Question

Solve tan2x=sqrt3 for 0<x<360?

Answer 1

`x in {15^circ, 105^circ, 195^circ,285^circ}`

Explanation:

Let `theta=2x`
So that the given `tan(2x)=sqrt(3)`
becomes `tan(theta)=sqrt(3)`

This is one of the standard reference angles, namely `30^circ`

Based on the CAST quadrant layout we know that this reference angle will apply to Quadrants 1 and 3;

that is to the angles `30^circ` and `180^circ+30^circ=210^circ`
for `theta in [0:360^circ]` i.e. for `x in [0^circ:180^circ]`
and
to the angles `360^circ+30^circ=390^circ` and `360^circ+180^circ+30^circ=570^circ`
for `theta in (360^circ:720^circ]` i.e. for `x in (180^circ:360^circ]`

Therefore `theta in {30^circ,210^circ,390^circ,570^circ}`
and since `2x=theta`
`color(white)("XXXXX")x in {15^circ,105^circ,195^circ,285^circ}`