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 
answers.

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 
radian. One radian equals approximately 57 degrees, 17 minutes, 
45 seconds.  Pi radians equals exactly 180 degrees. So, Pi/2 radians 
(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/