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

To find the exact values of $ \sin(x) $, $ \cos(x) $, and $ \tan(x) $ for $ x = \frac{7\pi}{6} $, follow these steps:

The angle $ \frac{7\pi}{6} $ is in the third quadrant.

For the sine function:

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

For the cosine function:

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

For the tangent function:

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

The angle $ frac{7pi}{6} $ lies in the third quadrant. For this angle, we can find the sine, cosine, and tangent values by using reference angles and the unit circle:

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

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

$ anleft(frac{7pi}{6}
ight) = frac{sinleft(frac{7pi}{6}
ight)}{cosleft(frac{7pi}{6}
ight)} = frac{-frac{1}{2}}{-frac{sqrt{3}}{2}} = frac{1}{sqrt{3}} = frac{sqrt{3}}{3} $

For $ x = 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{sqrt{3}}{3} $

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