762 CHAPTER 7 Trigonometric Identities and Equations Concepts Examples Evaluate y = cos-1 0. Write y = cos-1 0 as cos y = 0. Then y = p 2 because cos p 2 = 0 and p 2 is in the range of cos-1 x. Use a calculator to find y in radians if y = sec-11-32. With the calculator in radian mode, enter sec-11-32 as cos-1A 1 -3B to obtain y ≈1.9106332. Evaluate sinAtan-1A -3 4B B . Let u = tan-1A -3 4B. Then tanu = - 3 4. Because tan u is negative when u is in quadrant IV, sketch a triangle as shown. y x –3 5 4 u We want sinAtan-1A -3 4B B = sinu. From the triangle, we have the following. sinu = - 3 5 7.5 Inverse Circular Functions Range Inverse Function Domain Interval Quadrants of the Unit Circle y = sin-1 x 3-1, 14 C -p 2 , p 2 D I and IV y = cos-1 x 3-1, 14 30, p4 I and II y = tan-1 x 1-∞, ∞2 A -p 2 , p 2 B I and IV y = cot-1 x 1-∞, ∞2 10, p I and II y = sec-1 x 1-∞, -14 ´31, ∞2 C 0, p 2 B ´ A p 2 , pD I and II y = csc-1 x 1-∞, -14 ´31, ∞2 C -p 2 , 0B ´ A0, p 2 D I and IV x y y = sin–1x –1 0 1 – p 2 p 2 (1, ) 2 p (–1, – ) 2 p x y y = cos–1x –1 0 1 p (0, ) 2 p (–1, p) (1, 0) x y y = tan–1x –2 –1 0 1 2 – p 2 p 2 (1, ) 4 p (–1, – ) 4 p See the section for graphs of the other inverse circular (trigonometric) functions. Find the exact value of tan 67.5°. Use tanA 2 = 1 - cos A sin A with A = 135°. tan67.5° = tan 135° 2 = 1 - cos 135° sin135° = 1 - Q22 2 R 22 2 = 1 + 22 2 22 2 # 2 2 = 2 + 22 22 = 22 + 1 Rationalize the denominator and simplify. Half-Angle Identities cos A 2 = {B1 + cos A 2 sin A 2 = {B1 - cos A 2 tan A 2 = {B1 - cos A 1 + cos A tan A 2 = sinA 1 + cos A tan A 2 = 1 - cos A sinA AIn the identities involving radicals, the sign is chosen on the basis of the function under consideration and the quadrant of A 2.B
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