Question

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

Answer 1

`sqrt 2/3`.

Explanation:

Let `a = arc tan sqrt 2`. Then `tan a = sqrt 2`. Tangent is positive.

Both sine and cosine have the same sign. So,

either `sin a = sqrt2/ sqrt 3 and cos a = 1/sqrt 3`

or `sin a = - sqrt 2/ sqrt 3 and cos a = -1/sqrt 3`.

Now, `sin (2 arc tan sqrt 2)=sin 2a = 2 sin a cos a = sqrt2 /3`, in both the cases.