---
tag: SXQGHWHFRF
kind: concept
name: Spherical trigonometry
msc: [51M09, 51M10]
sources:
  - url: https://en.wikipedia.org/wiki/Spherical_trigonometry
same_as:
  wikidata: Q46463
  wikipedia: Spherical trigonometry
---

## Summary

Spherical trigonometry studies the relations between the sides and the angles of spherical triangles, the triangles on a sphere bounded by arcs of great circles.
Its main formulas are the spherical law of cosines, $\cos c = \cos a \cos b + \sin a \sin b \cos \gamma$, and the spherical law of sines, $\sin a / \sin \alpha = \sin b / \sin \beta = \sin c / \sin \gamma$.
It is the part of [[spherical geometry|SQ9RQVANMG]] used in astronomy, geodesy and navigation.
The area of a spherical triangle is given by [[Girard's theorem|Y9DM9B0RA3]].

## Explanations

### L1 human

On a globe, a triangle is made of three arcs of great circles, and its sides are measured as angles seen from the centre of the Earth.
Spherical trigonometry answers questions such as: given two sides and the angle between them, how long is the third side?
A pilot uses it to find the great-circle distance between two airports from their latitudes and longitudes.
For small triangles the formulas reduce to the plane ones, which is why plane trigonometry works on a field but not across an ocean.