Spherical law of cosines

In spherical trigonometry, the law of cosines (also called the cosine rule for sides) is a theorem relating the sides and angles of spherical triangles, analogous to the ordinary law of cosines from plane trigonometry. Given a unit sphere, a "spherical triangle" on the surface of the sphere is defined by the great circles connecting three points u, v, and w on the sphere (shown at right). If the lengths of these three sides are a (from u to v), b (from u to w), and c (from v to w), and the angle of the corner opposite c is C, then the (first) spherical law of cosines states: Since this is a unit sphere, the lengths a, b, and c are simply equal to the angles (in radians) subtended by those sides from the center of the sphere. (For a non-unit sphere, the lengths are the subtended angles times the radius, and the formula still holds if a, b and c are reinterpreted as the subtended angles). As a special case, for C = π/2, then cos C = 0, and one obtains the spherical analogue of the Pythagorean theorem: If the law of cosines is used to solve for c, the necessity of inverting the cosine magnifies rounding errors when c is small. In this case, the alternative formulation of the law of haversines is preferable. A variation on the law of cosines, the second spherical law of cosines, (also called the cosine rule for angles) states: where A and B are the angles of the corners opposite to sides a and b, respectively. It can be obtained from consideration of a spherical triangle dual to the given one. Let u, v, and w denote the unit vectors from the center of the sphere to those corners of the triangle. The angles and distances do not change if the coordinate system is rotated, so we can rotate the coordinate system so that is at the north pole and is somewhere on the prime meridian (longitude of 0). With this rotation, the spherical coordinates for are where θ is the angle measured from the north pole not from the equator, and the spherical coordinates for are The Cartesian coordinates for are and the Cartesian coordinates for are The value of is the dot product of the two Cartesian vectors, which is Let u, v, and w denote the unit vectors from the center of the sphere to those corners of the triangle.