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.

${\cos }^{-1}\left(\cos \frac{15\mathrm{\pi }}{7}\right)$

Short answer

The exact value is$\frac{\mathrm{\pi }}{7}$.

Step-by-step solution

Step 1. Given Information

We are given a composite function ${\cos }^{-1}\left(\cos \frac{15\mathrm{\pi }}{7}\right)$.

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 

The angle $\begin{array}{rcl}& & \frac{15\mathrm{\pi }}{7}\end{array}$does not lie in the interval $\left[0,\mathrm{\pi }\right]$, so we cannot use the property ${\cos }^{-1}\left(\cos x\right)=x$.

Now as cosine function is cyclic with a period of $2\mathrm{\pi }$so we have $\begin{array}{rcl}& & \cos \end{array}x\begin{array}{rcl}& =& \end{array}\begin{array}{rcl}& & \cos (2\mathrm{\pi }+x)\end{array}$. Now $\begin{array}{rcl}& & \cos \frac{15\mathrm{\pi }}{7}\end{array}$can be written as

$\begin{array}{rcl}\cos \frac{15\mathrm{\pi }}{7}& =& \cos (2\mathrm{\pi }+\frac{\mathrm{\pi }}{7})\\ & =& \cos \frac{\mathrm{\pi }}{7}\end{array}$

Now, $\frac{\mathrm{\pi }}{7}$lies in the interval $\left[0,\mathrm{\pi }\right]$. So using the property formed by cosine and its inverse function we have

$\begin{array}{rcl}{\cos }^{-1}\left(\cos \frac{15\mathrm{\pi }}{7}\right)& =& {\cos }^{-1}\left(\cos \frac{\mathrm{\pi }}{7}\right)\\ & =& \frac{\mathrm{\pi }}{7}\end{array}$