488 CHAPTER 7 Analytic Trigonometry Approximating the Value of Inverse Trigonometric Functions Use a calculator to approximate each expression in radians rounded to two decimal places. (a) − sec 31 (b) csc 4 1( ) − − (c) cot 1 2 1− (d) cot 2 1( ) − − EXAMPLE 2 The only angle θ for which 0 , 2 , θ π θ π ≤ ≤ ≠ whose secant is 1 is 0, so sec 1 0. 1 = − (b) Let csc 2. 1 θ = − Then θ is the angle, 2 2 , 0, π θ π θ − ≤ ≤ ≠ whose cosecant equals 2 ( ) or, equivalently, whose sine equals 1 2 . csc 2 2 2 , 0 csc 2 2 2 , 0 1 θ π θ π θ θ π θ π θ = − ≤ ≤ ≠ = − ≤ ≤ ≠ − The only angle θ for which 2 2 , 0, π θ π θ − ≤ ≤ ≠ whose cosecant is 2 is 6 , π so csc 2 6 . 1 π = − (c) Let cot 3 . 1 θ ( ) = − − Then θ is the angle, 0 , θ π < < whose cotangent equals 3. − cot 3 0 cot 3 0 1 θ θ π θ θ π ( ) = − < < = − < < − The only angle θ for which 0 θ π < < whose cotangent is 3 − is 5 6 , π so cot 3 5 6 . 1 π ( ) − = − Now Work PROBLEM 11 NOTE Remember that the range of y x sin 1 = − is 2 , 2 π π ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ and that the range of y x cos 1 = − is 0, . π [ ] j Most calculators do not have keys that evaluate the inverse cotangent, cosecant, or secant functions. The easiest way to evaluate them is to convert each to an inverse trigonometric function whose range is the same as the one to be evaluated. In this regard, notice that y x cot 1 = − and y x sec , 1 = − except where undefined, have the same range as y x cos 1 = − and that = − y x csc , 1 except where undefined, has the same range as = − y x sin . 1 Solution First, set the calculator to radian mode. (a) Let θ = − sec 3. 1 Then sec 3 θ = and 0 , 2 . θ π θ π ≤ ≤ ≠ Now find cosθ because = − y x cos 1 has the same range as y x sec , 1 = − except where undefined. Because sec 1 cos 3, θ θ = = this means cos 1 3 . θ = Then cos 1 3 , 1 θ = − and θ = = ≈ − − sec 3 cos 1 3 1.23 1 1 ↑ Use a calculator. (b) Let csc 4 . 1 θ ( ) = − − Then csc 4, 2 2 , 0. θ π θ π θ = − − ≤ ≤ ≠ Now find sinθ because y x sin 1 = − has the same range as y x csc , 1 = − except where undefined. Because csc 1 sin 4, θ θ = = − this means sin 1 4 . θ = − Then sin 1 4 , 1 θ ( ) = − − and csc 4 sin 1 4 0.25 1 1 θ ( ) ( ) − = = − ≈ − − −
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