In spherical trigonometry, the law of cosines 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, and on the sphere. If the lengths of these three sides are , and , and the angle of the corner opposite is, then the spherical law of cosines states: Since this is a unit sphere, the lengths, and are simply equal to the angles subtended by those sides from the center of the sphere.. As a special case, for, then, and one obtains the spherical analogue of the Pythagorean theorem: If the law of cosines is used to solve for, the necessity of inverting the cosine magnifies rounding errors when 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, states: where and are the angles of the corners opposite to sides and, respectively. It can be obtained from consideration of a spherical triangle dual to the given one.
Proofs
First proof
Let, and 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. 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.
Second proof
Let, and denote the unit vectors from the center of the sphere to those corners of the triangle. We have,,, and. The vectors and have lengths and respectively and the angle between them is, so using cross products, dot products, and the Binet–Cauchy identity.
Rearrangements
The first and second spherical laws of cosines can be rearranged to put the sides and angles on opposite sides of the equations:
Planar limit: small angles
For small spherical triangles, i.e. for small, and, the spherical law of cosines is approximately the same as the ordinary planar law of cosines, To prove this, we will use the small-angle approximation obtained from the Maclaurin series for the cosine and sine functions: Substituting these expressions into the spherical law of cosines nets: or after simplifying: The big O terms for and are dominated by as and get small, so we can write this last expression as:
