Question

How do you find sin 2x, given tan x = -2 and cos x > 0?

Answer 1

Find sin 2x, knowing tan x = -2, and cos x > 0

Ans: `sin 2x = 4/5`

Explanation:

3 Trig identities to be used:
`1 + tan^2 x = 1/cos^2 x`(1)
`sin^2 x + cos ^2 x = 1` (2)
`sin 2x = 2sin x.cos x` (3)
Given tan x = -2. First find cos x and sin x
(1) --> `1 + 4 = 1/(cos^2 x)` --> `cos^2 x = 1/5` --> `cos x = +- 1/sqrt5`.
Since cos x > 0, then `cos x = 1/sqrt5.`
(2) --> `sin^2 x = 1 - cos^2 x = 1 - 1/5 = 4/5` -->`sin x = +- 2/sqrt5.`
Since cos x > 0 then `sin x = 2/sqrt5`.
(3) --> `sin 2x = 2sin x.cos x = 2(1/sqrt5)(2/sqrt5) = 4/5`