702 CHAPTER 7 Trigonometric Identities and Equations Applications of the Sum and Difference Identities EXAMPLE 3 Finding Exact Sine andTangent Function Values Find the exact value of each expression. (a) sin 75° (b) tan 7p 12 (c) sin 40° cos 160° - cos 40° sin 160° SOLUTION (a) sin 75° = sin145° + 30°2 75° = 45° + 30° = sin 45° cos 30° + cos 45° sin 30° Sine sum identity = 22 2 # 23 2 + 22 2 # 1 2 Substitute known values. = 26 + 22 4 Multiply, and then add fractions. 1a + b21c + d2 = ac + ad + bc + bd; 1x - y21x + y2 = x2 - y2 Factor first. Then divide out the common factor. (b) tan 7p 12 = tan a p 3 + p 4b 7p 12 = 4p 12 + 3p 12 = p 3 + p 4 = tan p 3 + tan p 4 1 - tan p 3 tan p 4 Tangent sum identity = 23 + 1 1 - 23 # 1 Substitute known values. = 23 + 1 1 - 23 # 1 + 23 1 + 23 Rationalize the denominator. = 23 + 3 + 1 + 23 1 - 3 = 4 + 223 -2 Combine like terms. = 2A2 + 23 B 21-12 Factor out 2. = -2 - 23 Write in lowest terms. (c) sin 40° cos 160° - cos 40° sin 160° = sin140° - 160°2 Sine difference identity = sin1-120°2 Subtract. = - sin 120° Even-odd identity = - 23 2 Substitute the known value. S Now Try Exercises 41, 47, and 57.
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