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### Circular Functions

```
Date: 01/27/2001 at 13:36:39
From: Sheena
Subject: Circular Functions

How do you define circular functions? Can you give me an example, so I
can do them?
```

```
Date: 01/27/2001 at 14:09:41
From: Doctor Robert
Subject: Re: Circular Functions

Draw a unit circle: a circle with radius 1 with its center at the
origin, that is, the point (0,0). Now imagine a point on that circle
moving around in a counterclockwise direction. If you draw a ray from
the origin to that point, it makes an angle with the positive
horizontal axis.

That point on the circle has coordinates.  If we define the first
coordinate to be the cosine of the angle, and the second coordinate to
be the sine of the angle, we have defined a circular function.

If the angle is zero degrees, the point on the circle has coordinates
(1,0) so that cos(0) = 1 and sin(0) = 0. Now move the point on the
circle so that the ray from the origin makes an angle of 45 degrees.
You find using Pythagoras' theorem that the coordinates of that point
are (root2/2, root2/2) or (.707, .707) roughly.

If you consider the angle to be the independent variable, then for the
cosine function, the dependent variable takes on the values of the
x-coordiante of the point on the circle.  Starting at an angle of
zero, the value of the cosine function starts at 1 and then decreases
to zero (when the angle is 90 degrees) and decreases to -1 (when the
angle is 180 degrees). The cosine function is a circular function when
defined in this manner.

- Doctor Robert, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Functions

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