Let's walk through this problem step-by-step.
1. Identify the Reference Angle:
- The given angle is [tex]\(\frac{7\pi}{6}\)[/tex].
- The reference angle for [tex]\(\frac{7\pi}{6}\)[/tex] is [tex]\(\frac{\pi}{6}\)[/tex].
2. Determine the Terminal Point for the Reference Angle:
- The terminal point for the reference angle [tex]\(\frac{\pi}{6}\)[/tex] is [tex]\(\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)\)[/tex].
3. Determine the Quadrant of the Given Angle:
- [tex]\(\frac{7\pi}{6}\)[/tex] is an angle that lies in the third quadrant. Recall that angles in the third quadrant are between [tex]\(\pi\)[/tex] and [tex]\(\frac{3\pi}{2}\)[/tex].
4. Adjust the Signs for the Coordinates:
- In the third quadrant, both the x-coordinate and y-coordinate are negative.
- Therefore, we need to modify the terminal point [tex]\(\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)\)[/tex] accordingly:
- The x-coordinate becomes [tex]\(-\frac{\sqrt{3}}{2}\)[/tex].
- The y-coordinate becomes [tex]\(-\frac{1}{2}\)[/tex].
Combining these steps, the terminal point of [tex]\(\frac{7\pi}{6}\)[/tex] is [tex]\(\left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)\)[/tex].
So, the correct answer is:
B. [tex]\(\left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)\)[/tex].
