Determine the values of $ heta $ that satisfy both $ sin( heta) = -frac{1}{2} $ and $ cos( heta) = -frac{sqrt{3}}{2} $
Answer 1
Ava Martin
First, recognize that $ \sin(\theta) = -\frac{1}{2} $ in the third and fourth quadrants. The angles in these quadrants are $ \theta = \frac{7\pi}{6} $ and $ \theta = \frac{11\pi}{6} $.
Next, recognize that $ \cos(\theta) = -\frac{\sqrt{3}}{2} $ in the second and third quadrants. The angles in these quadrants are $ \theta = \frac{5\pi}{6} $ and $ \theta = \frac{7\pi}{6} $.
Therefore, the angle that satisfies both conditions is $ \theta = \frac{7\pi}{6} $.
Answer 2
Emma Johnson
To determine the values of $ heta $ that satisfy both $ sin( heta) = -frac{1}{2} $ and $ cos( heta) = -frac{sqrt{3}}{2} $:
1. Recognize that $ sin( heta) = -frac{1}{2} $ typically occurs at $ heta = frac{7pi}{6} $ and $ heta = frac{11pi}{6} $.
2. Recognize that $ cos( heta) = -frac{sqrt{3}}{2} $ typically occurs at $ heta = frac{5pi}{6} $ and $ heta = frac{7pi}{6} $.
The common angle is $ heta = frac{7pi}{6} $.
Answer 3
Matthew Carter
The solution to $ sin( heta) = -frac{1}{2} $ and $ cos( heta) = -frac{sqrt{3}}{2} $ is $ heta = frac{7pi}{6} $.
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