Calculate the distance between these waypoints using Mercator sailing:

Question

Solve for true course
Initial position (A) Lat 32°17.6’ S, Long 007°14.6’W
Final position (B)Lat 29°12.3’ S, Long 001°38.4’ E

Answer 1

To calculate the distance between the waypoints using Mercator sailing, you can follow these steps:

1. Convert the latitude and longitude values from degrees, minutes, and seconds to decimal degrees.

For initial position (A):
Latitude: 32°17.6’ S = -32.2933°
Longitude: 007°14.6’ W = -7.2433°

For final position (B):
Latitude: 29°12.3’ S = -29.205°
Longitude: 001°38.4’ E = 1.640°

2. Calculate the true course (TC) using the formula:
TC = atan((sin(longitudeB - longitudeA) * cos(latitudeB)) / (cos(latitudeA) * sin(latitudeB) - sin(latitudeA) * cos(latitudeB) * cos(longitudeB - longitudeA)))

Substituting the values:
TC = atan((sin(1.640 - (-7.2433)) * cos(-29.205)) / (cos(-32.2933) * sin(-29.205) - sin(-32.2933) * cos(-29.205) * cos(1.640 - (-7.2433))))

3. Convert the true course from radians to degrees.

4. The distance between the waypoints can be calculated using the formula:
Distance = 60 * 1.852 * d * cos(TC)

Where d is the difference in longitude between the waypoints.

Substituting the values:
Distance = 60 * 1.852 * abs(1.640 - (-7.2433)) * cos(TC)

Now, let's calculate the true course and distance:

Step 1:
Latitude A: -32.2933
Longitude A: -7.2433
Latitude B: -29.205
Longitude B: 1.640

Step 2:
TC = atan((sin(1.640 - (-7.2433)) * cos(-29.205)) / (cos(-32.2933) * sin(-29.205) - sin(-32.2933) * cos(-29.205) * cos(1.640 - (-7.2433))))
TC ≈ 65.513°

Step 3:
True Course: 65.513°

Step 4:
Distance = 60 * 1.852 * abs(1.640 - (-7.2433)) * cos(TC)
Distance ≈ 976.32 nautical miles

Therefore, the distance between the waypoints (A and B) is approximately 976.32 nautical miles.