Plane, parallel and mid-latitude sailing
1The lesson in plain words
Plane sailing links course, distance, d.lat and departure in one right-angled triangle. Mean latitude turns the departure into d.long.
Plane sailing
Over a short run, treat the Earth's surface as flat. Course C, distance d, d.lat and departure form a right-angled triangle: d.lat = d × cos C and dep = d × sin C. To find C, use C = tan⁻¹(dep ÷ d.lat), the calculator’s inverse-tangent function. Then d = d.lat ÷ cos C. For this magnitude-and-name routine, use the quadrant angle (0–90°), and set the calculator to degrees.
Parallel sailing
Sailing due east or west along one parallel, d.lat is zero and the whole distance is departure: d.long = dep ÷ cos lat. Example: 120 M due east along 45°N gives d.long = 120 ÷ cos 45° = 169.7′ = 2°49.7′ E. At 60°N the same 120 M would give 4°00′ of longitude.
Mid-latitude sailing
On a slanting course the latitude changes, so which latitude turns departure into d.long? Use the mean latitude, halfway between start and finish: d.long = dep ÷ cos (mean lat), and dep = d.long × cos (mean lat) going the other way. Spanish courses call it latitud media. This joins plane sailing, which works in miles, to positions in latitude and longitude.
When cos C misleads
For a course near 090° or 270°, cos C is small, and dividing a small d.lat by it can magnify rounding errors. Then find the distance as d = dep ÷ sin C instead. For a course near north or south, departure is small and the choice of latitude hardly matters.
A fixed routine
For equal latitudes, use parallel sailing directly. Otherwise, between two positions: find named d.lat and d.long, mean latitude, departure, then course and distance. For an arrival position: find d.lat and departure from course and distance, then arrival latitude, mean latitude, d.long and arrival longitude. Set your calculator to degrees. Keep full precision until the final answer; round course to 0.1° or 1° and distance to 0.1 M.
2Key data
- Plane sailing
- d.lat = d cos C; dep = d sin C
- Course and distance
- tan C = dep ÷ d.lat; d = d.lat ÷ cos C
- Near east or west
- d = dep ÷ sin C
- Parallel sailing
- d.long = dep ÷ cos lat
- Mid-latitude sailing
- d.long = dep ÷ cos (mean lat)
- Example
- 120 M east along 45°N → d.long 2°49.7′ E
3Key points
- 1
Plane sailing: d.lat = d cos C, dep = d sin C.
- 2
Parallel sailing: d.long = dep ÷ cos lat, when you sail along one parallel.
- 3
Mid-latitude sailing uses the mean latitude to convert between dep and d.long.
- 4
Near east or west, take the distance from dep ÷ sin C.
4Frequent mistakes
Wrong: Use 246° as the quadrant angle for S 66° W.
Right: The quadrant angle is 66°. Use it in this routine, then name d.lat S and departure W.
Wrong: Use the starting latitude to convert departure on a long slanting leg.
Right: Use the mean latitude, halfway between the start and arrival latitudes.
Wrong: Use rounded d.lat in d.lat ÷ cos C near east or west.
Right: Prefer dep ÷ sin C here: dividing by a small cosine can magnify rounding errors.
5Practice case & exam keys
You sail 90 M due west along 38°20′N from 038°10′W. A crew member gives arrival longitude 039°40′W. Accept it?
Show the answer
No. Convert departure to d.long first. This is parallel sailing: all 90 M is departure. d.long = 90 ÷ cos 38°20′ ≈ 114.7′ = 1°54.7′ W. Arrival longitude = 038°10.0′W + 1°54.7′ = 040°04.7′W, still at 38°20′N.
Exam keys
- d.lat = d cos C; dep = d sin C; tan C = dep ÷ d.lat.
- Parallel sailing: d.long = dep ÷ cos lat.
- Mid-latitude sailing: d.long = dep ÷ cos (mean lat).
- Course near 090°/270°: distance = dep ÷ sin C.
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