717 7.4 Double-Angle and Half-Angle Identities EXAMPLE 7 Using a Sum-to-Product Identity Write sin 2u - sin 4u as a product of two functions. SOLUTION sin 2u - sin 4u = 2 cos a 2u + 4u 2 b sin a 2u - 4u 2 b = 2 cos 6u 2 sin a -2u 2 b Simplify the numerators. = 2 cos 3u sin1-u2 Divide. = -2 cos 3u sin u sin1-u2 = - sin u S Now Try Exercise 43. Use the identity for sin A - sin B, with A = 2u and B = 4u. Half-Angle Identities From alternative forms of the identity for cos 2A, we derive identities for sin A 2 , cos A 2 , and tan A 2 , known as half-angle identities. We derive the identity for sin A 2 as follows. cos 2x = 1 - 2 sin2 x Cosine double-angle identity 2 sin2 x = 1 - cos 2x Add 2 sin2 x and subtract cos 2x. sin x = tB1 - cos 2 x 2 Divide by 2 and take square roots. sin A 2 = tÅ 1 −cos A 2 Let 2x = A, so x = A 2 . Substitute. The {symbol indicates that the appropriate sign depends on the quadrant of A 2 . For example, if A 2 is a quadrant III angle, we choose the negative sign because the sine function is negative in quadrant III. We derive the identity for cos A 2 using another double-angle identity. cos 2x = 2 cos2 x - 1 Cosine double-angle identity 1 + cos 2x = 2 cos2 x Add 1. cos2 x = 1 + cos 2x 2 Rewrite and divide by 2. cos x = {B1 + cos 2x 2 Take square roots. cos A 2 = tÅ 1 +cos A 2 Replace x with A 2 . An identity for tan A 2 comes from the identities for sin A 2 and cos A 2 . tan A 2 = sin A 2 cos A 2 = {B1 - cos A 2 {B1 + cos A 2 = tÅ 1 −cos A 1 +cos A Remember both the positive and negative square roots.
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