Associated Topics || Dr. Math Home || Search Dr. Math

### Sine numbers in Charts for Angles

```
Date: 12/22/97 at 21:21:15
From: Paul Oswalt
Subject: Where do the sine numbers come from in the charts for angles?

Hi. I am a 39-yr.-old toolmaker. I would like to know where the
numbers in sine charts or on a calculator come from. I probably
learned this in school, but that was a while back. I use these numbers
every day, and was curious how they are derived. I have looked in a
couple of books I have, but no luck. I  am guessing Pi has something
to do with it, but in trying I couldn't come up with the right

If you can help, thanks.

Paul Oswalt
```

```
Date: 01/05/98 at 09:24:54
From: Doctor Bruce
Subject: Re: Where do the sine numbers come from in the charts for
angles?

Hello Paul,

There is a formula to calculate the sine of an angle. The formula is
what we call a series: it is, formally, the sum of infinitely many
numbers. Fortunately, the numbers dwindle in size very rapidly, so we
can use just the first few terms of the infinite series as a good
approximation. Here are the first 5 terms of the series for sine.

sin(x)  =  x - x^3/3! + x^5/5! - x^7/7! + x^9/9! - ...

I'm sure you can see the pattern.  What may look strange are the
numbers 3!, 5!, 7!, 9! in the denominators.  These are equal to  6,
120, 5040, and 362880, respectively.  We calculate  n!  (read "n
factorial") by multiplying all the whole numbers from 1 up to n
(including n). So these numbers get large very fast. 100! is a number
with 158 digits, for example.

You guessed that Pi has something to do with all this, and you are
right. It turns out that if we substitute the value Pi/2 (about 1.57)
for x in the formula given above, the value will come out very close
to 1. If we include more terms we will come even closer to 1. In a
certain sense, the sum of "all" the infinitely many terms in the
series is exactly 1. So we find sin(Pi/2) = 1.  But what kind of
"angle" is Pi/2?

Most real-world applications of trigonometry use the standard system
of measuring angles in degrees. But in order to use this formula for
sine, we have to use a different unit of measure. It is called a
(the number I substituted above) equals 90 degrees. I am sure you know
that the sine of 90 degrees is 1.

Hope this helps,

-Doctor Bruce,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
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
Math Forum Home || Math Library || Quick Reference || Math Forum Search