Find all angles $ heta$ between Define the unit circle in trigonometry$ and $2pi$ such that $cos( heta) = -frac{1}{2}$

To find the angles $\theta$ such that $\cos(\theta) = -\frac{1}{2}$, we start by identifying the quadrants where $\cos(\theta)$ is negative. Cosine is negative in the second and third quadrants.

First, we find the reference angle:

$\cos^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{3}$

Now, we find the angles in the second and third quadrants:

Second quadrant: $\pi – \frac{\pi}{3} = \frac{2\pi}{3}$

Third quadrant: $\pi + \frac{\pi}{3} = \frac{4\pi}{3}$

Thus, the angles are $\frac{2\pi}{3}$ and $\frac{4\pi}{3}$.

To solve for $ heta$ where $cos( heta) = -frac{1}{2}$ within the interval

txt2

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le heta < 2pi$, we recognize that the cosine function is negative in the second and third quadrants.

We first find the reference angle:

$cos^{-1}left(frac{1}{2}
ight) = frac{pi}{3}$

The corresponding angles in the second and third quadrants are:

Second quadrant: $pi – frac{pi}{3} = frac{2pi}{3}$

Third quadrant: $pi + frac{pi}{3} = frac{4pi}{3}$

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

Given $cos( heta) = -frac{1}{2}$, find $ heta$ in $[0, 2pi)$. Cosine is negative in the second and third quadrants.

Reference angle: $cos^{-1}left(frac{1}{2}
ight) = frac{pi}{3}$

Second quadrant: $pi – frac{pi}{3} = frac{2pi}{3}$

Third quadrant: $pi + frac{pi}{3} = frac{4pi}{3}$

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

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