8.1.2 Trigonometric Ratios (continued) Example 4 Using Trigonometric Ratios to Find Lengths
The length of a side in a right triangle is obtained here using a trigonometric ratio. One angle is given as 35 ° and the length of its opposite side is given as 7.3 inches. To find PQ , recognize that the known length is opposite the known angle, and that the unknown length is adjacent to the known angle. Thus, the tangent is the appropriate trigonometric ratio to use. Write the trigonometric ratio for the tangent of 35 ° and substitute the given values and the unknown. Isolate the unknown, PQ , and use a calculator to find the value of tan 35 ° and solve for PQ . The value found for PQ is approximately 10.43 inches. The length of a side in a right triangle is obtained here using a trigonometric ratio. One angle is given as 66 ° and the length of the hypotenuse is given as 5.7 cm. To find AB , recognize that the known length is of the hypotenuse, and that the unknown length is of the side adjacent to the known angle. Thus, the cosine is the appropriate trigonometric ratio to use. Write the trigonometric ratio for the cosine of 66 ° and substitute the given values and the unknown. Isolate the unknown, AB , and use a calculator to find the value of cos 66 ° and solve for AB . The value found for AB is approximately 2.32 cm. The length of a side in a right triangle is obtained here using a trigonometric ratio. One angle is given as 49 ° and the length of its opposite side is given as 3.7 cm. To find YZ , recognize that the known length is of the side opposite the known angle, and that the unknown length is of the hypotenuse. Thus, the sine is the appropriate trigonometric ratio to use. Write the trigonometric ratio for the sine of 49 ° and substitute the given values and the unknown. Isolate the unknown, YZ , and use a calculator to find the value of sin 49 ° and solve for YZ . The value found for YZ is approximately 4.90 cm.
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