Using the Great Circle Sailing Formula.

Question

Calculate the Great Circle Distance (nautical miles)
Calculate the Initial course (T)
Calculate the Final course (T)

Initial Position, A: (11° 14’ N, 125° 03’ E)
Final Position, B: (08° 01’ S, 079° 34’ W)

Answer 1

To calculate the Great Circle Distance, Initial Course, and Final Course using the Great Circle Sailing Formula, we will use the following steps:

Step 1: Convert the given coordinates from degrees, minutes, and seconds to decimal degrees.

Initial Position A:
Latitude: 11° 14’ N = 11 + 14/60 = 11.2333° N
Longitude: 125° 03’ E = 125 + 3/60 = 125.05° E

Final Position B:
Latitude: 08° 01’ S = -(8 + 1/60) = -8.0167° S
Longitude: 079° 34’ W = -(79 + 34/60) = -79.5667° W

Step 2: Convert the decimal degrees to radians.

Initial Position A:
Latitude: 11.2333° N = 11.2333 * (π/180) = 0.1960 radians N
Longitude: 125.05° E = 125.05 * (π/180) = 2.1851 radians E

Final Position B:
Latitude: -8.0167° S = -8.0167 * (π/180) = -0.1397 radians S
Longitude: -79.5667° W = -79.5667 * (π/180) = -1.3864 radians W

Step 3: Apply the Great Circle Sailing Formula to calculate the Great Circle Distance (nautical miles).

Great Circle Distance = arc length = Radius of Earth * angular separation
Where the Radius of Earth is assumed to be 3440 nautical miles.

Great Circle Distance = 3440 * arccos(sin(Latitude A) * sin(Latitude B) + cos(Latitude A) * cos(Latitude B) * cos(Longitude A - Longitude B))

Great Circle Distance = 3440 * arccos(sin(0.1960) * sin(-0.1397) + cos(0.1960) * cos(-0.1397) * cos(2.1851 - (-1.3864)))

Great Circle Distance = 3440 * arccos((0.0034) + (0.9797) * (-0.0480))

Great Circle Distance = 3440 * arccos(0.3290)

Great Circle Distance ≈ 3440 * 1.2487

Great Circle Distance ≈ 4292.57 nautical miles

Therefore, the Great Circle Distance between Positions A and B is approximately 4292.57 nautical miles.

Step 4: Calculate the Initial Course (T).

Initial Course = arctan(sin(Longitude B - Longitude A) / (cos(Latitude A) * tan(Latitude B) - sin(Latitude A) * cos(Longitude B - Longitude A)))

Initial Course = arctan(sin(-1.3864 - 2.1851) / (cos(0.1960) * tan(-0.1397) - sin(0.1960) * cos(-1.3864 - 2.1851)))

Initial Course = arctan(sin(-3.5715) / (0.9797 * (-0.0480) - (0.0034) * cos(-3.5715)))

Initial Course = arctan(-0.1786 / (-0.0472 -0.0031))

Initial Course = arctan(-0.1786 / -0.0503)

Initial Course = arctan(3.5495)

Initial Course ≈ 74.99°

Therefore, the Initial Course from Position A to Position B is approximately 74.99°.

Step 5: Calculate the Final Course (T).

Final Course = arctan(sin(Longitude A - Longitude B) / (cos(Latitude B) * tan(Latitude A) - sin(Latitude B) * cos(Longitude A - Longitude B)))

Final Course = arctan(sin(2.1851 - (-1.3864)) / (cos(-0.1397) * tan(0.1960) - sin(-0.1397) * cos(2.1851 - (-1.3864))))

Final Course = arctan(sin(3.5715) / (-0.9797 * tan(0.1960) - (0.0480) * cos(3.5715 - 2.1851)))

Final Course = arctan(0.1786 / (-0.9554 - 0.0463))

Final Course = arctan(0.1786 / -1.0016)

Final Course = arctan(-0.1785)

Final Course ≈ -9.95°

Therefore, the Final Course from Position A to Position B is approximately -9.95°.