Question

How do you solve `3tan^3x = tan x` in the interval 0 to 2pi?

Answer 1

so
so `x = 30 ,210,150,330,0,180, 360`

Explanation:

Divide both sides by `tan x`

`3 tan^2 x = 1`

`tan^2 x = 1/3`

`sqrt(tan^2 x) = sqrt(1/3`

`tan x =+- 1/sqrt 3`

Now we have the interval `0 - 2pi = 0^@ - 360^@`

Lets look at our unit circle;

Look it up here

`tan x =+- 1/sqrt 3`
Lets see which values work for this
so `x = 30 ,210,150,330`

`Tanx` can also = 0

so `x = 30 ,210,150,330,0,180, 360`

Answer 2

We have that

#3tan^3x = tan x=>tanx*(3*tan^2x-1)=0=> tanx=0 or tanx=+-sqrt3/3#

From `tanx=0=>x=0,x=pi,x=2pi`

From `tanx=sqrt3/3=>x=pi/6, x=7/6pi`

From `tanx=-sqrt3/3=>x=5/6*pi , x=11*pi/6`

Hence the solutions for `x` in `[0,2pi]` are

`0,pi,2pi,pi/6,5/6*pi,7/6*pi,11/6*pi`