# How do you evaluate cos^-1 (-1/2)?

Jun 19, 2017

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

#### Explanation:

Note: $\cos \left(\frac{\pi}{3}\right) = \frac{1}{2} , \cos \left(\pi - \frac{\pi}{3}\right) = - \frac{1}{2}$

Let ${\cos}^{-} 1 \left(- \frac{1}{2}\right) = \theta \therefore \cos \theta = \left(- \frac{1}{2}\right)$ We know

$\cos \left(\frac{2 \pi}{3}\right) = - \frac{1}{2} \therefore \cos \theta = \cos \left(\frac{2 \pi}{3}\right) \therefore \theta = \frac{2 \pi}{3}$

Hence ${\cos}^{-} 1 \left(- \frac{1}{2}\right) = \frac{2 \pi}{3}$

Range of ${\cos}^{-} 1 \left(x\right)$ is $\left[0 , \pi\right] \mathmr{and} \frac{2 \pi}{3}$ is in the range.

So ${\cos}^{-} 1 \left(- \frac{1}{2}\right) = \frac{2 \pi}{3}$ [Ans]