Chapter 21: Navigational Mathematics
Navigation is applied trigonometry. Plane trigonometry handles short distances; spherical trigonometry handles the Earth as it actually is.
Plane and spherical triangles
In a plane right triangle the sine, cosine and tangent relations plus Pythagoras solve every sailing over short distances. On a sphere the sides are arcs measured in degrees and the angles are between great circles. The law of cosines for sides, cos a = cos b cos c + sin b sin c cos A, and the law of sines, sin a / sin A = sin b / sin B, solve the navigational triangle, great-circle distance, and course problems.
Napier's rules and haversines
For right spherical triangles, Napier's rules (the circular parts mnemonic) reduce ten relations to two simple statements, historically important for hand computation. The haversine form of the distance formula, hav d = hav(Δlat) + cos L1 cos L2 hav(Δlong), was adopted because it preserves precision for very small angles where the plain cosine formula loses significant figures.
- Law of cosines for sides — the workhorse for distance and altitude
- Law of sines — quick when two sides and an opposite angle are known
- Haversine — numerically stable for short distances
Formula cards
Spherical law of cosines
cos a = cos b cos c + sin b sin c cos A
Great-circle distance: cos D = sin L1 sin L2 + cos L1 cos L2 cos(ΔLo).
Haversine distance
hav D = hav(ΔL) + cos L1 cos L2 hav(ΔLo)
hav θ = (1 − cos θ)/2.
Key takeaways
- • Great-circle and altitude computations are the same spherical cosine formula.
- • Haversines exist for numerical stability, not tradition.
- • Plane trigonometry is adequate for runs of a few hundred miles in mid-latitudes.
Knowledge check
Immediate feedback with rationales. Answer each question to see the explanation.
Chapter 21 quiz
0 of 3 answered
1.The haversine formula is preferred over the plain cosine formula because it:
2.Great-circle distance between two points is found with:
3.Napier's rules apply to: