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     Date: 08-28-95    Time: 00:31a     Number: 2032   
     From: dr.math@forum.swarthmore.     Refer: 0       
       To: Bjørn Stærk                Board ID: ROLVSOY         Reply: 
  Subject: Sine/Cosine                     132: E-mail         Status: Private
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Date: Sun, 27 Aug 1995 17:13:54 -0400
From: "Dr. Math" 

Hello!

Well, it depends.  Sine and Cosine are defined in terms of geometric 
objects, so if you wanted to you could draw a circle of radius r on a pair 
of axes, lay your angle down on the circle, find where it intersects the 
circle, and if the coordinates of this point are (x,y) then the cosine of
the angle is x/r and the sine is y/r.  

But I gather that you're looking for a more algebraic, formulaic way?  Well,
there's no algebraic way to compute exactly the trigonometric values of 
every angle, only certain ones like Pi/2, Pi/3, Pi/4, and things like that.
You can, however, use an algebraic formula to find an _approximation_ to 
any angle.  To approximate Sine, use the formula 
     x^3      x^5       x^7
x - ----  +  -----  -  ----- + ....
     3!        5!        7! 

For cosine, use the formula

     x^2      x^4       x^6
1 - ----  +  -----  -  ----- + ....
     2!        4!        6! 

These both converge pretty quickly to the right value, and they converge 
most quickly when x isn't too far away from 0.

As Bj\xrn St\frk wrote to Dr. Math,
>
>Dear Dr. Math
>
>Is it possible to compute sine and cosine for a given angle
>without using a calculator or a sine table?
>
>
>
>---
> ■ SPEED 2.00 [NR] ■ 


-Doctor Ken,  The Geometry Forum
 Check out our web site!  
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