Determine the exact values of the trigonometric functions for angle $ frac{7pi}{6} $

To determine the exact values of the trigonometric functions for angle $ \frac{7\pi}{6} $, we follow these steps:

1. Recognize that $ \frac{7\pi}{6} $ is in the third quadrant.
2. Calculate the reference angle:
$ \pi – \frac{7\pi}{6} = \frac{\pi}{6} $.

The sine and cosine values in the third quadrant are negative:

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

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

3. Calculate the tangent value:

$ \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} $

To determine the exact values of the trigonometric functions for angle $ frac{7pi}{6} $:
The angle is in the third quadrant, and the reference angle is $ frac{pi}{6} $.

The sine and cosine are:

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

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

The tangent value is:

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

For angle $ frac{7pi}{6} $:
$sin = -frac{1}{2}, cos = -frac{sqrt{3}}{2}, an = frac{sqrt{3}}{3}$

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