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### Trig Based on the Unit Square

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Date: 10/22/98 at 10:31:28
From: ben barr
Subject: Trig

Hi,

Lately, this question has been puzzling me.

Why are the sine and cosine graphs defined in a unit circle?  Why not
a unit square or any other geometric shape?

Thanks for your time!  I hope to hear back soon.
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Date: 10/22/98 at 21:46:23
From: Doctor Sam
Subject: Re: Trig

Ben,

That is really a fabulous question! You can certainly define things
like the sine and the cosine using a unit square (or an ellipse or a
rectangle, or any other geometric figure). It's fun to see what some
of them look like.

For example, if you draw a unit square from (1,0) to (1,1) to (0,1) to
(0,0) and graph the y-coordinate as you move around the square
counter-clockwise you will get something very similar to the graph
y = sin x, except it will be made up of straight line segments rather
than a smooth curve.

But the reason that the trig functions are defined using the unit
circle is that scientists often want to describe circular motion (like
planets around the sun or satellites around the Earth). The trig
functions are easy to use for this purpose because they have circular
motion built into their definitions.

Of course, the trig functions aren't used only to describe motion
around a circle, but it turns out that they are very useful when
describing anything that repeats periodically, and many things in
nature occur in cycles.

I hope that helps.

- Doctor Sam, The Math Forum
http://mathforum.org/dr.math/
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