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.