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.

${\tan }^{-1}\left(\tan \frac{2\mathrm{\pi }}{3}\right)$

Short answer

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

Step-by-step solution

Step 1. Given Information

We are given a composite function ${\tan }^{-1}\left(\tan \frac{2\mathrm{\pi }}{3}\right)$.

We need to find its exact value.

Step 2. Concept Used

The tangent function and its inverse, form these properties

${\tan }^{-1}\left(\tan x\right)=x$ where $-\frac{\mathrm{\pi }}{2}<x<\frac{\mathrm{\pi }}{2}$

and

$\tan \left({\tan }^{-1}x\right)=x$where$-\infty <x<\infty$

Step 3. Find the exact value

The angle $\frac{2\mathrm{\pi }}{3}$is not in the interval $\left(-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right)$, so we cannot use the property ${\tan }^{-1}\left(\tan x\right)=x$.

Angle $\frac{2\mathrm{\pi }}{3}$is in the second quadrant so its tangent value is negative, and thus the corresponding reference angle is $\frac{2\mathrm{\pi }}{3}-\mathrm{\pi }=-\frac{\mathrm{\pi }}{3}$. So we get

$\tan \left(\frac{2\mathrm{\pi }}{3}\right)=\tan \left(-\frac{\mathrm{\pi }}{3}\right)$

Now, $-\frac{\mathrm{\pi }}{3}$lies in the interval $\left(-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right)$. So using the property formed by tangent and its inverse function we have

$\begin{array}{rcl}{\tan }^{-1}\left(\tan \frac{2\mathrm{\pi }}{3}\right)& =& {\tan }^{-1}\left(\tan -\frac{\mathrm{\pi }}{3}\right)\\ & =& -\frac{\mathrm{\pi }}{3}\end{array}$