# How do you simplify sin (tan^-1 (-3))?

Jul 8, 2016

$- \frac{3}{\sqrt{10}}$, for principal value of ${\tan}^{- 1} \left(- 3\right)$. The general solution is $\pm \frac{3}{\sqrt{10}}$.

#### Explanation:

Let $a = {\tan}^{- 1} \left(- 3\right)$. Then, $\tan a = - 3 < 0$.

So, the principal a is in the fourth quadrant and the general a is in

either fourth or second.

And so, $\sin a = - \frac{3}{\sqrt{10}}$ against principal a and the given

expression

$\sin a = - \frac{3}{\sqrt{10}}$

The general solution is $\pm \frac{3}{\sqrt{10}}$.