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     Date: 08-30-95    Time: 07:33p     Number: 2154   
     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: Wed, 30 Aug 1995 11:12:16 -0400
From: "Dr. Math" 

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].


As Bj\xrn St\frk wrote to Dr. Math,
>
>> Hello!
>Hello again! :)
>
>> 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.
>
>Thank you, very much. But I don't exactly see where to use the
>angle in this formula. Say that