Find the points on the unit circle where the $ sec( heta) = 2 $, and prove their coordinates.

To find points on the unit circle where $ \sec(\theta) = 2 $, recall that:

$ \sec(\theta) = \frac{1}{\cos(\theta)} $

Thus, we need:

$ \frac{1}{\cos(\theta)} = 2 $

So:

$ \cos(\theta) = \frac{1}{2} $

The angles on the unit circle with $ \cos(\theta) = \frac{1}{2} $ are:

$ \theta = \frac{\pi}{3} \text{ and } \theta = \frac{5\pi}{3} $

The corresponding points on the unit circle are:

For $ \theta = \frac{\pi}{3} $:

$ (\cos(\frac{\pi}{3}), \sin(\frac{\pi}{3})) = \left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right) $

For $ \theta = \frac{5\pi}{3} $:

$ (\cos(\frac{5\pi}{3}), \sin(\frac{5\pi}{3})) = \left(\frac{1}{2}, -\frac{\sqrt{3}}{2}\right) $

To find points on the unit circle where $ sec( heta) = 2 $, solve:

$ sec( heta) = frac{1}{cos( heta)} = 2 $

So:

$ cos( heta) = frac{1}{2} $

Angles on the unit circle with $ cos( heta) = frac{1}{2} $ are:

$ heta = frac{pi}{3} ext{ and } heta = frac{5pi}{3} $

Points on the unit circle:

$ left(frac{1}{2}, frac{sqrt{3}}{2}
ight) ext{ and } left(frac{1}{2}, -frac{sqrt{3}}{2}
ight) $

To find points on the unit circle where $ sec( heta) = 2 $:

$ frac{1}{cos( heta)} = 2 $

So:

$ cos( heta) = frac{1}{2} $

Angles $ heta = frac{pi}{3} $ and $ heta = frac{5pi}{3} $ give points:

$ left(frac{1}{2}, frac{sqrt{3}}{2}
ight) ext{ and } left(frac{1}{2}, -frac{sqrt{3}}{2}
ight) $

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