703 7.3 Sum and Difference Identities EXAMPLE 4 Writing Functions as Expressions Involving Functions of U Write each function as an expression involving functions of u alone. (a) cos130° + u2 (b) tan145° - u2 (c) sin1180° - u2 SOLUTION (a) cos130° + u2 = cos 30° cos u - sin 30° sin u Cosine sum identity = 23 2 cos u - 1 2 sin u cos 30° = 23 2 and sin 30° = 1 2 = 23 cos u - sin u 2 a b # c = ac b ; Subtract fractions. (b) tan145° - u2 = tan 45° - tan u 1 + tan 45° tan u Tangent difference identity = 1 - tan u 1 + 1 # tan u tan 45° = 1 = 1 - tan u 1 + tan u Multiply. (c) sin1180° - u2 = sin 180° cos u - cos 180° sin u Sine difference identity = 0 # cos u - 1-12 sin u sin 180° = 0 and cos 180° = -1 = sin u Simplify. S Now Try Exercises 65, 71, and 75. EXAMPLE 5 Finding Function Values and the Quadrant of A+B Suppose that A and B are angles in standard position, such that sin A = 4 5 , p 2 6A6p, and cos B = - 5 13 , p6B6 3p 2 . Find each of the following. (a) sin1A + B2 (b) tan1A + B2 (c) the quadrant of A + B Pay attention to signs. SOLUTION (a) The identity for sin1A + B2 involves sin A, cos A, sin B, and cos B. We are given values of sin A and cos B. We must find values of cos A and sin B. sin2 A + cos2 A = 1 Fundamental identity a4 5b 2 + cos2 A = 1 sin A = 4 5 16 25 + cos2 A = 1 Square 4 5 . cos2 A = 9 25 Subtract 16 25 . cos A = - 3 5 Take square roots. Because A is in quadrant II, cos A60.
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