608 CHAPTER 6 The Circular Functions and Their Graphs For example, the quadrant I real number p 3 is associated with the point Q 1 2 , 23 2 R on the unit circle. Therefore, we can use symmetry to identify the coordinates of points having p 3 as reference arc. (0, 1) (1, 0) (0, –1) (–1, 0) 0 0 ° 180° 60° 90° 150° 210° 300° 360° 315° 330° 2P P 135° 120° 225° 240° 270° 45° 30° 1 2 ( , ) √3 2 1 2( , – ) √3 2 1 2 (– , – ) √3 2 ( , ) √2 2 √2 2 1 2 ( , – ) √3 2 1 2 ( , ) √3 2 1 2 (– , ) √3 2 (– , ) √2 2 √2 2 ( , – ) √2 2 √2 2 1 2 (– , – ) √3 2 (– , – ) √2 2 √2 2 1 2 (– , ) √3 2 3P 2 5P 3 7P 4 4P 3 5P 4 7P 6 5P 6 3P 4 2P 3 11P 6 P 3 P 2 P 4 P 6 The unit circle x2 + y2 = 1 x y 0 Figure 13 Symmetry and Function Values for Real Numbers with Reference Arc P 3 s Quadrant of s Symmetry Type and Corresponding Point cos s sin s P 3 I not applicable; ¢ 1 2 , 23 2 ≤ 1 2 23 2 p - p 3 = 2P 3 II y-axis; ¢- 1 2 , 23 2 ≤ - 1 2 23 2 p + p 3 = 4P 3 III origin; ¢- 1 2 , - 23 2 ≤ - 1 2 - 23 2 2p - p 3 = 5P 3 IV x-axis; ¢ 1 2 , - 23 2 ≤ 1 2 - 23 2 The ordered pair 1x, y2 represents a point on the unit circle, and therefore -1 … x … 1 and -1 … y … 1, −1 "cos s "1 and −1 "sin s "1. For any value of s, both sin s and cos s exist, so the domain of these functions is the set of all real numbers. NOTE Because cos s = x and sin s = y, we can replace x and y in the equation of the unit circle x2 + y2 = 1 and obtain the following. cos2 s +sin2 s =1 Pythagorean identity
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