Question

In Problems 1– 8, find the exact value of each expression. Do not use a calculator.

${\tan }^{-1}(-\sqrt{3})$

Short answer

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

Step-by-step solution

Step 1. Definition 

The inverse tangent function $y={\tan }^{-1}x$means that $x=\tan y$where $-\infty <x<\infty$and $-\frac{\mathrm{\pi }}{2}<y<\frac{\mathrm{\pi }}{2}$.

Step 2. Find the exact value 

Let $\theta ={\tan }^{-1}(-\sqrt{3})$, so using the definition of inverse tangent function it can be written as $-\sqrt{3}=\tan \theta$where $-\frac{\mathrm{\pi }}{2}<\theta <\frac{\mathrm{\pi }}{2}$.

In this interval the angle for which the tangent value is $-\sqrt{3}$is $-\frac{\mathrm{\pi }}{3}$.

So the value of ${\tan }^{-1}(-\sqrt{3})$is $-\frac{\mathrm{\pi }}{3}$.