Question

Given `tanA=-3/4, 0<A<pi`, what is `sin(2A)`?

Answer 1

`- 12/25`

Explanation:

Use trig identity:
`sin^2 x = 1/(1 + cot^2 x)`
In this case:
`sin^2 A = 1/(1 + cot^2 A) = 1/(1 + 16/9) = 9/25` --> `sin A = +- 3/5`
Because 0 < A < pi, therefore, sin A is positive.
`sin A = 3/5`. Find cos A
`cos^2 A = 1- sin^2 A = 1 - 9/25 = 16/25` --> `cos A = +- 4/5`
Because tan A is negative, therefore, A is in Q. 2, cos A is negative.
`sin (2A) = 2sin A.cos A = (3/5)(- 4/5) = - 12/25`