# How do you evaluate tan(cos^-1(-2/3)) without a calculator?

Aug 21, 2016

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

#### Explanation:

Let a = cos&(-1)(-2/3) in Q2.

Tangent is negative in Q2, and sine is positive..

Now, $\cos a = - \frac{2}{3}$ and, the given expression is

$\tan a = \sin \frac{a}{\cos} a . = \frac{\sqrt{1 - {\cos}^{2} a}}{\cos} a$

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

$= - \frac{\sqrt{5}}{2}$