Find the general solutions for the equation $cos( heta) = 0.5$ on the unit circle.

To solve the equation $\cos(\theta) = 0.5$, we need to find the angles on the unit circle where the cosine value equals

txt1

txt1

txt1

.5$.

First, we know that $\cos(\theta) = 0.5$ at $\theta = \frac{\pi}{3}$ and $\theta = -\frac{\pi}{3}$.

In general, these solutions can be expressed as:

$ \theta = \frac{\pi}{3} + 2k\pi $

or

$ \theta = -\frac{\pi}{3} + 2k\pi $

where $k$ is any integer.

To find all solutions to $cos( heta) = 0.5$, we recognize that it holds at specific points on the unit circle.

The cosine of $ heta$ equals

txt2

txt2

txt2

.5$ at $ heta = frac{pi}{3}$ and $ heta = frac{5pi}{3}$.

Thus, the general solutions are:

$ heta = frac{pi}{3} + 2npi $

or

$ heta = frac{5pi}{3} + 2npi $

where $n$ is any integer.

The equation $cos( heta) = 0.5$ on the unit circle has solutions at:

$ heta = frac{pi}{3} + 2mpi $

and

$ heta = frac{5pi}{3} + 2mpi $

for integer $m$.

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