Chapter 4.7, Problem 4E

To determine

To fill: in the blank of the given statement.

Answer to Problem 4E

A trigonometric function has an inverse function only when its domain is restricted.

Explanation of Solution

Given information:

Given statement is

A trigonometric function has an -------------function only when its domain is restricted.

Calculation:

A trigonometric function has an inverse function only when its domain is restricted.

The sine function is not one to one over its whole domain, so in order to define an inverse. It must be restricted to an interval on which it is one to one.

  $y={\sin }^{-1}x$ is the value of $y$ such that $x=\sin y,$ where $-\frac{\pi }{2}\le y\le \frac{\pi }{2}$ .

  $y={\cos }^{-1}x$ is the value of $y$ such that $x=\cos y,$ where $-\frac{\pi }{2}\le y\le \frac{\pi }{2}$ .