tan^2 3x = 3
tan 3x = +/- sqrt(3)
tan u = +/- sqrt(3) at u = pi/3 , 2pi/3 , 4pi/3, and 5pi/3 (where y = +/- sqrt(3)/2 and x = +/- 1/2)
so the general solution is u = pi/3 + n * pi or u = 2pi/3 + n * pi (where n is an integer)
here, u = 3x
so one series of solutions is 3x = pi/3 + n * pi ==> x = pi/9 + n * pi/3
and the other is 3x = 2pi/3 + n * pi ==> x = 2pi/9 + n * pi/3
these are the general solutions
for the 12 specific solutions between 0 and 2pi, let n = 0 to 5
n = 0 ==> x = pi/9 or 2pi/9
n = 1 ==> x = 4pi/9 or 5pi/9 (note that pi/3 = 3pi/9)
n = 2 ==> x = 7pi/9 or 8pi/9
n = 3 ==> x = 10pi/9 or 11pi/9
n = 4 ==> x = 13pi/9 or 14pi/9
n = 5 ==> x = 16pi/9 or 17pi/9
n = 6 will take us out of the interval 0 to 2pi
so 12 total solutions: n = pi/9, 2pi/9 , 4pi/9 , 5pi/9 , 7pi/9 , 8pi/9 , 10pi/9 , 11pi/9 , 13pi/9 , 14pi/9 , 16pi/9, or 17pi/9
there are 2 solutions per period, and since the period of tan x is pi ==> the period of tan 3x is pi/3, there are 6 periods of tan 3x between 0 and 2pi