Your answer is correct.
√3 tan ( 3 x ) = - 1
Subtract 1 to both sides
√3 tan ( 3 x ) = - 1
Divide both sides by √3
tan ( 3 x ) = - 1/ √3
Take the invert tangent of both sides.
3 x = - π / 6 + n π
Because period of tangent is 2π
n = ±1 , ± 2 , ± 3...
x = ( - π / 6 ) / 3 + n π / 3
x = - π / 18 + n π / 3
Interval [0,2π) means x > 0 , x ⩽ 2π
x = 0 is not part of interval, x = 2 π is part of interval.
The solutions are:
for n = 1
x = - π / 18 + π / 3 =
- π / 18 + 6 π / 18 = 5 π / 18
for n = 2
x = - π / 18 + 2 π / 3 =
- π / 18 + 12 π / 18 = 11 π / 18
for n = 3
x = - π / 18 + 3 π / 3 =
- π / 18 + 18 π / 18 = 17 π / 18
for n = 4
x = - π / 18 + 4 π / 3 =
- π / 18 + 24 π / 18 = 23 π / 18
for n = 5
x = - π / 18 + 5 π / 3 =
- π / 18 + 30 π / 18 = 29 π / 18
for n = 6
x = - π / 18 + 6 π / 3 =
- π / 18 + 36 π / 18 = 35 π / 18
So:
x = 17 π / 18
x = 23 π / 18
x = 35 π / 18