718 CHAPTER 7 Trigonometric Identities and Equations We derive an alternative identity for tan A 2 using double-angle identities. tan A 2 = sin A 2 cos A 2 Definition of tangent = 2 sin A 2 cos A 2 2 cos2 A 2 = sin 2 AA 2B 1 + cos 2 AA 2B Double-angle identities tan A 2 = sin A 1 +cos A Simplify. From the identity tan A 2 = sin A 1 + cos A , we can also derive an equivalent identity. tan A 2 = 1 −cos A sin A Multiply by 2 cos A 2 in numerator and denominator. Half-Angle Identities In the following identities, the {symbol indicates that the sign is chosen on the basis of the function under consideration and the quadrant of A 2 . cos A 2 = tÅ 1 +cos A 2 sin A 2 = tÅ 1 −cos A 2 tan A 2 = tÅ 1 −cos A 1 +cos A tan A 2 = sin A 1 +cos A tan A 2 = 1 −cos A sin A Three of these identities require a sign choice. When using these identities, select the plus or minus sign according to the quadrant in which A 2 terminates. For example, if an angle A = 324°, then A 2 = 162°, which lies in quadrant II. So when A = 324°, cos A 2 and tan A 2 are negative, and sin A 2 is positive. EXAMPLE 8 Using a Half-Angle Identity to Find an Exact Value Find the exact value of cos 15° using the half-angle identity for cosine. SOLUTION cos 15° = cos 30° 2 = B1 + cos 30° 2 Choose the positive square root. = F1 + 23 2 2 = FQ 1 + 23 2 R # 2 2 # 2 = 32 + 23 2 Simplify the radicals. S Now Try Exercise 49.
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