Arccos Algorithms and Computer Calculators

Date: 03/21/98 at 05:51:04
From: gilson
Subject: finding angle from cos

I need to use value of the angle in a computer program and I have the 
value for the cos and sin only. I need to know which formula or 
algorithm to use. I know it is ambiguous, but the first value only 
would solve my problem. What I need is the algorithm used when you 
press arccos button on the calculator. Thanks.

Date: 03/21/98 at 06:26:50
From: Doctor Mitteldorf
Subject: Re: finding angle from cos

Dear Gilson,

I don't know how standard the arccos calculation has become.  It 
probably is similar but not identical from one brand of calculator to 
the next.  

The basis of the calculation is a Taylor series:

     arcsin(x) = x + 1/2 (x^3/3) + (1/2)(3/4)(x^5/5) + 
                (1/2)(3/4)(5/6)(x^7/7) + ... 

This series converges fast when x is small, slow when x is large.  
(It doesn't converge at all when x>1, of course.) So the periodicity 
of the sin function is used to make the series give an accurate result 
with fewer terms. If x is small, then the series is taken for 
arcsin(x). If x is close to 1, then the series is evaluated for
sqrt(1 - x^2), giving the arccos, then the result is subtracted from 
pi/2. In other words, the calculator uses the facts that sin^2+cos^2=1 
and that sin(x)=cos(pi/2-x).

For in-between values, the calculator probably uses the trig sum 
and difference formulas so that arcsin(sqrt(2)/2)=pi/4 can be used as 
a first approximation.

So there are lots of ways to skin the cat, and since the computation 
is so common, it's probably been highly optimized for speed and 
accuracy way beyond the basic Taylor series.

-Doctor Mitteldorf,  The Math Forum