Find the angles on the unit circle where the cosine value is equal to $-frac{1}{2}$.
Answer 1
Christopher Garcia
To find the angles on the unit circle where the cosine value is equal to $-\frac{1}{2}$, we start by considering the unit circle properties and the cosine function.
The cosine value $-\frac{1}{2}$ corresponds to specific angles whose coordinates on the unit circle have an x-value of $-\frac{1}{2}$. These angles are found in the second and third quadrants of the unit circle.
We first identify the reference angle associated with the cosine value of $\frac{1}{2}$, which is $60^\circ$ or $\frac{\pi}{3}$ radians. Hence, the angles where the cosine is $-\frac{1}{2}$ are as follows:
1. Second quadrant: $180^\circ – 60^\circ = 120^\circ$ or $\pi – \frac{\pi}{3} = \frac{2\pi}{3}$ radians.
2. Third quadrant: $180^\circ + 60^\circ = 240^\circ$ or $\pi + \frac{\pi}{3} = \frac{4\pi}{3}$ radians.
Therefore, the angles on the unit circle where the cosine value is $-\frac{1}{2}$ are $120^\circ$ and $240^\circ$ or $\frac{2\pi}{3}$ and $\frac{4\pi}{3}$ radians.
Answer 2
Olivia Lee
To determine the angles on the unit circle with a cosine value of $-frac{1}{2}$, we use the properties of the unit circle and cosine function.
Cosine values are negative in the second and third quadrants. The reference angle for $frac{1}{2}$ is $60^circ$ or $frac{pi}{3}$ radians. Thus, we have:
1. Second quadrant angle: $180^circ – 60^circ = 120^circ$ or $frac{2pi}{3}$ radians.
2. Third quadrant angle: $180^circ + 60^circ = 240^circ$ or $frac{4pi}{3}$ radians.
So, the angles where the cosine value is $-frac{1}{2}$ are $120^circ$, $240^circ$ or $frac{2pi}{3}$, $frac{4pi}{3}$ radians.
Answer 3
Daniel Carter
For $cos( heta) = -frac{1}{2}$, the angles are in the second and third quadrants.
1. $120^circ$ or $frac{2pi}{3}$ radians.
2. $240^circ$ or $frac{4pi}{3}$ radians.
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