Answer
The solutions of the equation are $x=\frac{2\pi }{3}+2n\pi $ or $x=\frac{4\pi }{3}+2n\pi $, where n is any integer.
Work Step by Step
The x in $\cos x=-\frac{1}{2}$. Because $\cos \frac{\pi }{3}=\frac{1}{2}$, and the solutions of $\cos x=-\frac{1}{2}$ in [0,2 $\pi $ ) are:
$\begin{align}
& x=\pi -\frac{\pi }{3} \\
& =\frac{3\pi }{3}-\frac{\pi }{3} \\
& =\frac{2\pi }{3}
\end{align}$
$\begin{align}
& x=\pi +\frac{\pi }{3} \\
& =\frac{3\pi }{3}+\frac{\pi }{3} \\
& =\frac{4\pi }{3}
\end{align}$
The period of the sine function is $2\pi $, and the solutions are given by:
$x=\frac{2\pi }{3}+2n\pi $ or $x=\frac{4\pi }{3}+2n\pi $, where n is any integer.
