Calculate the exact values of the trigonometric functions for an angle of $ frac{7pi}{6} $ radians on the unit circle.

To find the trigonometric functions for the angle $ \frac{7\pi}{6} $, locate the angle on the unit circle.

First, convert $ \frac{7\pi}{6} $ to degrees: $ \frac{7\pi}{6} \times \frac{180^\circ}{\pi} = 210^\circ $

Next, find the reference angle: $ 210^\circ – 180^\circ = 30^\circ $

Using the reference angle and the unit circle values, we have:

$ \sin\left(\frac{7\pi}{6}\right) = -\sin\left(30^\circ\right) = -\frac{1}{2} $

$ \cos\left(\frac{7\pi}{6}\right) = -\cos\left(30^\circ\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} $

For the angle $ frac{7pi}{6} $ on the unit circle, its corresponding degree measure is 210°.

The reference angle is 30°.

From 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{sqrt{3}}{3} $

For the angle $ frac{7pi}{6} $, the values are:

$ 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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