How do you evaluate Sin(2 arctan(sqrt 2))?

May 7, 2016

$\frac{\sqrt{2}}{3}$.

Explanation:

Let $a = a r c \tan \sqrt{2}$. Then $\tan a = \sqrt{2}$. Tangent is positive.

Both sine and cosine have the same sign. So,

either $\sin a = \frac{\sqrt{2}}{\sqrt{3}} \mathmr{and} \cos a = \frac{1}{\sqrt{3}}$

or $\sin a = - \frac{\sqrt{2}}{\sqrt{3}} \mathmr{and} \cos a = - \frac{1}{\sqrt{3}}$.

Now, $\sin \left(2 a r c \tan \sqrt{2}\right) = \sin 2 a = 2 \sin a \cos a = \frac{\sqrt{2}}{3}$, in both the cases.