Spherical trigonometry

Spherical trigonometry studies relationships among angles and distances in triangles drawn on a sphere, whose sides are arcs of great circles.

Spherical law of cosines: A relation connecting a spherical triangle’s side lengths and the angle opposite one side. It gives a central way to calculate an unknown side or angle from the other measurements.
Spherical geometry: The geometry of figures on the surface of a sphere. Spherical trigonometry is the triangle-focused part of this broader geometry.
Celestial navigation: Navigation that determines position using observations of celestial bodies. Spherical triangles connect observed sky angles with positions on Earth.
Plane trigonometry: The study of relationships among angles and side lengths in plane triangles. Its triangle angle sums and side formulas differ because the plane has zero curvature.
Spherical law of sines: A spherical-triangle relation stating that each side’s sine divided by its opposite angle’s sine has the same value. It complements the cosine law when a triangle’s angles and a side are known.
Spherical triangle: A region on a sphere bounded by three arcs of great circles. Its sides and angles are the basic objects measured by spherical trigonometry.
Astronomy: The study of celestial objects, space, and the universe. Astronomical position calculations use spherical triangles on the celestial sphere.
Euclidean geometry: Geometry based on Euclid’s postulates, including the parallel postulate. Spherical geometry violates the parallel postulate, changing triangle properties.
Napier’s rules: Mnemonic rules for finding relations among the parts of a right spherical triangle. They simplify spherical calculations when one angle is a right angle.
Geodesic: A locally shortest path between points on a curved surface. On a sphere, great-circle arcs are geodesics and supply spherical-triangle sides.

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