Determine the values of $sin$, $cos$, and $ an$ for an angle of $frac{7pi}{6}$ on the unit circle

First, locate the angle $\frac{7\pi}{6}$ on the unit circle. This angle corresponds to 210 degrees.

The coordinates of the point on the unit circle at this angle are:

$ \left( -\frac{\sqrt{3}}{2}, -\frac{1}{2} \right) $

Thus:

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

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

$ \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} $

At angle $frac{7pi}{6}$ (210 degrees), the coordinates are:

$ -frac{sqrt{3}}{2}, -frac{1}{2} $

So:

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

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

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

For $frac{7pi}{6}$ (210°):

$ 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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