Trigonometric equation $\sec(3\theta/2) = -2$ - brain dead Find $\theta$ with $\sec(3\theta/2)=-2$ on the interval $[0, 2\pi]$. I started off with $\cos(3 \theta/2)=-1/2$, thus $3\theta/2 = 2\pi/3$, but I don't know what to do afterwards, the answer should be a huge list of $\theta$s, which I cannot seem to get.

$$\sec\left(\frac{3\theta}{2}\right)=-2 \implies\cos\left(\frac{3\theta}{2}\right)=-\frac{1}{2} \implies \cos\left(\frac{3\theta}{2}\right)=\cos\left(\frac{2\pi}{3}\right)$$ 
Thus, we have a general solution as follows$$\frac{3\theta}{2}=2n\pi \pm \frac{2\pi}{3} \implies \theta=\frac{2}{3}\left(2n\pi\pm\frac{2\pi}{3}\right)=\frac{4\pi}{9}\left(3n\pm1 \right)$$Where, $n$ is any integer. All the values of $\theta$ in the given interval $[0, 2\pi]$ can be easily determined by setting different positive values of the integer $n=0, 1, 2, ..$ & selecting the correct values under given constraints, we get following three values of $\theta$ $$\theta=\frac{4\pi}{9}, \frac{8\pi}{9}, \frac{16\pi}{9}$$
