Question

Evaluate sin $\left({\tan }^{-1}\frac{3}{4}\right)$.

Answer 1

We know that
${\tan }^{-1}x={\sin }^{-1}\frac{x}{\sqrt{1+x^2}}$

$$\begin{gathered} \therefore \sin \left({\tan }^{-1}\frac{3}{4}\right)=\sin \left\{{\sin }^{-1}\left(\frac{\frac{3}{4}}{\sqrt{1+\frac{9}{16}}}\right)\right\} \\ =\sin \left\{{\sin }^{-1}\left(\frac{\frac{3}{4}}{{\displaystyle \frac{5}{4}}}\right)\right\} \\ =\sin \left({\sin }^{-1}\frac{3}{5}\right) \\ =\frac{3}{5}\left[\because \sin \left({\sin }^{-1}x\right)=x\right] \end{gathered}$$

​∴ $\sin \left({\tan }^{-1}\frac{3}{4}\right)=\frac{3}{5}$