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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 \( \left( -2\sqrt{3}, -2 \right) \).

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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.

Find the exact coordinates of the point where the terminal side of the angle θ = 7π/6 (in standard position) intersects a circle of radius 4 centered at the origin.

Learn how to find the exact coordinates where the terminal side of the angle θ = 7π/6 intersects a circle of radius 4 centered at the origin. The problem explores trigonometry and the unit circle, providing a step-by-step solution suitable for students in Grades 9-12.

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