Answer
The solution set is $$\{\frac{5\pi}{3}+2n\pi,n\in Z\}$$
Work Step by Step
$$2\sqrt3\cos\frac{x}{2}=-3$$
1) Solve the equation over the interval $[0,2\pi)$
The interval for $x$ is $[0,2\pi)$
As a result, the interval for $\frac{x}{2}$ is $[0,\pi)$
$$2\sqrt3\cos\frac{x}{2}=-3$$
$$\cos\frac{x}{2}=-\frac{3}{2\sqrt3}=-\frac{\sqrt3}{2}$$
Over the interval $[0,\pi)$, there is only one value of $\frac{x}{2}$ where $\cos\frac{x}{2}=-\frac{\sqrt3}{2}$, which is $\{\frac{5\pi}{6}\}$
Therefore, $$\frac{x}{2}=\{\frac{5\pi}{6}\}$$
We would stop here and not solve for $x$.
2) Solve the equation for all solutions
Cosine function has period $2\pi$, so we would add $2\pi$ to all solutions found in part 1) for $\frac{x}{2}$.
$$\frac{x}{2}=\{\frac{5\pi}{6}+2n\pi, n\in Z\}$$
Thus, $$x=\{\frac{5\pi}{3}+2n\pi,n\in Z\}$$
