Find the exact values of $ sin, cos, $ and $ an $ for $ frac{7pi}{6} $ using the unit circle.

To find the exact values of $ \sin, \cos, $ and $ \tan $ for $ \frac{7\pi}{6} $ using the unit circle, we need to determine the coordinates of the point corresponding to this angle.

Since $ \frac{7\pi}{6} $ is in the third quadrant, both sine and cosine values will be negative:

$ \sin(\frac{7\pi}{6}) = -\frac{1}{2} $

$ \cos(\frac{7\pi}{6}) = -\frac{\sqrt{3}}{2} $

Now, using the identity $ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} $:

$ \tan(\frac{7\pi}{6}) = \frac{-\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3} $

To find the exact values of $ sin, cos, $ and $ an $ for $ frac{7pi}{6} $, note that $ frac{7pi}{6} $ is in the third quadrant:

$ sin(frac{7pi}{6}) = -frac{1}{2} $

$ cos(frac{7pi}{6}) = -frac{sqrt{3}}{2} $

$ an(frac{7pi}{6}) = frac{sin(frac{7pi}{6})}{cos(frac{7pi}{6})} = frac{-frac{1}{2}}{-frac{sqrt{3}}{2}} = frac{sqrt{3}}{3} $

For $ frac{7pi}{6} $ in the third quadrant:

$ sin(frac{7pi}{6}) = -frac{1}{2} $

$ cos(frac{7pi}{6}) = -frac{sqrt{3}}{2} $

$ an(frac{7pi}{6}) = frac{sqrt{3}}{3} $

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