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

To find the exact values of $\sin(\frac{7\pi}{6})$, $\cos(\frac{7\pi}{6})$, and $\tan(\frac{7\pi}{6})$ using the unit circle, we follow these steps:

1. Identify the reference angle: The reference angle for $\frac{7\pi}{6}$ is $\frac{\pi}{6}$.

2. Determine the quadrant: Since $\frac{7\pi}{6}$ is in the third quadrant, both sine and cosine are negative.

3. Evaluate sine and cosine: $ \sin(\frac{\pi}{6}) = \frac{1}{2}, \cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2} $

Thus, $ \sin(\frac{7\pi}{6}) = -\frac{1}{2}, \cos(\frac{7\pi}{6}) = -\frac{\sqrt{3}}{2} $

4. Compute tangent: $ \tan(\frac{7\pi}{6}) = \frac{\sin(\frac{7\pi}{6})}{\cos(\frac{7\pi}{6})} = \frac{-\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3} $

Thus, the exact values are: $ \sin(\frac{7\pi}{6}) = -\frac{1}{2}, \cos(\frac{7\pi}{6}) = -\frac{\sqrt{3}}{2}, \tan(\frac{7\pi}{6}) = \frac{\sqrt{3}}{3} $

To find the exact values of $sin(frac{7pi}{6})$, $cos(frac{7pi}{6})$, and $ an(frac{7pi}{6})$ using the unit circle, follow these steps:

1. Reference angle for $frac{7pi}{6}$ is $frac{pi}{6}$.

2. $frac{7pi}{6}$ is in the third quadrant, making sine and cosine negative.

3. Since $ sin(frac{pi}{6}) = frac{1}{2}, cos(frac{pi}{6}) = frac{sqrt{3}}{2} $

Thus, $ sin(frac{7pi}{6}) = -frac{1}{2}, cos(frac{7pi}{6}) = -frac{sqrt{3}}{2} $

4. Compute tangent: $ an(frac{7pi}{6}) = frac{sin(frac{7pi}{6})}{cos(frac{7pi}{6})} = frac{1}{sqrt{3}} = frac{sqrt{3}}{3} $

So, $ sin(frac{7pi}{6}) = -frac{1}{2}, cos(frac{7pi}{6}) = -frac{sqrt{3}}{2}, an(frac{7pi}{6}) = frac{sqrt{3}}{3} $

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