The Math Forum

Ask Dr. Math - Questions and Answers from our Archives
Associated Topics || Dr. Math Home || Search Dr. Math

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:

A similar calculation is used to find the other trig functions,
particularly the cosine.

- Doctor Jerry, The Math Forum 
Associated Topics:
High School Calculators, Computers
High School Trigonometry

Search the Dr. Math Library:

Find items containing (put spaces between keywords):
Click only once for faster results:

[ Choose "whole words" when searching for a word like age.]

all keywords, in any order at least one, that exact phrase
parts of words whole words

Submit your own question to Dr. Math

[Privacy Policy] [Terms of Use]

Math Forum Home || Math Library || Quick Reference || Math Forum Search

Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.