Question

How do you evaluate `sin [2 Tan^-1 (-2)]`?

Answer 1

-4/5

Explanation:

Consider the arc T in Quadrant 4, representing by the right triangle
OBM, with:
tan T = -2 --> OB = 1 ; BM = -2
Consequently, the radius hypotenuse `OM = sqrt(1 + 4) = sqrt5`
In this right triangle OBM,
`sin T = 1/sqrt5` and `cos T = -2/sqrt5`

`sin ((2tan^-2(-2)) -> sin (2arctan (-2)) = sin (2T)`
Use trig identity: sin 2a = 2sin a.cos a -->

`sin 2T = 2sin T.cos T = 2(1/sqrt5)(-2/sqrt5) = -4/5`