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### Sine/Cosine Equations

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Date: 10/23/96 at 16:54:51
From: Tyler Durrett
Subject: Sine/Cosine equations

Hi.  Is there any equation  for sine or cosine?  If so, what are they?

Thanks.

Tyler Durrett
```

```
Date: 10/24/96 at 20:17:21
From: Doctor Tom
Subject: Re: Sine/Cosine equations

Do you mean to calculate them from scratch?  Sure.

If x is the angle in radians (3.14159265... radians = 180 degrees),
then here's a formula for sin(x) and cos(x):

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

cos(x) = 1 - x^2/2! + x^4/4! - x^6/6! + x^8/8! - ...

7! means "7 factorial", or 7x6x5x4x3x2x1, and so on.

You have to add together an infinite number of terms in the
formulas above to get the exact values of sin and cos, but
in practical situations, the terms get small very rapidly,
and if you use just the first 5 or 6 terms you will have
a very accurate result.

For reasons of numerical stability and for slightly better
accuracy, computers use different formulas, but those formulas
only work for the numbers a computer can represent.  My
formulas above can be used to get as exact a value as you
want, simply by using enough terms.

So if you wanted to find the sin of 30 degrees, that's 1/6
of pi, so x = pi/6.  Plug that into the sin(x) formula and
work out as many terms as you need.

-Doctor Tom,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Trigonometry

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