`2cos x+1=0`
`<=>cos x=-1/2`
`<=>x=[+-2\pi]/3+k2\pi` `(k in ZZ)`
\(2\cos x+1=0\)
\(\Leftrightarrow2\cos x=-1\)
\(\Leftrightarrow\cos x=-\dfrac{1}{2}\)
\(\Leftrightarrow\cos x=\cos\left(\dfrac{2\pi}{3}\right)\)
\(\Leftrightarrow x=\pm\dfrac{2\pi}{3}+k2\pi\) \((k\in \mathbb Z)\)