Find all the solutions for $cos( heta) = -frac{1}{2}$ on the unit circle.

$ \text{We need to find all } \theta \text{ such that } \cos(\theta) = -\frac{1}{2}. $

$ \text{The values of } \theta \text{ where } \cos(\theta) = -\frac{1}{2} \text{ are at } \theta = \frac{2\pi}{3} + 2k\pi \text{ and } \theta = \frac{4\pi}{3} + 2k\pi \text{ for any integer } k. $

$ \text{Thus, all solutions are: } \theta = \frac{2\pi}{3} + 2k\pi \text{ and } \theta = \frac{4\pi}{3} + 2k\pi. $

$ ext{To solve for } cos( heta) = -frac{1}{2}, ext{ first find the reference angle.} $

$ ext{The reference angle where } cos( heta) = frac{1}{2} ext{ is } heta = frac{pi}{3}. $

$ ext{Thus, } heta = pi – frac{pi}{3} = frac{2pi}{3} ext{ and } heta = pi + frac{pi}{3} = frac{4pi}{3}. $

$ ext{Including the periodic nature of cosine, the general solutions are: } heta = frac{2pi}{3} + 2kpi ext{ and } heta = frac{4pi}{3} + 2kpi. $

$ ext{Given } cos( heta) = -frac{1}{2}, ext{ we know the reference angle is } heta = frac{pi}{3}. $

$ ext{Thus, the solutions are: } heta = frac{2pi}{3} + 2kpi ext{ and } heta = frac{4pi}{3} + 2kpi, ext{ where } k ext{ is an integer.} $

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