# What does arccos(cos ((-2pi)/3))  equal?

Dec 24, 2015

$\frac{2 \pi}{3}$

#### Explanation:

It would look strange how is that possible! A question usually which pops up isn't $\arccos \left(\cos \left(A\right)\right) = A$.

To understand this we can use $\cos \left(- \theta\right) = \cos \left(\theta\right)$
Therefore $\cos \left(- \frac{2 \pi}{3}\right) = \cos \left(\frac{2 \pi}{3}\right)$

Following it up with $\arccos \left(\cos \left(- \frac{2 \pi}{3}\right)\right) = \arccos \left(\cos \left(\frac{2 \pi}{3}\right)\right)$

That leads us to our answer $\frac{2 \pi}{3}$.

Let us understand the same in a different manner.
The range of $\arccos \left(x\right)$ is $\left[0 , \pi\right]$.
$\cos \left(\frac{- 2 \pi}{3}\right) = - \frac{1}{2}$

The angle $\frac{- 2 \pi}{3}$ is not in the range of the function. So we select the angle in the range $\left[0 , \pi\right]$ which gives cos(x) = -1/2 that works out to $\frac{2 \pi}{3}$.