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
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