535 5.2 Trigonometric Functions EXAMPLE 1 Finding Function Values of an Angle The terminal side of an angle u in standard position passes through the point 18, 152. Find the values of the six trigonometric functions of angle u. SOLUTION Figure 14 shows angle u and the triangle formed by dropping a perpendicular from the point 18, 152 to the x-axis. The point 18, 152 is 8 units to the right of the y-axis and 15 units above the x-axis, so x = 8 and y = 15. Now use r = 2x2 + y2 . r = 282 + 152 = 264 + 225 = 2289 = 17 We can now use these values for x, y, and r to find the values of the six trigonometric functions of angle u. sin u = y r = 15 17 cos u = x r = 8 17 tan u = y x = 15 8 csc u = r y = 17 15 sec u = r x = 17 8 cot u = x y = 8 15 S Now Try Exercise 13. x y 0 8 17 15 (8, 15) x = 8 y = 15 r = 17 u Figure 14 EXAMPLE 2 Finding Function Values of an Angle The terminal side of an angle u in standard position passes through the point 1-3, -42. Find the values of the six trigonometric functions of angle u. SOLUTION As shown in Figure 15, x = -3 and y = -4. r = 21-322 + 1-422 r = 2x2 + y2 r = 225 Simplify the radicand. r = 5 r 70 Now we use the definitions of the trigonometric functions. sin u = -4 5 = - 4 5 cos u = -3 5 = - 3 5 tan u = -4 -3 = 4 3 csc u = 5 -4 = - 5 4 sec u = 5 -3 = - 5 3 cot u = -3 -4 = 3 4 S Now Try Exercise 17. x y 0 5 –3 –4 x = –3 y = –4 r = 5 (–3, –4) u Figure 15 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 1 y 302 sec U = r x 1 x 302 cot U = x y 1 y 302
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