Computing Sine and Cosine

Date: 8/27/95 at 9:0:54
Subject: Sine/Cosine
From: Bj\xrn St\frk
Date: Sun, 27 Aug 1995 13:15:09 +0100

Dear Dr. Math

Is it possible to compute sine and cosine for a given angle
without using a calculator or a sine table?

Date: 8/27/95 at 17:13:53
From: Doctor Ken
Subject: Re: Sine/Cosine

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.

-Doctor Ken,  The Geometry Forum
 Check out our web site!  http://mathforum.org/dr.math/dr-math.html   

Date: 8/29/95 at 12:22:13
Subject: Sine/Cosine
From: Bj\xrn St\frk
Date: Tue, 29 Aug 1995 17:28:41 +0100

Hello again! :)

Thank you, very much. But I don't exactly see where to use the
angle in this formula. Say that I have an angle of 45 degrees,
(or the same value in radians). How do I calculate sine/cosine
for this angle?

Date: 8/30/95 at 11:12:16
From: Doctor Ken
Subject: Re: Sine/Cosine

Bjorn-

The way you'd use these formulas is to plug in the angle (in radians) into
the formula.  So for the angle .25 radians, you'd get

      .25^3     .25^5     .25^7
.25 - ------ + ------- - -------     =   .247404...
        3!        5!        7!

Keep in mind that this formula is only very accurate from about 
-Pi/2 to Pi/2.  So to do an angle outside this range, use instead an angle
that would have the same sine: instead of Sin[2.7] use Sin[Pi-2.7].
Do a similar thing for cosine, for which the formula is also good between
-Pi/2 and Pi/2: instead of Cos[2.7] use -Cos[Pi-2.7].

-Doctor Ken,  The Geometry Forum