# How do you evaluate  sin(tan^-1(7/9))?

Jan 4, 2017

$\sin \left({\tan}^{- 1} \left(\frac{7}{9}\right)\right) = \frac{7}{\sqrt{130}}$

#### Explanation:

Let $\theta = {\tan}^{- 1} \left(\frac{7}{9}\right)$

then $\tan \theta = \frac{7}{9}$ or $\cot \theta = \frac{9}{7}$

and hence $\sin \left({\tan}^{- 1} \left(\frac{7}{9}\right)\right)$

= $\sin \theta = \frac{1}{\csc} \theta$

= $\frac{1}{\sqrt{{\csc}^{2} \theta}}$

= $\frac{1}{\sqrt{1 + {\cot}^{2} \theta}}$

= $\frac{1}{\sqrt{1 + {\left(\frac{9}{7}\right)}^{2}}}$

= 1/sqrt(1+81/49)=1/sqrt(130/49

= $\frac{7}{\sqrt{130}}$