Calculating Sine Without Using the Sine Key or a Table
Date: 01/02/2004 at 01:41:28 From: Jod Subject: Basic Trigonometric Functions I'm curious what the arithmetic is behind the trig functions. For example, to evaluate sin(48), what math process could I use if I didn't have a calculator?
Date: 01/02/2004 at 10:02:24 From: Doctor Jerry Subject: Re: Basic Trigonometric Functions Hello Jod, The mathematical processes behind the trig functions are, except for special values like 0, 30, 45, 60, 90, and the like, not finite processes. Specifically, one uses the series sin(x) = (x) - (x^3)/3! + (x^5)/5! - ... where x is in radians. So, if you wanted sin(5 deg), you would convert 5 degrees to radians: x = 5*pi/180 Then, using the above formula sin(5*pi/180) = (5*pi/180) - (5*pi/180)^3/3! + (5*pi/180)^5/5! - ... If you evaluate just the first three terms you will find (for small angles) a pretty good approximation. From above, our approximation is sin(5*pi/180) = .0872664626 - .0001107620 + .0000000422 = 0.0871557428 Compare that with the result from my calculator, which is sin(5deg) = 0.087155742... For bigger angles you would need to calculate more terms in the formula to maintain that level of accuracy. Calculators, however, don't use the series because it would take too much time. They use an algorithm called the CORDIC algorithm. It is very accurate and can be done rapidly. If you do a Google search on CORDIC you will find a description of this algorithm. One such link is here: CORDIC FAQ http://www.dspguru.com/info/faqs/cordic.htm A similar calculation is used to find the other trig functions, particularly the cosine. - Doctor Jerry, The Math Forum http://mathforum.org/dr.math/
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