-2cosx=1

To solve the equation \(-2\cos(x) = 1\), follow these steps:

1. Isolate the cosine term: \[ \cos(x) = -\frac{1}{2} \]

2. Determine the angles where \(\cos(x) = -\frac{1}{2}\): The cosine function equals \(-\frac{1}{2}\) at specific angles in the unit circle:

   • In the second quadrant: \(x = \frac{2\pi}{3} + 2k\pi\)
   • In the third quadrant: \(x = \frac{4\pi}{3} + 2k\pi\)

   Here, \(k\) is any integer that accounts for the periodic nature of the cosine function.

3. Final solutions: So, the general solutions are: \[ x = \frac{2\pi}{3} + 2k\pi \quad \text{and} \quad x = \frac{4\pi}{3} + 2k\pi \quad (k \in \mathbb{Z}) \]

This represents all the angles where \(\cos(x) = -\frac{1}{2}\).