Trigonometry

Firstly, consider the range of 3x.
Since -π < x < π, then -3π < 3x < 3π

4 cos(3x) + 2 = 0
4 cos(3x) = -2
cos(3x) = -1/2
3x = (π - π/3), (π + π/3), (π - π/3 + 2π), (π - π/3 - 2π), (π + π/3 - 2π), (π +π/3 - 4π)
3x = 2π/3, 4π/3, 8π/3, -4π/3,-2π/3, -8π/3
x = -8π/9, -4π/9, -2π/9, 2π/9, 4π/9, 8π/9

2015-05-19 00:10:40 補充：
By using calculator, cos⁻¹(-1/2) only gives one answer 2π/3, the "smallest positive angle of cos⁻¹(-1/2) in the range 0 ≤ 3x ≤ 2π".

If you want to find more answers, you have to use your calculator to find cos⁻¹(+1/2) which gives π/3 as the answer.

2015-05-19 00:11:03 補充：
When cos(3x) = -ve, 3x lies in the 2nd and 3nd quadrants. Then 3x = π ± π/3 in the range 0 ≤ 3x ≤ 2π.

Finally, ± 2π, ± 4π ...... until all the answers of 3x are found in the range -3π ≤ 3x ≤ 3π.

Calculator: Radian measure
4 cos3x +2=0
4 cos 3x =-2
cos 3x = -1/2
3x = cos^-1(-1/2)
x = cos^-1(-1/2)/3
x = 0.6981317 rad

*calculator: cos^-1 : "shift" + "cos"