310e6fff2785_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#103;&#125;&#123;&#102;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"9\" style=\"vertical-align: -9px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol>\n<li value=\"13\">sin \u03b8\u00a0 = <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-49ea7b808323d2ec5714576c1f379ec4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#115;&#125;&#123;&#114;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"7\" style=\"vertical-align: -6px;\" \/>, cos \u03b8\u00a0 = <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-a516c17086be584012181d5158fb6f96_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#116;&#125;&#123;&#114;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/>, tan \u03b8\u00a0 = <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-81d72b08f166863a2918d4a618e2b66f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#115;&#125;&#123;&#116;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"7\" style=\"vertical-align: -6px;\" \/><\/li>\n<li value=\"15\">b = 19.8<\/li>\n<li value=\"17\">c = 12<\/li>\n<li value=\"19\">y = 19.3, z = 8.2<\/li>\n<li value=\"21\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-6150531321f68bd917536a4352de756f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: 0px;\" \/> = 61\u00b0<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-199478770c6257562ee854658a4e1f50_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: 0px;\" \/> = 29\u00b0<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-f4dcfcb6aad74038f145771931316aa8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#68;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"28\" style=\"vertical-align: 0px;\" \/> = 90\u00b0b = 38.5c = 21.3d = 44<\/li>\n<li value=\"23\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-4233306bbe710ee1fc36720ac41656d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#84;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"26\" style=\"vertical-align: 0px;\" \/> = 36.9\u00b0<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0088aa20e1c8f289eee85038005e4707_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#82;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: 0px;\" \/> = 90\u00b0<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-14865fa6e637d75143fed43981c32354_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#110;&#103;&#108;&#101;&#32;&#83;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"25\" style=\"vertical-align: 0px;\" \/> = 53.1\u00b0t = 15r = 25s = 20<\/li>\n<li value=\"25\">83.3 m<\/li>\n<li value=\"27\">10.6 ft<\/li>\n<li value=\"29\">20.9\u00b0<\/li>\n<\/ol>\n<h1><strong>Attribution:<\/strong><\/h1>\n<p>This chapter has been adapted from \u201cSolve Applications: Sine, Cosine and Tangent Ratios\u201d in <a href=\"https:\/\/opentextbc.ca\/introalgebra\/\">Introductory Algebra<\/a>\u00a0by Izabela Mazur, which is under a <a href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY 4.0 Licence<\/a>. See the Adaptation Statement for more information.<\/p>\n","protected":false},"author":125,"menu_order":2,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1590","chapter","type-chapter","status-publish","hentry"],"part":1593,"_links":{"self":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapters\/1590","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/wp\/v2\/users\/125"}],"version-history":[{"count":4,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapters\/1590\/revisions"}],"predecessor-version":[{"id":1907,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapters\/1590\/revisions\/1907"}],"part":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/parts\/1593"}],"metadata":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapters\/1590\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/wp\/v2\/media?parent=1590"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapter-type?post=1590"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/wp\/v2\/contributor?post=1590"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/wp\/v2\/license?post=1590"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}