SECTION 7.1 The Inverse Sine, Cosine, and Tangent Functions 475 (Refer to Table 1 and Figure 4 on the previous page, if necessary.) The only angle in the interval π π ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ 2 , 2 whose sine is − 1 2 is π − 6 , so π ( ) − = − − sin 1 2 6 1 Finding an Approximate Value of an Inverse Sine Function Find an approximate value of each of the following functions: (a) − sin 1 3 1 (b) ( ) − − sin 1 4 1 Express the answer in radians rounded to two decimal places. EXAMPLE 3 Figure 5(a) Figure 5(b) Now Work PROBLEM 17 For most numbers x, the value = − y x sin 1 must be approximated using technology. See Example 3. Solution Because the angle is to be measured in radians, first set the mode of the graphing utility to radians. (a) Figure 5(a) shows the solution using a TI-84 Plus CE graphing calculator. Rounded to two decimal places, = − sin 1 3 0.34. 1 (b) Figure 5(b) shows the solution using Desmos. Rounded to two decimal places, ( ) − = − − sin 1 4 0.25. 1 Now Work PROBLEM 27 3 Define the Inverse Cosine Function Figure 6 shows the graph of = y x cos . Because every horizontal line = y b, where b is between −1 and 1, inclusive, intersects the graph of = y x cos infinitely many times, it follows that the cosine function is not one-to-one. Figure 6 y x x y cos , , 1 1 = −∞< <∞ − ≤ ≤ 2p– 2 p– 2 x p 3p ––– 2 2p 2p 5p ––– 2 y 21 1 y 5 b 21 # b # 1 However, if the domain of = y x cos is restricted to the interval π [ ] 0, , the restricted function π = ≤ ≤ y x x cos 0 is one-to-one and has an inverse function.* See Figure 7. * This is the generally accepted restriction to define the inverse cosine function. Figure 7 y x x y cos , 0 , 1 1 π = ≤ ≤ − ≤ ≤ p– 2 x p y 21 (0, 1) (p, 21)
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