Evaluate $\tan \left(\frac{7\pi }{6}\right)$ using the unit circle.

• A. $-\sqrt{3}$
• B. $-\frac{\sqrt{3}}{3}$
• C. $\frac{1}{\sqrt{3}}$
• D. $\frac{\sqrt{3}}{3}$

Answer: (D)

Verified step by step guidance

1. Recognize that the angle given is \( \frac{7\pi}{6} \). Since angles on the unit circle are periodic with period \( 2\pi \), this angle lies in the third quadrant because \( \pi < \frac{7\pi}{6} < \frac{3\pi}{2} \).
2. Find the reference angle for \( \frac{7\pi}{6} \). The reference angle is the acute angle formed with the x-axis, calculated as \( \frac{7\pi}{6} - \pi = \frac{7\pi}{6} - \frac{6\pi}{6} = \frac{\pi}{6} \).
3. Recall the value of \( \tan \) for the reference angle \( \frac{\pi}{6} \). From the unit circle, \( \tan \left( \frac{\pi}{6} \right) = \frac{1}{\sqrt{3}} \).
4. Determine the sign of \( \tan \left( \frac{7\pi}{6} \right) \) based on the quadrant. In the third quadrant, both sine and cosine are negative, so their ratio (tangent) is positive.
5. Therefore, \( \tan \left( \frac{7\pi}{6} \right) = + \frac{1}{\sqrt{3}} \). This matches the simplified form \( \frac{\sqrt{3}}{3} \) given in the options.