Chapter 7, Problem 17RE

To determine

To calculate: The exact value of the expression $\cos \left[{\cos }^{-1}\left(-0.3\right)\right]$ .

Answer to Problem 17RE

The exact value of the expression $\cos \left[{\cos }^{-1}\left(-0.3\right)\right]$ is $-0.3$ .

Explanation of Solution

Given information:

The expression is $\cos \left[{\cos }^{-1}\left(-0.3\right)\right]$ .

Formula used:

For the cosine function and its inverse, the following holds:

  $\begin{array}{c}f\left(f^{-1}\left(x\right)\right)=\cos \left({\cos }^{-1}\left(x\right)\right)\\ =x\end{array}$

Where, $-1\le x\le 1$

Calculation:

Consider the expression $\cos \left[{\cos }^{-1}\left(-0.3\right)\right]$ .

Observe that the value $-0.3$ is lies between $-1$ and $1$ .

Recall that for the cosine function and its inverse is related as,

  $\begin{array}{c}f\left(f^{-1}\left(x\right)\right)=\cos \left({\cos }^{-1}\left(x\right)\right)\\ =x\end{array}$

Where, $-1\le x\le 1$

Therefore, the exact value of the expression $\cos \left[{\cos }^{-1}\left(-0.3\right)\right]$ is,

  $\cos \left[{\cos }^{-1}\left(-0.3\right)\right]=-0.3$

Thus, the exact value of the expression $\cos \left[{\cos }^{-1}\left(-0.3\right)\right]$ is $-0.3$ .