The problem asks for calculations related to the orbit of planet XYZ around star ABC in a circular orbit with a radius of $82,000,000$ miles. Let's break down each part:

(a) Finding the angle formed by XYZ’s movement in one day (in radians):

Given:

• The period of revolution of the planet XYZ around the star ABC is 378 days.
• One full revolution (the angle around the orbit) is $2\pi$ radians (since it's a circle).

To find the angle the planet moves in one day, we need to determine the fraction of the full revolution completed in one day.

The fraction is $\frac{1 \text{ day}}{378 \text{ days}}$.

So, the angle in one day is: $\theta = \frac{1}{378} \times 2\pi = \frac{2\pi}{378} = \frac{\pi}{189} \text{ radians}.$

Thus, the angle formed by XYZ’s movement in one day is $\frac{\pi}{189}$ radians.

Would you like to proceed with parts (b) and (c) next, or do you have any questions on part (a)?

Here are some related questions to expand your understanding:

1. How would the answer change if the orbit period was shorter or longer?
2. Can we convert the angle in radians to degrees for better intuition?
3. What would happen to the orbit angle if XYZ orbited in half the time?
4. Can this formula be applied to objects in elliptical orbits?
5. How can we express this angle per hour instead of per day?

Tip: In circular motion, angles in radians are proportional to the time taken compared to the orbital period.