Using the Great Circle Sailing Formula.
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)
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°.
To calculate the great circle distance, initial course, and final course using the Great Circle Sailing Formula, follow these steps:
Step 1: Convert the latitude and longitude values 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: Use the decimal degree values to calculate the great circle distance.
The formula to calculate the great circle distance is:
Distance = 60 * arccos(sin(lat1)*sin(lat2) + cos(lat1)*cos(lat2)*cos(lon1-lon2))
Distance = 60 * arccos(sin(11.2333)*sin(-8.0167) + cos(11.2333)*cos(-8.0167)*cos(125.05-(-79.5667)))
Distance = 60 * arccos(0.4819)
Distance ≈ 4739.29 nautical miles
The great circle distance between positions A and B is approximately 4739.29 nautical miles.
Step 3: Calculate the initial course (T) using the following formula:
cos(T) = (sin(lat2) - sin(lat1)*cos(D))/(sin(D)*cos(lat1))
T = arccos(cos(T))
where D is the angular distance (i.e., the great circle distance divided by the Earth's radius, which is approximately 3440 nautical miles).
D = Distance / 3440
T = arccos((sin(-8.0167) - sin(11.2333)*cos(D))/(sin(D)*cos(11.2333)))
T ≈ 160.48°
The initial course (T) is approximately 160.48°.
Step 4: Calculate the final course (T) using the following formula:
cos(T) = (sin(lat1) - sin(lat2)*cos(D))/(sin(D)*cos(lat2))
T = arccos(cos(T))
T = arccos((sin(11.2333) - sin(-8.0167)*cos(D))/(sin(D)*cos(-8.0167)))
T ≈ 282.01°
The final course (T) is approximately 282.01°.