Consider the triangle formed by 3 points on the surface of a sphere: A, B, and C. By convention, we label the sides opposite the points with the lower case letter corresponding to the vertex opposite the side. So the side we might want to call "BC" actually gets labelled "a". Let "O" be the origin, or the center of the sphere. The length of the side "a" is given by the angle BOC. Similarly the side "b" has length AOC and side "c" has length AOB. The angle at the vertex A is the dihedral angle between the planes AOB and AOC. Given these definitions, we have completely described a spherical triangle and can now discuss the handful of important theorems that make their analysis useful:

The cosine rule

cos(a) = cos(b) * cos(c) + sin(b) * sin(c) * cos(A)

The transposed cosine rule

sin(a) * cos(B) = cos(b) * sin(c) - sin(b) * cos(c) * cos(A)

The sine rule

sin(a) / sin(A) = sin(b) / sin(B) = sin(c) / sin(C)

The four parts rule (usually written with cotangents)

cos(a) * cos(C) = sin(a) * cos(b)/sin(b) - sin(C) * cos(B)/sin(B)

The cosine rule for the angles (also known as the polar cosine rule

cos(A) = -cos(B) * cos(C) + sin(B) * sin(C) * cos(a)

The polar transposed cosine rule:

sin(A) * cos(b) = cos(B) * sin(C) + sin(B) * cos(C) * cos(a)

Trigonometric Identities

tan(x) = sin(x) / cos(x)
sin(x)*sin(x) + cos(x)*cos(x) = 1
cos(90-x) = sin(x)
sin(90-x) = cos(x)
sin(90+x) = cos(x)
cos(90+x) = -sin(x)
sin(-x) = -sin(x)
cos(-x) = cos(x)
sin(180-x) = sin(x)
cos(180-x) = -cos(x)
sin(x+180) = -sin(x)
cos(x+180) = -cos(x)