Spherical Law Of CosinesEdit

On the surface of a sphere, triangles behave differently from their flat-plane counterparts. The spherical law of cosines provides a precise relationship among the three sides and three angles of a spherical triangle, encapsulating how curvature alters basic trigonometric intuition. If a, b, c denote the arc-lengths of the sides (measured as angles on a unit sphere, or as arc lengths with radius R scaled to those angles), and A, B, C are the angles opposite those sides, then one of the central forms of the law reads cos a = cos b cos c + sin b sin c cos A. The same relation can be solved for any angle, for example cos A = (cos a − cos b cos c) / (sin b sin c). These equations generalize the familiar Euclidean law of cosines to curved surfaces and reduce to the ordinary formula in the limit of vanishing curvature or for infinitesimely small triangles.

The spherical law of cosines sits at the core of Spherical geometry and has wide-ranging consequences for how we think about distances and angles on curved surfaces. It underpins practical calculations in navigation and geodesy, and it informs algorithms in computer graphics and astronomy that must operate on curved spaces such as the Earth. In particular, it underlies methods for determining the shortest route between two points on a sphere (a great-circle path) and for converting between angular measurements and surface distances on curved manifolds.

Mathematical statement

The primary form (sides a, b, c opposite angles A, B, C) on a sphere of radius R is cos(a/R) = cos(b/R) cos(c/R) + sin(b/R) sin(c/R) cos(A), which, for the unit sphere (R = 1), simplifies to cos a = cos b cos c + sin b sin c cos A.
An equivalent form expresses an angle in terms of the other two sides: cos A = (cos a − cos b cos c) / (sin b sin c), with analogous expressions obtained by cyclic permutation of the variables.
The sides a, b, c are measured as arc lengths along great circles, or equivalently as the central angles subtended by those arcs (a = s_a/R, etc.). The curvature of the surface is what distinguishes these relationships from their flat-plane counterparts.

Notation and conventions

A spherical triangle is defined by three points on the surface of a sphere and the three great-circle arcs that connect them.
Angles A, B, C are the interior angles at the triangle’s vertices, measured in radians (or degrees) as appropriate.
The law applies to any triangle formed by great-circle segments; it reflects the intrinsic curvature of the sphere rather than any embedding in a higher-dimensional space.

Derivation (sketch)

One standard derivation places the triangle on a unit sphere, assigns unit vectors to the triangle’s vertices, and uses the dot product to relate angular distances. If the vertices are unit vectors u, v, w with angular distances a, b, c between them, then cos a = u · v, cos b = v · w, cos c = w · u. By expressing A as the angle between the tangents to the great circles at a common vertex and manipulating the vector relations, one arrives at cos a = cos b cos c + sin b sin c cos A. Alternative derivations exploit spherical coordinates or projections onto tangent planes and reach the same identity. For further variants and proofs, see Spherical geometry and Napier's rules for spherical triangles.

Relationships to other spherical formulas

Spherical law of sines: sin A / sin a = sin B / sin b = sin C / sin c.
Right-angled spherical triangles have their own set of rules (Napier’s rules) that are particularly convenient for computations.
The law of cosines for the angle (cos A, cos B, cos C) is related to the side form by symmetry and algebraic rearrangement.
The law can be used to derive the central angle between two points on a sphere given their latitudes and longitudes, yielding the familiar great-circle distance formula, cos d = sin φ1 sin φ2 + cos φ1 cos φ2 cos Δλ, when expressed in terms of lat/long coordinates. See Haversine formula and Great-circle distance for related approaches.

Applications

Navigation and geodesy: The spherical law of cosines, together with its siblings in spherical trigonometry, enables calculations of distances and bearings between points on the surface of a globe. It is foundational in early and modern navigation computations that rely on great-circle routes.
Astronomy: Spherical trigonometry describes celestial triangles formed by the apparent positions of stars, planets, and the observer’s zenith, where curvature of the celestial sphere plays a role in angle-distance relations.
Computer graphics and global rendering: When simulating or projecting curved surfaces, the spherical law of cosines provides exact relationships that help manage angular measures and surface distances on a sphere.
Geodetic practice and spherical models of the Earth: Even as many practitioners use ellipsoidal models for high-precision work, the spherical law of cosines remains a crucial reference point and a stepping-stone to more complex ellipsoidal corrections.

Generalizations and limitations

On a perfect sphere, the law is exact and widely applicable. Real-world applications often model the Earth as an oblate spheroid (ellipsoid) rather than a sphere, in which case the simple cosine law is only an approximation. For ellipsoidal models, practitioners turn to more general geodesic formulas, such as those embedded in Vincenty’s formulae, or to numerical methods for solving geodesic problems.
Numerical stability: In computational contexts, especially for small distances, the direct use of cosines can lead to precision loss. The haversine form and related trigonometric identities are preferred in many libraries to maintain numerical accuracy.
Higher-dimensional analogs: The spherical law of cosines has extensions to polygons on curved surfaces of constant curvature and to more general Riemannian manifolds, where trigonometric relations are replaced by tools from differential geometry.