Which expression is equivalent to $\sin \left(\frac{7\pi }{6}\right)$ on the unit circle?

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

Answer: (C)

Verified step by step guidance

1. First, recognize that the angle given is \(\frac{7\pi}{6}\) radians. This angle is in the third quadrant of the unit circle because \(\pi < \frac{7\pi}{6} < \frac{3\pi}{2}\).
2. Recall that the reference angle for \(\frac{7\pi}{6}\) is found by subtracting \(\pi\) from it: \(\frac{7\pi}{6} - \pi = \frac{7\pi}{6} - \frac{6\pi}{6} = \frac{\pi}{6}\).
3. Since sine corresponds to the y-coordinate on the unit circle, and sine is negative in the third quadrant, the value of \(\sin\left(\frac{7\pi}{6}\right)\) will be the negative of \(\sin\left(\frac{\pi}{6}\right)\).
4. Recall the exact value of \(\sin\left(\frac{\pi}{6}\right)\), which is \(\frac{1}{2}\).
5. Therefore, \(\sin\left(\frac{7\pi}{6}\right) = -\frac{1}{2}\). This matches the expression \(-\frac{1}{2}\) from the given options.