The given angle is:

$\theta = \frac{7\pi}{6}$

This is an angle in radians. To understand this better, let's convert this into degrees and analyze it in the context of the unit circle.

1. Converting to degrees: $\theta = \frac{7\pi}{6} \times \frac{180^\circ}{\pi} = \frac{7 \times 180^\circ}{6} = 210^\circ$ Thus, $\theta = 210^\circ$.

2. Position on the Unit Circle: An angle of $210^\circ$ lies in the third quadrant of the unit circle.

3. Reference Angle: The reference angle for $210^\circ$ is: $\text{Reference Angle} = 210^\circ - 180^\circ = 30^\circ$ This implies that $\theta$ is $30^\circ$ past $180^\circ$, in the third quadrant.

4. Sine and Cosine Values: For $\theta = \frac{7\pi}{6}$:

• $\sin(\theta) = -\sin(30^\circ) = -\frac{1}{2}$
• $\cos(\theta) = -\cos(30^\circ) = -\frac{\sqrt{3}}{2}$

Would you like more details on the unit circle or any specific trigonometric functions related to this?

Here are 5 related questions you might explore:

1. What are the sine and cosine of other angles in the third quadrant?
2. How can we convert between radians and degrees more generally?
3. What is the tangent of $\theta = \frac{7\pi}{6}$?
4. How do the reference angles affect the trigonometric values?
5. How do we graph $\theta = \frac{7\pi}{6}$ on the unit circle?

Tip: In the unit circle, angles measured counterclockwise from the positive x-axis are positive, while those measured clockwise are negative.