Answer
The solution in the interval $[0,2\pi )$ is $\frac{3\pi }{2}$.
Work Step by Step
We know that the period of the cosine function is $2\pi $. So, in the interval $(0,\,\,\pi ]$, the only value for which the cosine function is $-1$ is $\pi $.
Therefore, all the solutions to $\cos \frac{2\theta }{3}=-1$ are given by:
$\begin{align}
& \frac{2\theta }{3}=\pi +2n\pi \\
& \theta =\frac{3\pi }{2}+3n\pi
\end{align}$
Where, n is any integer. And the solution in the interval $[0,2\pi )$ is obtained by letting $n=0$. Thus, the equation is calculated by taking first $n$ as 0. It can be further simplified as follows.
$\begin{align}
& \theta =\frac{3\pi }{2}+3n\pi \\
& =\frac{3\pi }{2}+3\times 0\times \pi \\
& =\frac{3\pi }{2}+0 \\
& =\frac{3\pi }{2}
\end{align}$
