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

2 Answers
Mar 13, 2016

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

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