Question

Calculate: $\sin \left(2\mathrm{arc}\tan 3\right)$

Answer 1

The given expression is $\sin \left(2\mathrm{arc}\tan 3\right)$,

which can be written as $\sin \left(2{\tan }^{-1}3\right)$

Now if${\tan }^{-1}3=\theta$

then $\tan \left(\theta \right)=3$

and $\sin \left(2{\tan }^{-1}3\right)=\sin \left(2\theta \right)$

$=\frac{2\tan \left(\theta \right)}{1+{\tan }^2\left(\theta \right)}$

$$\begin{gathered} =\frac{2\times 3}{1+9}[\because \tan \theta =3] \\ =\frac{6}{10} \\ =\frac{3}{5} \end{gathered}$$

Hence, the required value is $\frac{3}{5}$.