Question

Find the x

`0 \leqslant x \leqslant 360`

`3tan {}^(3 ) = tan \: x`

Answer 1

substitute z = tan x

rewrite

`3z^3 = z\\3z^3-z = 0\\z(3z^2-1)=0`

this has three solutions:

(1) z = 0

(2+3)

`3z^2 = 1\\|z| = (1)/(√(3))\\`

`z = \pm \frac {1}{ √(3)}`

So, in the range
`0\leq x\leq 360` these all are solutions:

for (1) tan x is 0 for x=0, 180, and 360 degrees

for (2) tan x is 1/sqrt(3) for π/6 and 7π/6

for (3) tan x is -1/sqrt(3) for -π/6 and 5π/6