Question

In Problems 13–24, find the exact value of each expression.

${\tan }^{-1}\frac{\sqrt{3}}{3}$

Short answer

The exact value of ${\tan }^{-1}\frac{\sqrt{3}}{3}is\frac{\mathrm{\pi }}{6}$.

Step-by-step solution

Step 1. Given Information:   

Given expression: ${\tan }^{-1}\frac{\sqrt{3}}{3}$

We want to find the exact value of given expression.

Step 2. Solution: 

The given expression can be rewritten as:

${\tan }^{-1}\frac{\sqrt{3}}{3}=x;-\frac{\mathrm{\pi }}{2}<x<\frac{\mathrm{\pi }}{2}$

Hence we get $\tan x=\frac{\sqrt{3}}{3}$

Since we have tan^-1 function initially, the bounds on the tan function are going to be in the interval $\left(-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right)$ which represents the range of tan^-1 function.

Using the unit circle, we know that in order to get $\frac{\sqrt{3}}{3}$, x must be equal to $\frac{\mathrm{\pi }}{6}$.

Hence, we get ${\tan }^{-1}\frac{\sqrt{3}}{3}=\frac{\mathrm{\pi }}{6}$.