Chapter 9.3, Problem 3DC

Solution

To verify: Whether the domains found in exercises 1 and 2 are the only domains for which the trigonometric functions have an inverse functions.

No.

Given:

The restricted domain for the sine function to have an inverse function is $\left[-\frac{\pi }{2},\frac{\pi }{2}\right]$ , and the restricted domain for the cosine function to have an inverse function is $\left[0,\pi \right]$ .

Calculation:

The domain of sine function can be restricted to any of the intervals $\left[-\frac{3\pi }{2},-\frac{\pi }{2}\right],\left[-\frac{\pi }{2},\frac{\pi }{2}\right],\left[\frac{\pi }{2},\frac{3\pi }{2}\right]$ etc., is one-one and its range is $\left[-1,1\right]$ as shown in the graph below.

  

Similarly, the domain of cosine function can be restricted to any of the intervals $\left[-\pi ,0\right],\left[0,\pi \right],\left[\pi ,2\pi \right]$ etc., is one-one and its range is $\left[-1,1\right]$ as shown in the graph below.

  

Hence, there exist infinite restricted domains in which the trigonometric functions have inverse functions.