When working with spherical coordinates I have come

Rubi Reid

Rubi Reid

Answered question


When working with spherical coordinates I have come across
and was wondering if there was some kind of identity for this that I could use instead of computing the inside of the acrccos function first.

Answer & Explanation



Beginner2022-03-16Added 1 answers

The problem with cos(x) is that near x=0, cosx=1x22+O(x4). Thus cos1(x) loses accuracy near x=1. An equivalent formula without such accuracy loss is the following:
Notice that near x=0, sin(x)=x+O(x3) .Thus for x small enough, sin(x)x and we can use the simple approximation cos1(cos(a)cos(b))a2+b2. Notice that in a spherical right triangle with legs a,b and hypoteneuse c, the formula cosc=cosacosb holds and if the triangle has small enough sides, the Pythagorean approximation c2a2+b2 holds.

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