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5.5 Inverse Trigonometric Functions and Their Graphs Copyright © Cengage Learning. All rights reserved. Inverse Trigonometric Functions and Their Graphs The inverse of a function f is a function f –1 that reverses the rule of f. For a function to have an inverse, it must be one-to-one. Since the trigonometric functions are not one-to-one, they do not have inverses. However, we can restrict the domains of the trigonometric functions in such a way that the resulting functions are oneto-one! 2 The Inverse Sine Function 3 The Inverse Sine Function Let's first consider the sine function. There are many ways to restrict the domain of sine so that the new function is one-to-one. What we are going to do is to restrict the domain to the interval [– /2, /2]. The reason for this choice is that sine is one-to-one on this interval and moreover attains each of the values in its range on this interval. 4 The Inverse Sine Function From Figure 1 we see that sine is one-to-one on this restricted domain (by the Horizontal Line Test) and so has an inverse. y = sin x y = sin x, Graphs of the sine function and the restricted sine function Figure 1 5 The Inverse Sine Function We can now define an inverse sine function on this restricted domain. The graph of y = sin–1x is shown in Figure 2; it is obtained by reflecting the graph of y = sin x, – /2 x /2, in the line y = x. Graphs of y = sin–1x 6 The Inverse Sine Function Thus y = sin–1x is the number in the interval [– /2, /2] whose sine is x. In other words, sin(sin–1x) = x. 7 The Inverse Sine Function In fact, from the general properties of inverse functions, we have the following cancellation properties. When evaluating expressions involving sin–1, we need to remember that the range of sin–1 is the interval [– /2, /2]. 8 Example 3 – Evaluating Expressions with Inverse Sine Find each value. (a) sin–1 (b) sin–1 9 The Inverse Cosine Function If the domain of the cosine function is restricted to the interval [0, ], the resulting function is one-to-one and so has an inverse. We choose this interval because on it, cosine attains each of its values exactly once (see Figure 3). y = cos x y = cos x, Graphs of the cosine function and the restricted cosine function 10 The Inverse Cosine Function Thus y = cos–1x is the number in the interval [0, ] whose cosine is x. The following cancellation properties follow from the inverse function properties. 11 The Inverse Cosine Function The graph of y = cos–1x is shown in Figure 4; it is obtained by reflecting the graph of y = cos x, 0 x , in the line y = x. Graph of y = cos–1 x Figure 4 12 Example 5 – Evaluating Expressions with Inverse Cosine Find each value. (a) cos–1 (b) cos–1 13 The Inverse Tangent Function We restrict the domain of the tangent function to the interval (– /2, /2) (JUST LIKE SINE!!!) to obtain a one-toone function. Thus y = tan–1x is the number in the interval (– /2, /2) whose tangent is x. 14 The Inverse Tangent Function The following cancellation properties follow from the inverse function properties. 15 The Inverse Tangent Function Figure 5 shows the graph of y = tan x on the interval (– /2, /2) and the graph of its inverse function, y = tan–1x. y = tan x, y = tan–1x Graphs of the restricted tangent function and the inverse tangent function 16 Example 6 – Evaluating the Inverse Tangent Function Find each value. (a) tan–11 (b) tan–1 (c) tan–1(20) 17 Summary When evaluating sin-1, cos-1, and tan-1, always look at the following domain for each function. Sine 𝜋 𝜋 − , 2 2 Cosine 0, 𝜋 Tangent 𝜋 𝜋 − , 2 2 18 Example 7 – Evaluating inverse Trig. Functions Find each value. (a) Given tan-1( (b) Given sin-1(− ) = t, find cos t. 3 ) 3 = t, find cot t. (c) Evaluate 𝑠𝑖𝑛−1 𝑡𝑎𝑛 (d) Evaluate 𝑐𝑜𝑠 𝑠𝑖𝑛−1 𝜋 4 1 − 2 . . 19