Find the angle $ heta $ in the unit circle where $ cos( heta) = -frac{1}{2} $

First, recall that cosine represents the x-coordinate on the unit circle.

The cosine of $ \theta $ equals $ -\frac{1}{2} $ at the angles $ \theta = \frac{2\pi}{3} $ and $ \theta = \frac{4\pi}{3} $ in radians.

We can find these angles by considering the unit circle symmetry: for $ \cos(\theta) = -\frac{1}{2} $:

$ \theta = \pi – \frac{\pi}{3} = \frac{2\pi}{3} $

and

$ \theta = \pi + \frac{\pi}{3} = \frac{4\pi}{3} $

Therefore, the angle $ \theta $ where $ \cos(\theta) = -\frac{1}{2} $ is $ \frac{2\pi}{3} $ and $ \frac{4\pi}{3} $ radians.

Let’s start by understanding where the cosine value $ -frac{1}{2} $ occurs on the unit circle.

The cosine function is negative in the second and third quadrants. Specifically, $ cos( heta) = -frac{1}{2} $ at two angles within one period:

First solution:

$ heta = pi – frac{pi}{3} = frac{2pi}{3} $

Second solution:

$ heta = pi + frac{pi}{3} = frac{4pi}{3} $

Therefore, the angles are $ frac{2pi}{3} $ and $ frac{4pi}{3} $ radians.

The cosine of an angle on the unit circle represents its x-coordinate.

For $ cos( heta) = -frac{1}{2} $, the angles are:

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

and

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

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