# How do you evaluate cos(tan^-1(1/2)) without a calculator?

Sep 23, 2016

$\pm \frac{2}{\sqrt{5}}$

#### Explanation:

let ${\tan}^{-} 1 \left(\frac{1}{2}\right) = a \implies \tan a = \frac{1}{2}$

$H = \sqrt{{2}^{2} + {1}^{2}} = \sqrt{5}$

$\cos \left({\tan}^{-} 1 \left(\frac{1}{2}\right)\right) = \cos a = \frac{2}{H} = \frac{2}{\sqrt{5}}$

As tangent is positive in both the first and the third quadrants,
$\implies \cos a = \pm \frac{2}{\sqrt{5}}$

positive sign for $a$ in the first quadrant and negative sign for $a$ in the third quadrant.