584 CHAPTER 5 Trigonometric Functions Concepts Examples Trigonometric Functions Trigonometric Functions Let 1x, y2 be a point other than the origin on the terminal side of an angle u in standard position. The distance from the point to the origin is r = 2x2 + y2. The six trigonometric functions of u are defined as follows. sin U = y r cos U = x r tan U = y x 1 x 302 csc U = r y sec U = r x cot U = x y 1 y 302 1x 302 1 y 302 See the summary table of trigonometric function values for quadrantal angles in this section. Reciprocal Identities sin U = 1 csc U cos U = 1 sec U tan U = 1 cot U csc U = 1 sin U sec U = 1 cos U cot U = 1 tan U Pythagorean Identities sin2 U +cos2 U =1 tan2 U +1 =sec2 U 1 +cot2 U =csc2 U Quotient Identities sin U cos U =tan U cos U sin U =cot U Signs of the Trigonometric Functions II Sine and cosecant positive x < 0, y > 0, r > 0 I All functions positive x > 0, y > 0, r > 0 III Tangent and cotangent positive x < 0, y < 0, r > 0 IV Cosine and secant positive x > 0, y < 0, r > 0 x y 0 If the point 1-2, 32 is on the terminal side of an angle u in standard position, find the values of the six trigonometric functions of u. Here x = -2 and y = 3, so r = 21-222 + 32 = 24 + 9 = 213. sin u = 3213 13 cos u = - 2213 13 tan u = - 3 2 csc u = 213 3 sec u = - 213 2 cot u = - 2 3 If cot u = - 2 3 , find tan u. tan u = 1 cot u = 1 - 2 3 = - 3 2 Find sin u and tan u, given that cos u = 23 5 and sin u 60. sin2 u + cos2 u = 1 Pythagorean identity sin2 u + ¢ 23 5 ≤ 2 = 1 Replace cos u with 23 5 . sin2 u + 3 25 = 1 Square 23 5 . sin2 u = 22 25 Subtract 3 25 . sin u = - 222 5 Choose the negative root. To find tan u, use the values of sin u and cos u from above and the quotient identity tan u = sin u cos u . tan u = sin u cos u = - 222 523 5 = - 222 23 # 23 23 = - 266 3 Simplify the complex fraction, and rationalize the denominator. Identify the quadrant(s) of any angle u that satisfies sin u 60, tan u 70. Because sin u 60 in quadrants III and IV, and tan u 70 in quadrants I and III, both conditions are met only in quadrant III. 5.2
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