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.

Your procedure is correct, except you forgot to include the many other values of $\cos^{-1}(-\frac12)$.
$\cos{x} = -\frac12$ for $x = \frac{2\pi}{3}\pm2\pi n, \frac{4\pi}{3}\pm2\pi n$, where $n$ is an integer.
So you would have:
$$ \frac{3\theta}{2} = \frac{2\pi}{3} \to \theta = \frac{4\pi}{9} $$
$$ \frac{3\theta}{2} = \frac{8\pi}{3} \to \theta = \frac{16\pi}{9} $$
$$ \frac{3\theta}{2} = \frac{14\pi}{3} \to \theta = \frac{28\pi}{9} $$
$$ \frac{3\theta}{2} = \frac{20\pi}{3} \to \theta = \frac{40\pi}{9} $$
etc.
AND:
$$ \frac{3\theta}{2} = \frac{4\pi}{3} \to \theta = \frac{8\pi}{9} $$
$$ \frac{3\theta}{2} = \frac{10\pi}{3} \to \theta = \frac{20\pi}{9} $$
$$ \frac{3\theta}{2} = \frac{16\pi}{3} \to \theta = \frac{32\pi}{9} $$
$$ \frac{3\theta}{2} = \frac{22\pi}{3} \to \theta = \frac{44\pi}{9} $$
etc.
The full list of solutions would be:
$$\theta = \frac{4\pi}{9}\pm\frac{4\pi}{3}$$
$$\theta = \frac{8\pi}{9}\pm\frac{4\pi}{3}$$
