541 5.2 Trigonometric Functions EXAMPLE 6 Determining Signs of Functions of Nonquadrantal Angles Determine the signs of the trigonometric functions of an angle in standard position with the given measure. (a) 87° (b) 300° (c) -200° SOLUTION (a) An angle of 87° is in the first quadrant, with x, y, and r all positive, so all of its trigonometric function values are positive. (b) A 300° angle is in quadrant IV, so the cosine and secant are positive, while the sine, cosecant, tangent, and cotangent are negative. (c) A -200° angle is in quadrant II. The sine and cosecant are positive, and all other function values are negative. S Now Try Exercises 97, 99, and 103. EXAMPLE 7 Identifying the Quadrant of an Angle Identify the quadrant (or possible quadrants) of an angle u that satisfies the given conditions. (a) sin u 70, tan u 60 (b) cos u 60, sec u 60 SOLUTION (a) Because sin u 70 in quadrants I and II and tan u 60 in quadrants II and IV, both conditions are met only in quadrant II. (b) The cosine and secant functions are both negative in quadrants II and III, so in this case u could be in either of these two quadrants. S Now Try Exercises 113 and 119. Figure 24(a) shows an angle u as it increases in measure from near 0° toward 90°. In each case, the value of r is the same. As the measure of the angle increases, y increases but never exceeds r, so y … r. Dividing both sides by the positive number r gives y r … 1. In a similar way, angles in quadrant IV as in Figure 24(b) suggest that -1 … y r , so -1 … y r … 1 and −1 "sin U "1. y r = sin u for any angle u. Similarly, −1 "cos U "1. x r = cos u for any angle u. The tangent of an angle is defined as y x . It is possible that x 6y, x = y, or x 7y. Thus, y x can take any value, so tan U can be any real number, as can cot U. The functions sec u and csc u are reciprocals of the functions cos u and sin u, respectively, making sec U "−1 or sec U #1 and csc U "−1 or csc U #1. x y 0 x y r (r, 0) (0, –r) u (b) Figure 24 x y 0 x y r (r, 0) (0, r) u (a)
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