Find the values of $ sin( heta) $, $ cos( heta) $, and $ an( heta) $ for $ heta = frac{7pi}{6} $ using the unit circle

To find the values of $ \sin(\theta) $, $ \cos(\theta) $, and $ \tan(\theta) $ for $ \theta = \frac{7\pi}{6} $ using the unit circle, we start by locating the angle on the unit circle:

$ \theta = \frac{7\pi}{6} $ corresponds to an angle in the third quadrant, where both sine and cosine values are negative.

In the unit circle, for $ \theta = \frac{7\pi}{6} $:

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

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

To find the tangent, use: $ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} $

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

To find the values of $ sin( heta) $, $ cos( heta) $, and $ an( heta) $ for $ heta = frac{7pi}{6} $:

Locate $ frac{7pi}{6} $ on the unit circle, which lies in the third quadrant:

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

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

Use the formula for tangent:

$ an( heta) = frac{sin( heta)}{cos( heta)} = frac{-frac{1}{2}}{-frac{sqrt{3}}{2}} = frac{1}{sqrt{3}} = frac{sqrt{3}}{3} $

For $ heta = frac{7pi}{6} $:

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

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

$ anleft( frac{7pi}{6}
ight) = frac{sin( heta)}{cos( heta)} = frac{1}{sqrt{3}} = frac{sqrt{3}}{3} $

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