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/