8.1.3 Solving Right Triangles Key Objectives • Use trigonometric ratios to find angle measures in right triangles and to solve real-world problems. Example 1 Identifying Angles from Trigonometric Ratios
An angle is identified from its trigonometric ratio in this example. The lengths of the legs and hypotenuse of the triangle are given. The cosine of the angle is given as 4/5. The cosine of ∠ 1 is 6/10, or 3/5. The cosine of ∠ 2 is 8/10, or 4/5. The calculated cosine of ∠ 2 matches the cosine of ∠ A , so ∠ A is ∠ 2.
Example 2 Calculating Angle Measures from Trigonometric Ratios
Inverse trigonometric functions are equal to the measure of the angle that has a given trigonometric ratio. For example, if sin 30 ° = 0.50, then sin − 1 0.5 = 30 ° . A calculator is used to calculate inverse trigonometric functions in this example. Make sure the calculator is set to calculate with angles in degrees and not radians. To calculate the inverse sine of 0.5, enter 0.5 on the keypad and push the “sin − 1 ” key. This key may also be labeled “asin” or “arcsine,” as this is another name for the inverse sine. “Arccosine” and “arctangent” are the other names for the other ratios. The calculator returns the value of 30, which is the answer in degrees. Thus, sin − 1 (0.5) = 30 ° . The inverse cosine of 0.25 is the measure of the angle whose cosine is 0.25. Enter 0.25 on the keypad and push “cos − 1 .” The result, rounded to hundredths, is 75.52 ° . Similarly, tan − 1 (5.1) = 78.91 ° , rounded to hundredths.
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