How Accurate Is the Spherical Law of Cosines for Calculating Distances?

[moonman239]

Let's say I want to use the spherical law of cosines to determine the distance between two points. How do I calculate the error in the approximation (ignoring ellipsoidal effects)?

 

[NobodySpecial]

Is the law of cosines an approximation for a sphere?

If you have an uncertainty in the position of each point and you want the uncertainty in the distance - you could do some horrible differentiation of the formula, OR you could just re-calcuate the distance for one position +/- the error and use that