Find the value of $ sec( heta) $ for $ heta $ in the unit circle

To find the value of $ \sec(\theta) $ for $ \theta $ in the unit circle, we need to recall the definition of secant. The secant function is the reciprocal of the cosine function:

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

Given that $ \theta $ is an angle in the unit circle, let’s consider $ \theta = \frac{\pi}{4} $ as an example. For this angle:

$ \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} $

Thus,

$ \sec\left(\frac{\pi}{4}\right) = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2} $

To find the value of $ sec( heta) $ for $ heta = frac{pi}{3} $ in the unit circle, we use the definition of secant:

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

For $ heta = frac{pi}{3} $, we have:

$ cosleft(frac{pi}{3}
ight) = frac{1}{2} $

Thus,

$ secleft(frac{pi}{3}
ight) = frac{1}{frac{1}{2}} = 2 $

To find $ sec( heta) $ for $ heta = frac{pi}{6} $, recall:

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

For $ heta = frac{pi}{6} $:

$ cos left( frac{pi}{6}
ight) = frac{sqrt{3}}{2} $

Thus,

$ sec left( frac{pi}{6}
ight) = frac{2}{sqrt{3}} = frac{2sqrt{3}}{3} $

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