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/