{"id":13831,"date":"2018-06-14T23:51:56","date_gmt":"2018-06-14T23:51:56","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/chapter\/using-right-triangles-to-evaluate-trigonometric-functions\/"},"modified":"2021-10-05T16:07:08","modified_gmt":"2021-10-05T16:07:08","slug":"using-right-triangles-to-evaluate-trigonometric-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/chapter\/using-right-triangles-to-evaluate-trigonometric-functions\/","title":{"raw":"Using Right Triangles to Evaluate Trigonometric Functions","rendered":"Using Right Triangles to Evaluate Trigonometric Functions"},"content":{"raw":"In earlier sections, we used a unit circle to define the trigonometric functions<\/strong>. In this section, we will extend those definitions so that we can apply them to right triangles. The value of the sine or cosine function of [latex]t[\/latex] is its value at [latex]t[\/latex] radians. First, we need to create our right triangle. Figure 1\u00a0shows a point on a unit circle<\/strong> of radius 1. If we drop a vertical line segment from the point [latex]\\left(x,y\\right)\\\\[\/latex] to the x<\/em>-axis, we have a right triangle whose vertical side has length [latex]y[\/latex] and whose horizontal side has length [latex]x[\/latex]. We can use this right triangle to redefine sine, cosine, and the other trigonometric functions as ratios of the sides of a right triangle.\r\n<\/span>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph Figure 1<\/b>[\/caption]\r\n\r\n \r\n\r\nWe know\r\n
[latex]\\cos \\text{ }t=\\frac{x}{1}=x\\\\[\/latex]<\/div>\r\nLikewise, we know\r\n
[latex]\\sin \\text{ }t=\\frac{y}{1}=y\\\\[\/latex]<\/div>\r\nThese ratios still apply to the sides of a right triangle when no unit circle is involved and when the triangle is not in standard position and is not being graphed using [latex]\\left(x,y\\right)\\\\[\/latex] coordinates. To be able to use these ratios freely, we will give the sides more general names: Instead of [latex]x[\/latex], we will call the side between the given angle and the right angle the adjacent side<\/strong> to angle [latex]t[\/latex]. (Adjacent means \"next to.\") Instead of [latex]y[\/latex], we will call the side most distant from the given angle the opposite side<\/strong> from angle [latex]\\text{}t[\/latex]. And instead of [latex]1[\/latex], we will call the side of a right triangle opposite the right angle the hypotenuse<\/strong>. These sides are labeled in Figure 2.\r\n\r\n \"A<\/span>\r\n

Figure 2.\u00a0<\/strong>The sides of a right triangle in relation to angle [latex]t[\/latex].<\/p>\r\n\r\n

Understanding Right Triangle Relationships<\/h2>\r\nGiven a right triangle with an acute angle of [latex]t[\/latex],\r\n
[latex]\\begin{array}{l}\\sin \\left(t\\right)=\\frac{\\text{opposite}}{\\text{hypotenuse}}\\hfill \\\\ \\cos \\left(t\\right)=\\frac{\\text{adjacent}}{\\text{hypotenuse}}\\hfill \\\\ \\tan \\left(t\\right)=\\frac{\\text{opposite}}{\\text{adjacent}}\\hfill \\end{array}\\\\[\/latex]<\/div>\r\nA common mnemonic for remembering these relationships is SohCahToa, formed from the first letters of \"S<\/u>ine is o<\/u>pposite over h<\/u>ypotenuse, C<\/u>osine is a<\/u>djacent over h<\/u>ypotenuse, T<\/u>angent is o<\/u>pposite over a<\/u>djacent.\"\r\n
\r\n

How To: Given the side lengths of a right triangle and one of the acute angles, find the sine, cosine, and tangent of that angle.<\/h3>\r\n
    \r\n \t
  1. Find the sine as the ratio of the opposite side to the hypotenuse.<\/li>\r\n \t
  2. Find the cosine as the ratio of the adjacent side to the hypotenuse.<\/li>\r\n \t
  3. Find the tangent is the ratio of the opposite side to the adjacent side.<\/li>\r\n<\/ol>\r\n<\/div>\r\n
    \r\n

    Example 1: Evaluating a Trigonometric Function of a Right Triangle<\/h3>\r\nGiven the triangle shown in Figure 3, find the value of [latex]\\cos \\alpha \\\\[\/latex].\r\n<\/span>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"A Figure 3<\/b>[\/caption]\r\n\r\n \r\n\r\n<\/div>\r\n
    \r\n

    Solution<\/h3>\r\nThe side adjacent to the angle is 15, and the hypotenuse of the triangle is 17, so:\r\n
    [latex]\\begin{array}{l}\\cos \\left(\\alpha \\right)=\\frac{\\text{adjacent}}{\\text{hypotenuse}}\\hfill \\\\ =\\frac{15}{17}\\hfill \\end{array}\\\\[\/latex]<\/div>\r\n<\/div>\r\n
    \r\n

    Try It 1<\/h3>\r\nGiven the triangle shown in Figure 4, find the value of [latex]\\text{sin}t[\/latex].\r\n<\/span>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"A Figure 4<\/b>[\/caption]\r\n\r\nSolution<\/a>\r\n\r\n<\/div>\r\n

    Relating Angles and Their Functions<\/h2>\r\nWhen working with right triangles, the same rules apply regardless of the orientation of the triangle. In fact, we can evaluate the six trigonometric functions of either of the two acute angles in the triangle in Figure 5. The side opposite one acute angle is the side adjacent to the other acute angle, and vice versa.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Right Figure 5.<\/b> The side adjacent to one angle is opposite the other.[\/caption]\r\n\r\nWe will be asked to find all six trigonometric functions for a given angle in a triangle. Our strategy is to find the sine, cosine, and tangent of the angles first. Then, we can find the other trigonometric functions easily because we know that the reciprocal of sine is cosecant, the reciprocal of cosine is secant, and the reciprocal of tangent is cotangent.\r\n
    \r\n

    How To: Given the side lengths of a right triangle, evaluate the six trigonometric functions of one of the acute angles.<\/h3>\r\n
      \r\n \t
    1. If needed, draw the right triangle and label the angle provided.<\/li>\r\n \t
    2. Identify the angle, the adjacent side, the side opposite the angle, and the hypotenuse of the right triangle.<\/li>\r\n \t
    3. Find the required function:\r\n