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Chapter 1 | Functions and Graphs
When evaluating an inverse trigonometric function, the output is an angle. For example, to evaluate cos −1 ⎛ ⎝ 1 2 ⎞ Clearly, many angles have this property. However, given the definition of cos −1 , we need the angle θ that not only solves this equation, but also lies in the interval [0, π ]. We conclude that cos −1 ⎛ ⎝ 1 2 ⎞ ⎠ = π 3 . We now consider a composition of a trigonometric function and its inverse. For example, consider the two expressions sin ⎛ ⎝ sin −1 ⎛ ⎝ 2 2 ⎠ , weneed to find an angle θ such that cos θ = 1 2 . ⎞ ⎠ ⎞ ⎠ and sin −1 (sin( π )). For the first one, we simplify as follows:
⎛ ⎝ sin −1
⎛ ⎝ 2 2
⎞ ⎠
⎞ ⎠ = sin
⎛ ⎝ π 4
⎞ ⎠ = 2 2 .
sin
For the second one, we have
sin −1 ⎛ ⎠ = sin −1 (0) =0. The inverse function is supposed to “undo” the original function, so why isn’t sin −1 ⎛ ⎝ sin( π ) ⎞ ⎝ sin( π ) ⎞ of inverse functions, a function f and its inverse f −1 satisfy the conditions f ⎛ ⎝ f −1 ( y ) ⎞
⎠ = π ? Recalling our definition ⎠ = y for all y in the domain of
f −1 and f −1 ⎛ ⎝ f ( x ) ⎞ ⎠ = x for all x in the domain of f , so what happened here? The issue is that the inverse sine function, sin −1 , is the inverse of the restricted sine function defined on the domain ⎡ ⎣ − π 2 , ⎤ ⎦ , it is true that sin −1 (sin x ) = x . However, for values of x outside this interval, the equation does not hold, even though sin −1 (sin x ) is defined for all real numbers x . π 2 ⎤ ⎦ . Therefore, for x in the interval ⎡ ⎣ − π 2 , π 2 What about sin(sin −1 y )? Does that have a similar issue? The answer is no . Since the domain of sin −1 is the interval [−1, 1], we conclude that sin(sin −1 y ) = y if −1≤ y ≤1 and the expression is not defined for other values of y . To summarize, sin(sin −1 y ) = y if−1≤ y ≤1 and sin −1 (sin x ) = x if − π 2 ≤ x ≤ π 2 . Similarly, for the cosine function, cos(cos −1 y ) = y if−1≤ y ≤1 and cos −1 (cos x ) = x if 0≤ x ≤ π . Similar properties hold for the other trigonometric functions and their inverses. Example 1.32 Evaluating Expressions Involving Inverse Trigonometric Functions
Evaluate each of the following expressions. a. sin −1 ⎛ ⎝ − 3 2 ⎞ ⎠
⎛ ⎝ tan −1
⎛ ⎝ − 1 3
⎞ ⎠
⎞ ⎠
b. tan
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