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Answer 1
Henry Green
To find all solutions for $ \cos(\theta) = -\frac{1}{2} $ in the range $ [0, 2\pi] $, we need to determine where the cosine function is -1/2 on the unit circle:
Cosine is negative in the second and third quadrants. The reference angle for $ \cos^{-1}(-\frac{1}{2}) $ is $ \frac{\pi}{3} $.
Therefore, the solutions are:
$ \theta_1 = \pi – \frac{\pi}{3} = \frac{2\pi}{3} $
$ \theta_2 = \pi + \frac{\pi}{3} = \frac{4\pi}{3} $
Thus, the solutions are $ \theta = \frac{2\pi}{3} $ and $ \theta = \frac{4\pi}{3} $.
Answer 2
Emily Hall
To solve $ cos( heta) = -frac{1}{2} $ in the range $ [0, 2pi] $, identify where cosine is negative and has a value of -1/2:
– It occurs in the second quadrant at $ heta = pi – frac{pi}{3} = frac{2pi}{3} $.
– It also occurs in the third quadrant at $ heta = pi + frac{pi}{3} = frac{4pi}{3} $.
The solutions are $ heta = frac{2pi}{3} $ and $ heta = frac{4pi}{3} $.
Answer 3
Maria Rodriguez
Solve $ cos( heta) = -frac{1}{2} $ in $ [0, 2pi] $:
The solutions are $ heta = frac{2pi}{3} $ and $ heta = frac{4pi}{3} $.
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