542 CHAPTER 5 Trigonometric Functions In summary, the ranges of the trigonometric functions are as follows. Ranges ofTrigonometric Functions Trigonometric Function of U Range (Set-Builder Notation) Range (Interval Notation) sin u, cos u 5 y 0 $ y$ … 16 3-1, 14 tan u, cot u 5 y 0 y is a real number6 1-∞, ∞2 sec u, csc u 5 y 0 $ y$ Ú 16 1-∞, -14 ´31, ∞2 EXAMPLE 8 Determining Whether a Value Is in the Range of a Trigonometric Function Determine whether each statement is possible or impossible. (a) sin u = 2.5 (b) tan u = 110.47 (c) sec u = 0.6 SOLUTION (a) For any value of u, we know that -1 … sin u … 1. Here 2.5 71, so it is impossible to find a value of u that satisfies sin u = 2.5. (b) The tangent function can take on any real number value. Thus, tan u = 110.47 is possible. (c) Because $ sec u$ Ú 1 for all u for which the secant is defined, the statement sec u = 0.6 is impossible. S Now Try Exercises 121, 125, and 127. The six trigonometric functions are defined in terms of x, y, and r, where the Pythagorean theorem shows that r2 = x2 + y2 and r 70. With these relationships, knowing the value of only one function and the quadrant in which the angle lies makes it possible to find the values of the other trigonometric functions. EXAMPLE 9 Finding All Function Values Given One Value and the Quadrant Suppose that angle u is in quadrant II and sin u = 2 3 . Find the values of the five remaining trigonometric functions. SOLUTION Choose any point on the terminal side of angle u. For simplicity, since sin u = y r , choose the point with r = 3. s in u = 2 3 Given value y r = 2 3 Substitute y r for sin u.
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