474 CHAPTER 7 Analytic Trigonometry 2 Find the Value of an Inverse Sine Function For some numbers x, it is possible to find the exact value of = − y x sin . 1 Finding the Exact Value of an Inverse Sine Function Find the exact value of: − sin 11 Solution EXAMPLE 1 Let θ = − sin 1. 1 Then θ is the angle, π θ π − ≤ ≤ 2 2 , whose sine equals 1. θ π θ π θ π θ π = − ≤ ≤ = − ≤ ≤ − sin 1 2 2 sin 1 2 2 1 By definition of y x sin 1 = − Now look at Table 1 and Figure 4. θ 2 π − 3 π − 4 π − 6 π − 0 6 π 4 π 3 π 2 π sinθ 1− 3 2 − 2 2 − 1 2 − 0 1 2 2 2 3 2 1 Table 1 Figure 4 2p– 2 p– 2 u p 3p –– 2 2p 2p 5p –– 2 21 1 # u # p– 2 p– 2 2 Finding the Exact Value of an Inverse Sine Function Find the exact value of: ( ) − − sin 1 2 1 Solution EXAMPLE 2 Let θ ( ) = − − sin 1 2 . 1 Then θ is the angle, π θ π − ≤ ≤ 2 2 , whose sine equals − 1 2 . θ θ π θ π π θ π ( ) = − =− − ≤ ≤ − ≤ ≤ − sin 1 2 sin 1 2 2 2 2 2 1 The only angle θ in the interval π π ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ 2 , 2 whose sine is 1 is π 2 . Note that π sin 5 2 also equals 1, but π5 2 is not in the range of the inverse sine function; that is, it lies outside the interval π π ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ 2 , 2 , which is not allowed. So π = − sin 1 2 1 Now Work PROBLEM 11
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