719 7.4 Double-Angle and Half-Angle Identities 0 p 3p 2 p 2 s 2 s Figure 8 Rationalize all denominators. EXAMPLE 10 Finding Function Values of s 2 Given Information about s Given cos s = 2 3 , with 3p 2 6s 62p, find sin s 2 , cos s 2 , and tan s 2 . SOLUTION The angle associated with s 2 terminates in quadrant II because 3p 2 6s 62p and 3p 4 6 s 2 6p. Divide each part by 2. See Figure 8. In quadrant II, the values of cos s 2 and tan s 2 are negative and the value of sin s 2 is positive. Use the appropriate half-angle identities and simplify. sin s 2 = D1 - 2 3 2 = B1 6 = 21 26 # 26 26 = 26 6 cos s 2 = - D1 + 2 3 2 = - B5 6 = - 25 26 # 26 26 = - 230 6 tan s 2 = sin s 2 cos s 2 = 26 6 - 230 6 = 26 - 230 = - 26 230 # 230 230 = - 2180 30 = - 625 6 # 5 = - 25 5 Notice that it is not necessary to use a half-angle identity for tan s 2 once we find sin s 2 and cos s 2 . However, using this identity provides an excellent check. S Now Try Exercise 55. EXAMPLE 9 Using a Half-Angle Identity to Find an Exact Value Find the exact value of tan 22.5° using the identity tan A 2 = sin A 1 + cos A . SOLUTION Because 22.5° = 45° 2 , replace A with 45°. tan 22.5° = tan 45° 2 = sin 45° 1 + cos 45° = 22 2 1 + 22 2 = 22 2 1 + 22 2 # 2 2 = 22 2 + 22 = 22 2 + 22 # 2 - 22 2 - 22 = 222 - 2 2 Rationalize the denominator. = 2 A 22 - 1B 2 = 22 - 1 Factor first, and then divide out the common factor. S Now Try Exercise 51. EXAMPLE 11 Simplifying Expressions Using Half-Angle Identities Simplify each expression. (a) {B1 + cos 12x 2 (b) 1 - cos 5a sin 5a SOLUTION (a) Use the identity cos A 2 = {3 1 + cos A 2 with A = 12x. {B1 + cos 12x 2 = cos 12x 2 = cos 6x
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