The problem in the image asks:

"Find the exact coordinates of the point where the terminal side of the angle $\theta = \frac{7\pi}{6}$ (in standard position) intersects a circle of radius 4 centered at the origin."

Step-by-Step Solution:

1. Interpretation of the problem:

   • The angle given is $\theta = \frac{7\pi}{6}$, which is in radians.
   • The terminal side of this angle intersects a circle with a radius of 4 units, centered at the origin.
   • We need to find the coordinates of the point of intersection.

2. Find the coordinates on the unit circle:

   • On the unit circle, the coordinates of a point corresponding to an angle $\theta$ are $(\cos(\theta), \sin(\theta))$.
   • For $\theta = \frac{7\pi}{6}$, this angle lies in the third quadrant.
   • Using the reference angle $\pi - \frac{7\pi}{6} = \frac{\pi}{6}$, we know: $\cos\left(\frac{7\pi}{6}\right) = -\cos\left(\frac{\pi}{6}\right) = -\frac{\sqrt{3}}{2}$ $\sin\left(\frac{7\pi}{6}\right) = -\sin\left(\frac{\pi}{6}\right) = -\frac{1}{2}$
   • So, the coordinates on the unit circle are $\left( -\frac{\sqrt{3}}{2}, -\frac{1}{2} \right)$.

3. Adjust for the circle of radius 4:

   • Since the radius of the circle is 4, we multiply the coordinates by 4. $\text{Coordinates} = 4 \times \left( -\frac{\sqrt{3}}{2}, -\frac{1}{2} \right)$ $\text{Coordinates} = \left( -2\sqrt{3}, -2 \right)$

Final Answer:

The exact coordinates of the point where the terminal side of the angle $\theta = \frac{7\pi}{6}$ intersects the circle of radius 4 are $\left( -2\sqrt{3}, -2 \right)$.

Would you like more details on any part of the solution or have further questions?

Here are 5 related questions for practice:

1. What are the coordinates of the point where $\theta = \frac{\pi}{4}$ intersects a circle of radius 5 centered at the origin?
2. Find the coordinates where $\theta = \frac{5\pi}{4}$ intersects a unit circle.
3. Determine the terminal coordinates for $\theta = \frac{11\pi}{6}$ on a circle of radius 6.
4. What is the reference angle for $\theta = \frac{5\pi}{3}$?
5. How do you convert an angle given in degrees to radians?

Tip: When dealing with angles on the unit circle, always identify the reference angle and quadrant to easily determine the sine and cosine values.