Question

In Problems 9–23, find the exact value, if any, of each composite function. If there is no value, say it is “not defined.” Do not use a calculator.

${\cos }^{-1}\left(\cos \frac{3\mathrm{\pi }}{4}\right)$

Short answer

The exact value is$\frac{3\mathrm{\pi }}{4}$.

Step-by-step solution

Step 1. Given Information

We are given a composite function ${\cos }^{-1}\left(\cos \frac{3\mathrm{\pi }}{4}\right)$.

We need to find its exact value.

Step 2. Concept used

The cosine function and its inverse, form these properties

${\cos }^{-1}\left(\cos x\right)=x$where$0\le x\le \mathrm{\pi }$

and

$\cos \left({\cos }^{-1}x\right)=x$where$-1\le x\le 1$

Step 3. Find the exact value

As $\frac{3\mathrm{\pi }}{4}$lies in the interval $\left[0,\mathrm{\pi }\right]$. So using the property formed by cosine and its inverse function we have

${\cos }^{-1}\left(\cos \frac{3\mathrm{\pi }}{4}\right)=\frac{3\mathrm{\pi }}{4}$