610 CHAPTER 6 The Circular Functions and Their Graphs EXAMPLE 1 Finding Exact Circular Function Values Find the exact values of sin 3p 2 , cos 3p 2 , and tan 3p 2 . SOLUTION Evaluating a circular function at the real number 3p 2 is equivalent to evaluating it at 3p 2 radians. An angle of 3p 2 radians intersects the unit circle at the point 10, -12, as shown in Figure 15. Because sin s = y, cos s = x, and tan s = y x , it follows that sin 3p 2 = -1, cos 3p 2 = 0, and tan 3p 2 is undefined. S Now Try Exercises 13 and 15. x y (0, –1) (0, 1) (–1, 0) (1, 0) U = 3P 2 0 Figure 15 (c) An angle of 2p 3 radians corresponds to an angle of 120°. In standard position, 120° lies in quadrant II with a reference angle of 60°. Cosine is negative in quadrant II. cos 2p 3 = cos 120° = -cos 60° = - 1 2 Reference angle S Now Try Exercises 19, 25, 29, and 33. EXAMPLE 2 Finding Exact Circular Function Values Find each exact function value using the specified method. (a) Use Figure 13 to find the exact values of cos 7p 4 and sin 7p 4 . (b) Use Figure 13 and the definition of the tangent to find the exact value of tan A - 5p 3 B. (c) Use reference angles and radian-to-degree conversion to find the exact value of cos 2p 3 . SOLUTION (a) In Figure 13, we see that the real number 7p 4 corresponds to the unit circle point Q 22 2 , - 22 2 R . cos 7p 4 = 22 2 and sin 7p 4 = - 22 2 (b) Moving around the unit circle 5p 3 units in the negative direction yields the same ending point as moving around p 3 units in the positive direction. Thus, - 5p 3 corresponds to Q 1 2 , 23 2 R. tan a- 5p 3 b = tan p 3 = 23 2 1 2 = 23 2 , 1 2 = 23 2 # 2 1 = 23 tan s = y x Simplify this complex fraction.
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