Question

Tanx=(2sqrt3/3)sinx What are all solutions?

Answer 1

Solve `tan x = ((2sqrt3)/3).sin x`

Ans: `0, pi, 2pi, +- pi/6`

Explanation:

`tan x = ((2sqrt3)/3).sin x`
`(sin x)/(cos x) = ((2sqrt3)/3).sin x`
`sin x = ((2sqrt3)/3).sin x.cos x`
`3sin x - (2sqrt3cos x.sin x) = 0`
`sin x (3 - 2sqrt3.cos x) = 0`

a. sin x = 0 --> x = 0; `x = pi`; and `x = 2pi`
b. `3 - (2sqrt3)cos x = 0` --> `(2sqrt3cos x) = 3`
`cos x = 3/(2sqrt3) = sqrt3/2` --> `x = +- pi/6`