Find the angles on the unit circle where $cos( heta) = -frac{1}{2}$.
Answer 1
Thomas Walker
To find the angles where $\cos(\theta) = -\frac{1}{2}$ on the unit circle, we need to consider the unit circle properties and the cosine function.
1. The cosine of an angle represents the x-coordinate on the unit circle.
2. $\cos(\theta) = -\frac{1}{2}$ corresponds to the x-coordinate -1/2.
3. The angles with $\cos(\theta) = -\frac{1}{2}$ are in the second and third quadrants because cosine is negative in these quadrants.
4. The reference angle for $\cos(\theta) = \frac{1}{2}$ is $\theta = \frac{\pi}{3}$.
5. Therefore, the angles are:
$\theta = \pi – \frac{\pi}{3} = \frac{2\pi}{3}$
$\theta = \pi + \frac{\pi}{3} = \frac{4\pi}{3}$
Thus, the angles are $\frac{2\pi}{3}$ and $\frac{4\pi}{3}$.
Answer 2
Alex Thompson
To determine the angles where $cos( heta) = -frac{1}{2}$, follow these steps:
1. Locate the unit circle and identify where the cosine value is -1/2.
2. The cosine value represents the horizontal coordinate on the unit circle.
3. Cosine is negative in the second and third quadrants.
4. The reference angle for $cos( heta) = frac{1}{2}$ is $frac{pi}{3}$.
5. Therefore, the angles can be calculated as:
$ heta = pi – frac{pi}{3} = frac{2pi}{3}$
$ heta = pi + frac{pi}{3} = frac{4pi}{3}$
Hence, the angles where cosine equals -1/2 are $frac{2pi}{3}$ and $frac{4pi}{3}$.
Answer 3
Emily Hall
To find the angles where $cos( heta) = -frac{1}{2}$:
1. Identify the unit circle sections where cosine is -1/2.
2. These sections are the second and third quadrants.
3. The reference angle is $frac{pi}{3}$.
4. Calculate the angles:
$ heta = pi – frac{pi}{3} = frac{2pi}{3}$
$ heta = pi + frac{pi}{3} = frac{4pi}{3}$
The angles are $frac{2pi}{3}$ and $frac{4pi}{3}$.
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