# How do you evaluate the equation cos(arctan(sqrt(3))) ?

Mar 6, 2018

$\frac{1}{2}$.

#### Explanation:

The Definition of $a r c \tan$ function :

$a r c \tan x = \theta , x \in \mathbb{R} \Leftrightarrow \tan \theta = x , \theta \in \left(- \frac{\pi}{2} , \frac{\pi}{2}\right)$.

We know that, $\tan \left(\frac{\pi}{3}\right) = \sqrt{3} , \mathmr{and} , \frac{\pi}{3} \in \left(- \frac{\pi}{2} , \frac{\pi}{2}\right)$.

$\therefore a r c \tan \sqrt{3} = \frac{\pi}{3}$.

Consequently, $\cos \left(a r c \tan \sqrt{3}\right) = \cos \left(\frac{\pi}{3}\right) = \frac{1}{2}$.