SECTION 7.2 The Inverse Trigonometric Functions (Continued) 489 (c) Let cot 1 2 . 1 θ = − Then cot 1 2 , 0 . θ θ π = < < Since cot 0, θ > it follows that θ lies in quadrant I. Now find cosθ because y x cos 1 = − has the same range as y x cot , 1 = − except where undefined. Use Figure 19 to find that cos 1 5 , 0 2 . θ θ π = < < So, cos 1 5 , 1 θ ( ) = − and cot 1 2 cos 1 5 1.11 1 1 θ ( ) = = ≈ − − (d) Let cot 2 . 1 θ ( ) = − − Then cot 2, 0 . θ θ π = − < < Since cot 0, θ < θ lies in quadrant II. Use Figure 20 to find that θ π θ π = − < < cos 2 5 , 2 . This means cos 2 5 , 1 θ ( ) = − − and cot 2 cos 2 5 2.68 1 1 θ ( ) ( ) − = = − ≈ − − Figure 20 θ θ π = − < < cot 2, 0 O u y x x2 1 y2 5 5 P 5 (22, 1) 5 Figure 19 θ θ π = < < cot 1 2 , 0 O u y x x2 1 y2 5 5 P 5 (1, 2) 5 Now Work PROBLEM 21 3 Find the Exact Value of Composite Functions Involving the Inverse Trigonometric Functions Figure 21 θ = tan 1 2 O u y x x2 1 y2 5 5 P 5 (2, 1) 5 Figure 22 θ =− sin 1 3 O u y x x2 1 y2 5 9 P 5 (x, 21) 3 Finding the Exact Value of an Expression Involving Inverse Trigonometric Functions Find the exact value of: sin tan 1 2 1 ( ) − EXAMPLE 3 Solution Let tan 1 2 . 1 θ = − Then tan 1 2 , θ = where 2 2 . π θ π − < < Because tan 0, θ > it follows that 0 2 , θ π < < so θ lies in quadrant I. Because y x tan 1 2 , θ = = let x 2 = and y 1. = Since r d O P , 2 1 5, 2 2 ( ) = = + = the point P x y , 2, 1 ( ) ( ) = = is on the circle x y 5. 2 2 + = See Figure 21. Then sin tan 1 2 sin 1 5 5 5 1 θ ( ) = = = − ↑ y r sinθ = Finding the Exact Value of an Expression Involving Inverse Trigonometric Functions Find the exact value of: cos sin 1 3 1( ) ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ − EXAMPLE 4 Solution Let sin 1 3 . 1 θ ( ) = − − Then sin 1 3 θ = − and 2 2 . π θ π − ≤ ≤ Because sin 0, θ < it follows that 2 0, π θ − ≤ < so θ lies in quadrant IV. Since y r sin 1 3 , θ = − = let y 1 = − and r 3. = The point P x y x x , , 1, 0, ( ) ( ) = = − > is on a circle of radius 3, x y 9. 2 2 + = See Figure 22. Then x 8 2 = and x 2 2 = so cos sin 1 3 cos 2 2 3 1 θ ( ) ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ = = − ↑ θ = x r cos
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