Suppose a planet has a radius of 5,000 km. Starting at the point on its surface at 0 degrees longitude and 0 degrees latitude,find the distance to the point on its surface at 30 degrees longitude and 60 degrees latitude.

cos c = cos b * cos a

so, he have
cos c = √3/2 * 1/2 = √3/4
c = 64.34° = 1.123 radians

so, since s=rθ, the arc length of c is 5000 * 1.123 = 5615 km

To find the distance between two points on the surface of a planet, we can use the haversine formula. The haversine formula is an equation used in navigation to calculate distances between two points on the surface of a sphere. Here's how you can use it to find the distance between the two points on the planet's surface:

1. Convert the latitude and longitude from degrees to radians. For this problem, we have:
- Point A: 0 degrees longitude, 0 degrees latitude.
- Point B: 30 degrees longitude, 60 degrees latitude.

To convert degrees to radians, multiply the degree value by π/180.

- Point A: (0 * π/180, 0 * π/180) = (0, 0)
- Point B: (30 * π/180, 60 * π/180) = (π/6, π/3)

2. Calculate the differences in latitude and longitude between the two points.
- Δφ = φ₂ - φ₁ (latitude)
- Δλ = λ₂ - λ₁ (longitude)

In our case:
- Δφ = (π/3) - 0 = π/3
- Δλ = (π/6) - 0 = π/6

3. Apply the haversine formula:

a. Calculate the haversine of half the difference in latitude and longitude:
- Haversine of Δφ/2: haversin(Δφ/2) = sin²(Δφ/2)
- Haversine of Δλ/2: haversin(Δλ/2) = sin²(Δλ/2)

b. Calculate the square of the haversine:
- h = haversin(Δφ/2) + cos(φ₁) * cos(φ₂) * haversin(Δλ/2)

c. Calculate the inverse haversine:
- c = 2 * atan2(√h, √(1-h))

d. Finally, calculate the distance by multiplying the radius of the planet by c:
- Distance = R * c, where R is the radius of the planet.

In this case, the radius of the planet is given as 5,000 km. So you can substitute R = 5,000 km into the formula and calculate the final distance.