# How do you find the exact value of sin(tan^-1 -5/7)?

Jan 13, 2017

$- \frac{5}{\sqrt{74}}$

#### Explanation:

${\tan}^{- 1} \left(- \frac{5}{7}\right) = {\sin}^{- 1} \left(- \frac{5}{\sqrt{{5}^{2} + {7}^{2}}}\right) = {\sin}^{- 1} \left(- \frac{5}{\sqrt{74}}\right)$

The given expression is

$\sin \left({\sin}^{- 1} \left(- \frac{5}{\sqrt{74}}\right)\right) = - \frac{5}{\sqrt{74}}$.

Note that , if arc tan is negative, the angle is in ${Q}_{4}$, and so,

$a r c \sin \in {Q}_{4}$ is also negative.