Haversine law

Suppose you want to solve a triangle. You know two sides and the angle between them. Then you can solve for the third side using the law of cosines. Now suppose you want to solve a big triangle, a triangle on the surface of the earth so large that the curvature of the earth matters. You can still use the law of cosines, but you’ll need the spherical law of cosines: cos(c) = cos(a) cos(b) + sin(a) sin(b) cos(C). If you know the (angular) lengths of sides a and b, and (tangential) angle C between the two sides, you can solve for c by taking the inverse cosine of the right hand side above. Now suppose you want to solve this big triangle because you’re a navigator on a ship a couple centuries ago, doing calculations by looking up trig functions and inverse trig functions in a table. You’re interested in triangles that are so big that you have to account for the fact that you’re living on a sphere. But at the same time, you’re triangles are still fairly small relative to the size of the…