Computers and Trigonometric Functions

Date: 02/11/97 at 21:37:20
From: Brian Hicks
Subject: Sine, Cosine, Tangent

Hi Dr. Math, 

I was wondering, what actual functions find the values for sine, 
cosine, and tangent? Let's say that:

sin 48  = 0.74314482547739
cos 48  = 0.66913060635886
tan 48  = 1.1106125148292

How does a calculator or computer come up with these numbers? What 
does the computer actually do to the number 48 (in this case) that 
produces these answers?  Thanks for all your help.  :)

-Brian

Date: 02/13/97 at 14:15:43
From: Doctor Ceeks
Subject: Re: Sine, Cosine, Tangent

Hi,

Nowadays, with computer memory so readily available, the values of 
sine, cosine, and tangent are often stored in a table in memory.  
(This memory is often located inside the same chip that carries the 
Central Processing Unit.) This makes it possible to produce answers 
very quickly. 

When the computer or calculator is asked to produce a value of sine, 
cosine, or tangent which is not in the table, the computer exploits 
the fact that small changes in the angle only create small changes in 
the value of the function to extrapolate an answer. Because computers 
and calculators are only expected to give approximate answers anyway 
(e.g. to 10 decimal places, or what have you), this extrapolation need 
be done only to the necessary accuracy.

However, before there were tables, the functions had to be computed
using some other method.

Consider the function sin(x). Thanks in large part to Newton, and also 
his student Taylor, it was discovered that when x is measured in 
radians:

   sin(x) = x - x^3/3! + x^5/5! - x^7/7! + ...

The equations for cosine and sine are similarly derived using Taylor 
series.  Also, you can get cos(x) by taking the square root
of 1 - sin^2(x). You can get tan(x) by dividing sin(x) by cos(x):

   cos(x) = 1 - x^2/2! + x^4/4! - x^6/6! + ...

   tan(x) = x + 2x^3/3! + 16x^5/5! + 272x^7/7! + 7936x^9/9! + ...

The explanation for why this is true is rather long, and I hope you 
don't mind if I refer you to a book.  You can find this in any good 
calculus book, for instance, by Apostol or by Courant and John.

You can think of this (infinite) sum as a series of correction 
factors. The first term (which is x) says: sin(x) is about equal to x.
Well this is not very accurate, but if you want more accuracy, you can 
go to the next term: sin(x) is more like x - x^3/3!. The further you 
go, the more accurate you get.

It's possible to know the error from the actual value of sin(x) if you 
stop at a certain point in the summation. Computers and calculators
need only add out to the point where the error is smaller than the
desired accuracy.

One last thing: Note that for most values of x (i.e. most angles), the 
decimal value of sin(x) cannot be given in complete accuracy because,
for most values of x, sin(x) will have a non-repeating decimal 
expansion.

-Doctor Ceeks,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/