557 5.3 Trigonometric Function Values and Angle Measures Calculator Approximations of Angle Measures To find the measure of an angle having a certain trigonometric function value, calculators have three inverse functions (denoted sin−1 , cos−1 , and tan−1). If x is an appropriate number, then sin−1 x, cos−1 x, or tan−1 x gives the measure of an angle whose sine, cosine, or tangent, respectively, is x. For applications in this chapter, these functions will return angles in quadrant I. Degree mode Figure 39 EXAMPLE 8 Finding Function Values with a Calculator Approximate the value of each expression. (a) sin 49° 12′ (b) sec 97.977° (c) 1 cot 51.4283° (d) sin1-246°2 SOLUTION See Figure 39. We give values to eight decimal places below. (a) We may begin by converting 49° 12′ to decimal degrees. 49° 12′ = 49 12 60 ° = 49.2° However, some calculators allow direct entry of degrees, minutes, and seconds. (The method of entry varies among models.) Entering either sin149° 12′2 or sin 49.2° gives the same approximation. sin 49° 12′ = sin 49.2° ≈0.75699506 (b) There are no dedicated calculator keys for the secant, cosecant, and cotangent functions. However, we can use reciprocal identities to evaluate them. Recall that sec u = 1 cos u for all angles u, where cos u ≠0. Therefore, we use the reciprocal of the cosine function to evaluate the secant function. sec 97.977° = 1 cos 97.977° ≈ -7.20587921 (c) Use the reciprocal identity 1 cot u = tan u to simplify the expression first. 1 cot 51.4283° = tan 51.4283° ≈ 1.25394815 (d) sin1-246°2 ≈0.91354546 S Now Try Exercises 109, 111, 115, and 119. EXAMPLE 9 Using InverseTrigonometric Functions to Find Angles Find an angle u in the interval 30°, 90°2 that satisfies each condition. (a) sin u = 0.96770915 (b) sec u = 1.0545829 SOLUTION (a) Using degree mode and the inverse sine function, we find that an angle u having sine value 0.96770915 is 75.399995°. (There are infinitely many such angles, but the calculator gives only this one.) u = sin-1 0.96770915 ≈75.399995° See Figure 40. Degree mode Figure 40
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