Spherical law of cosines

The spherical law of cosines relates the side lengths and angles of a triangle on a sphere. Unlike its planar counterpart, it accounts for the sphere’s curvature.

Spherical triangle: A figure on a sphere bounded by three arcs of great circles. The law relates the sides and angles of this curved triangle.
Spherical geometry: The study of geometric figures and relationships on the surface of a sphere. The law belongs to a geometry where straight paths are great-circle arcs.
Great-circle distance: The length of the shortest path along a sphere’s surface between two points. The law computes this distance from the points’ angular coordinates.
Law of cosines: A Euclidean triangle identity relating a side’s squared length to the other sides and the cosine of their included angle. Its flat-space formula is the limiting counterpart of the spherical law.
Great circle: A circle on a sphere whose center coincides with the sphere’s center. Spherical triangle sides are arcs of great circles, measured by their central angles.
Radians: A unit of angle equal to the angle subtended by an arc as long as a circle’s radius. Spherical side lengths are often expressed as angles in radians.
Great-circle navigation: Route planning that follows great-circle arcs to minimize distance on a spherical Earth. Its route distances can be obtained from spherical triangle relations.
Hyperbolic law of cosines: A relation among sides and angles of triangles in hyperbolic geometry. It describes constant negative curvature, in contrast with the sphere’s positive curvature.
Central angle: An angle whose vertex is at the center of a circle or sphere. It measures each spherical side as an angle at the sphere’s center.
Trigonometric functions: Functions that relate angles to ratios or coordinates, including sine and cosine. Sine and cosine encode the sides and angles in the spherical formula.

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