490 CHAPTER 7 Analytic Trigonometry Finding the Exact Value of an Expression Involving Inverse Trigonometric Functions Find the exact value of: tan cos 1 3 1( ) ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ − Solution EXAMPLE 5 Let cos 1 3 . 1 θ ( ) = − − Then cos 1 3 θ = − and 0 . θ π ≤ ≤ Because cos 0, θ < it follows that 2 , π θ π < ≤ so θ lies in quadrant II. Since x r cos 1 3 , θ = − = let x 1 = − and r 3. = The point P x y y , 1, , ( ) ( ) = = − y 0, > is on a circle of radius = + = r x y 3, 9. 2 2 See Figure 23. Then = y 8 2 and =y 2 2. Since =− x 1, we have tan cos 1 3 tan 2 2 1 2 2 1 θ ( ) ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ = = − = − − ↑ θ = y x tan Now Work PROBLEMS 33 AND 51 Figure 23 θ =− cos 1 3 O y x x2 1 y2 5 9 P 5 (21, y) 3 u 4 Write a Trigonometric Expression as an Algebraic Expression Now Work PROBLEM 61 EXAMPLE 6 Writing a Trigonometric Expression as an Algebraic Expression Write u sin tan 1 ( ) − as an algebraic expression containing u . Solution Option 1 Let u tan 1 θ = − so that u u tan , 2 2 , . θ π θ π = − < < −∞< <∞ This means sec 0. θ > Then u u u sin tan sin sin cos cos tan cos tan sec tan 1 tan 1 1 2 2 θ θ θ θ θ θ θ θ θ θ ( ) = = ⋅ = = = + = + − ↑ ↑ ↑ θ θ = Multiply by 1 cos cos θ θ θ = sin cos tan θ θ = + sec 1 tan 2 2 θ > sec 0 Option 2 Let u tan , 1 θ = − 2 2 . π θ π − < < Then u tan , θ = u . −∞< <∞ Since 2 2 , π θ π − < < then θ lies in either quadrant I or IV. If u 0, > then θ lies in quadrant I. See Figure 24(a). If u 0, < then θ lies in quadrant IV. See Figure 24(b). In either case, r u 1 . 2 = + Therefore, u u r u u sin tan sin 1 . 1 2 θ ( ) = = = + − Figure 24 (a) μ >0 O y x x2 1 y2 5 r2 P 5 (1, u) u r Figure 24 (b) μ <0 O y x x2 1 y2 5 r2 P 5 (1, u) u r ‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. 7.2 Assess Your Understanding 1. What are the domain and the range of y x sec ? = (pp. 413–414) 2. True or False The graph of y x sec = is one-to-one on the set 0, 2 2 , . π π π ) ( ⎡ ⎣ ⎢ ∪ ⎤ ⎦ ⎥ (p. 447) 3. If tan 1 2 , 2 2 , θ π θ π = − < < then sinθ = . (pp. 419–422) 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure
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