Using your knowledge of the unit circle solve: A string with initial point P (0, -1) and length 4π/3 is wound around the unit circle in a clockwise direction. What are the coordinates of the end point of the string? °.

To solve the problem, we need to understand the movement of the point on the unit circle when the string is wound in a clockwise direction.

### Step-by-Step Breakdown:

1. **Initial Position:**
   - The string starts at point \( P(0, -1) \). On the unit circle, this corresponds to the angle \( 3\pi/2 \), which is the standard angle measured counterclockwise from the positive x-axis.

2. **Length of the String:**
   - The string is \( 4\pi/3 \) units long. On the unit circle, this length is the same as an arc length measured along the circle.
   
   Since we are moving in the **clockwise** direction, the angle will decrease from \( 3\pi/2 \).

3. **Arc Length and Angle:**
   - The relationship between the arc length \( s \) on the unit circle and the angle \( \theta \) (in radians) is:
   \[
   s = r \theta
   \]
   For the unit circle, \( r = 1 \), so the arc length is simply equal to the angle in radians. This means we need to subtract the angle corresponding to the arc length \( 4\pi/3 \) from the initial angle \( 3\pi/2 \).

4. **Subtract the Arc Length from the Starting Angle:**
   The initial angle is \( 3\pi/2 \), and we subtract \( 4\pi/3 \):
   \[
   \text{New angle} = 3\pi/2 - 4\pi/3
   \]
   To subtract these, we first find a common denominator:
   \[
   3\pi/2 = 9\pi/6, \quad 4\pi/3 = 8\pi/6
   \]
   So, the new angle is:
   \[
   9\pi/6 - 8\pi/6 = \pi/6
   \]

5. **Find the Coordinates:**
   The angle \( \pi/6 \) corresponds to the coordinates on the unit circle. Using standard values for the unit circle at \( \theta = \pi/6 \), we have:
   \[
   \left( \cos \frac{\pi}{6}, \sin \frac{\pi}{6} \right) = \left( \frac{\sqrt{3}}{2}, \frac{1}{2} \right)
   \]

### Final Answer:
The coordinates of the end point of the string are:
\[
\left( \frac{\sqrt{3}}{2}, \frac{1}{2} \right)
\]

Would you like a detailed explanation of any part of this process?

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Here are some related questions for further practice:

1. What is the relationship between arc length and angle on any circle, not just the unit circle?
2. How do you calculate the coordinates of a point on the unit circle for a given angle?
3. What happens if the string is wound counterclockwise instead of clockwise?
4. What are the coordinates of the endpoint if the string length were \( 2\pi/3 \)?
5. How can we apply this process to a unit circle with a different radius?

**Tip:** Always visualize the unit circle when dealing with angles and arc lengths to make it easier to track positions and movements.

Using your knowledge of the unit circle solve: A string with initial point P (0, -1) and length 4π/3 is wound around the unit circle in a clockwise direction. What are the coordinates of the end point of the string?

This problem involves calculating the coordinates of a point after a string of length 4π/3 is wound around the unit circle in a clockwise direction. The solution covers key trigonometric concepts such as the unit circle, radian measure, and arc length. Suitable for high school students in Grades 10-12.

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