Question

`4sin(2x)=tanx`, solve for x (for `0<=x<=360`). I'm getting 7 answers, but only 4 answers are listed, and they are between `0` and `180`. Can somebody solve this?

Answer 1

Solve `4sin 2x = tan x`

Ans: `0; pi; 2pi; +- 69^@23`

Explanation:

4sin 2x - tan x = 0.
`4sin 2x - sin x/(cos x) = 0`
4sin 2x.cos x - sin x = 0
8sinx.cos^2 x - sin x = 0
(condition `cos x != 0 -> x != pi/2 and x != (3pi)/2`)
sin x(8cos^2 x - 1) = 0.
a. sin x = 0 --> `x = 0; x = pi; and x = 2pi`
b. `cos^2 x = 1/8` --> `cos x = +- 1/(2sqrt2) = +- 0.35`
a. cos x = 0.35 --> `x = +- 69^@23`
b. cos x = - 0.35 --> `x = +- 110^@49`
Check by calculator.
x = 69.33 --> sin 2x = 0.66 --> 4sin 2x = 2.64
tan x = tan 69.33 = 2.64. OK
x = 110.49 --> sin 2x = sin (220.98) = -0.66 --> 4sin 2x = -2.64 --> tan 110.49 = -2.67. OK