”Solve

To find the exact values of all angles $ \theta $ in the interval $ [0, 2\pi) $ that satisfy $ \cos(\theta) = -\frac{1}{2} $, we use the unit circle. The cosine value of $ -\frac{1}{2} $ corresponds to angles in the second and third quadrants. The reference angle is $ \frac{\pi}{3} $.

The angles are:

• In the second quadrant: $ \pi – \frac{\pi}{3} = \frac{2\pi}{3} $
• In the third quadrant: $ \pi + \frac{\pi}{3} = \frac{4\pi}{3} $

Thus, the solutions are:

$ \theta = \frac{2\pi}{3}, \frac{4\pi}{3} $

To find $ heta $ in $ [0, 2pi) $ such that $ cos( heta) = -frac{1}{2} $, note the reference angle $ frac{pi}{3} $ lies in the second and third quadrants:

• Second quadrant: $ pi – frac{pi}{3} = frac{2pi}{3} $
• Third quadrant: $ pi + frac{pi}{3} = frac{4pi}{3} $

Hence, $ heta = frac{2pi}{3}, frac{4pi}{3} $

The angles $ heta $ in $ [0, 2pi) $ for $ cos( heta) = -frac{1}{2} $ are:

$ heta = frac{2pi}{3}, frac{4pi}{3} $

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