Question

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

${\cos }^{-1}\left(-\frac{\sqrt{3}}{2}\right)$

Short answer

The exact value is $\frac{5\mathrm{\pi }}{6}$.

Step-by-step solution

Step 1. Definition 

The inverse cosine function $y={\cos }^{-1}x$means that $x=\cos y$where $-1\le x\le 1$and $0\le y\le \mathrm{\pi }$.

Step 2. Find the exact value 

Let $\theta ={\cos }^{-1}\left(-\frac{\sqrt{3}}{2}\right)$, so using the definition of inverse cosine function it can be written as $-\frac{\sqrt{3}}{2}=\cos \theta$where $0\le \theta \le \mathrm{\pi }$.

In this interval the angle for which the cosine value is $-\frac{\sqrt{3}}{2}$is $\frac{5\mathrm{\pi }}{6}$.

So the value of ${\cos }^{-1}\left(-\frac{\sqrt{3}}{2}\right)$is $\frac{5\mathrm{\pi }}{6}$.