# How do you evaluate tan(cos^-1((-1/sqrt2))) without a calculator?

Feb 3, 2017

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

#### Explanation:

From the definition of ${\cos}^{- 1}$, if $\cos \theta = x$, then ${\cos}^{- 1} x = \theta$, but within range $\left[0 , \pi\right]$

As $\cos \left(\frac{3 \pi}{4}\right) = - \frac{1}{\sqrt{2}}$,

${\cos}^{- 1} \left(- \frac{1}{\sqrt{2}}\right) = \frac{3 \pi}{4}$

Hence $\tan \left({\cos}^{- 1} \left(- \frac{1}{\sqrt{2}}\right)\right) = \tan \left(\frac{3 \pi}{4}\right) = - 1$