Let's analyze the problem.

We have a carousel that takes one minute (60 seconds) to complete one full revolution. The child starts at the position $(0, 1)$, which is due north on the unit circle. The carousel revolves counterclockwise.

To solve this problem, we need to determine the child's position after 75 seconds. Since the carousel revolves once every 60 seconds, the angle the child has moved through after 75 seconds can be found as a fraction of a full circle (which is $2\pi$ radians):

$\text{Fraction of a revolution} = \frac{75}{60} = 1.25 \text{ revolutions}.$

This means the child has gone through one full revolution and an additional 0.25 revolution. A full revolution brings the child back to the starting point $(0, 1)$, so we now need to calculate the position after 0.25 of a revolution, which corresponds to $0.25 \times 2\pi = \frac{\pi}{2}$ radians.

Since the carousel moves counterclockwise, we are looking at a $\frac{\pi}{2}$ (or 90-degree) rotation from the starting point $(0, 1)$. A counterclockwise 90-degree rotation moves the child to the position $(-1, 0)$, which is due west on the unit circle.

Thus, after 75 seconds, the child’s coordinates are:

$\boxed{(-1, 0)}.$

Would you like further clarification on this? Here are five related questions:

1. What would the coordinates of the child be after 30 seconds?
2. How would the solution change if the carousel revolved clockwise instead of counterclockwise?
3. What are the coordinates after 90 seconds, and why?
4. How can we generalize this to a carousel that takes $T$ seconds to complete one full revolution?
5. What is the relationship between angular velocity and time in circular motion?

Tip: When solving problems involving circular motion, always relate the angle moved to the fraction of the full revolution, then use the unit circle to find the exact coordinates.