554 CHAPTER 5 Trigonometric Functions The preceding example suggests the following diagram. x y 2108 308 608 O P r x y y = –1 r = 2 x = –Ë3 Figure 35 Special Angles as Reference Angles We can now find exact trigonometric function values of angles with reference angles of 30°, 45°, or 60°. *The authors would like to thank Bethany Vaughn and Theresa Matick, of Vincennes Lincoln High School, for their suggestions concerning this diagram. EXAMPLE 5 FindingTrigonometric Function Values of a Quadrant III Angle Find exact values of the six trigonometric functions of 210°. SOLUTION An angle of 210° is shown in Figure 35. The reference angle is 210° - 180° = 30°. To find the trigonometric function values of 210°, choose point P on the terminal side of the angle so that the distance from the origin O to P is 2. (Any positive number would work, but 2 is most convenient.) By the results from 30°960° right triangles, the coordinates of point P become A -23, -1B , with x = -23, y = -1, and r = 2. Then, by the definitions of the trigonometric functions, we obtain the following. sin 210° = -1 2 = - 1 2 csc 210° = 2 -1 = -2 cos 210° = -23 2 = - 23 2 sec 210° = 2 -23 = - 22 3 # 23 23 = - 223 3 tan 210° = -1 -23 = 12 3 # 23 23 cot 210° = -23 -1 = 23 = 23 3 S Now Try Exercise 71. Reference Angle U′ for U, where 0° *U *360°* 0 x y Q I 0 x y Q II 0 x y Q III 0 x y Q IV u9 u9 u9 u9 u9 = 1808 – u u9 = u u9 = u – 1808 u9 = 3608 – u u u u u
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