Which expression is equivalent to $\sin \frac{7\pi }{6}$?

• A. $-\frac{\sqrt{3}}{2}$
• B. $-\frac{1}{2}$
• C. $\frac{1}{2}$
• D. $\frac{\sqrt{3}}{2}$

Answer: (B)

Verified step by step guidance

1. Identify the given expression: \(\sin\left(\frac{7\pi}{6}\right)\).
2. Recall that \(\frac{7\pi}{6}\) radians is an angle in the third quadrant of the unit circle, where sine values are negative.
3. Use the reference angle for \(\frac{7\pi}{6}\), which is \(\frac{7\pi}{6} - \pi = \frac{\pi}{6}\).
4. Evaluate \(\sin\left(\frac{\pi}{6}\right)\), which is a well-known value: \(\frac{1}{2}\).
5. Since the angle is in the third quadrant where sine is negative, the value of \(\sin\left(\frac{7\pi}{6}\right)\) is \(-\frac{1}{2}\).