761 CHAPTER 7 Test Prep Concepts Examples Sum and Difference Identities cos1A - B2 = cos A cos B + sin A sin B cos1A + B2 = cos A cos B - sin A sin B sin1A + B2 = sin A cos B + cos A sin B sin1A - B2 = sin A cos B - cos A sin B tan1A + B2 = tan A + tan B 1 - tan A tan B tan1A - B2 = tan A - tan B 1 + tan A tan B Find the exact value of cos1-15°2. cos1-15°2 = cos130° - 45°2 -15° = 30° - 45° = cos 30° cos 45° + sin 30° sin 45° Cosine difference identity = 23 2 # 22 2 + 1 2 # 22 2 = 26 + 22 4 Simplify. Given cos u = - 5 13 and sin u 70, find sin 2u. Sketch a triangle in quadrant II because cos u 60 and sin u 70. Use it to find that sin u = 12 13 . sin 2u = 2 sin u cos u = 2 a 12 13b a- 5 13b = - 120 169 Write sin1-u2 sin 2u as the difference of two functions. sin1-u2 sin 2u = 1 2 3cos1-u - 2u2 - cos1-u + 2u24 = 1 2 3cos1-3u2 - cos u4 = 1 2 cos1-3u2 - 1 2 cos u = 1 2 cos 3u - 1 2 cos u Write cos u + cos 3u as a product of two functions. cos u + cos 3u = 2 cos a u + 3u 2 b cos a u - 3u 2 b = 2 cos a 4u 2 b cos a -2u 2 b = 2 cos 2u cos1-u2 = 2 cos 2u cos u y x 12 –5 13 U Substitute known values. 7.4 Double-Angle and Half-Angle Identities Double-Angle Identities cos 2A = cos2 A - sin2 A cos 2A = 1 - 2 sin2 A cos 2A = 2 cos2 A - 1 sin 2A = 2 sin A cos A tan 2A = 2 tan A 1 - tan2 A Product-to-Sum Identities cos A cos B = 1 2 3cos1A + B2 + cos1A - B24 sin A sin B = 1 2 3cos1A - B2 - cos1A + B24 sin A cos B = 1 2 3sin1A + B2 + sin1A - B24 cos A sin B = 1 2 3sin1A + B2 - sin1A - B24 Sum-to-Product Identities sin A + sin B = 2 sin a A + B 2 b cos a A - B 2 b sin A - sin B = 2 cos a A + B 2 b sin a A - B 2 b cos A + cos B = 2 cos a A + B 2 b cos a A - B 2 b cos A - cos B = -2 sin a A + B 2 b sin a A - B 2 b
RkJQdWJsaXNoZXIy NjM5ODQ=