# How do you evaluate cos(tan^-1(3/7)) without a calculator?

Oct 7, 2016

$\cos \theta = \pm \frac{7}{\sqrt{58}}$ depending on
whether $\theta$ is in $I$ quadrant or $I I I$ quadrant.

#### Explanation:

Let ${\tan}^{- 1} \left(\frac{3}{7}\right) = \theta$, then $\tan \theta = \frac{3}{7}$

Note that we are seeking $\cos \left({\tan}^{- 1} \left(\frac{3}{7}\right)\right)$ ie. $\cos \theta$

As $\tan \theta = \frac{3}{7}$,

${\sec}^{2} \theta = 1 + {\left(\frac{3}{7}\right)}^{2} = 1 + \frac{9}{49} = \frac{58}{49}$

Hence $\sec \theta = \pm \frac{\sqrt{58}}{7}$

and $\cos \theta = \pm \frac{7}{\sqrt{58}}$ depending on whether $\theta$ is in $I$ quadrant or $I I I$ quadrant.