Definition: The unit circle c is the circle with center (0,0) and radius R=1.

This means  \color {green}{c: \,\, x^2+y^2=1}

 
But our calculators and computers don't like functions in this form where y is inside.
Problem:  There is no nice y=f(x) function definition for a circle.
Solution: Do you understand a little bit about the sine and cosine functions? Just the definitions is enough.
Then there is a very nice way to define a circle using these functions and it will help you understand circles, sine and cosine too and maybe even help you in physics. A win-win-win.
The equations for the unit circle: \left\{ \matrix{ x = \cos \theta \cr y = \sin \theta \cr } \right. .
θ is called the parameter and x and y are called parametric functions.
  • With parametric equations - you use the parameter to find the values of x and of y .
  • You don't plot the parameter - just the points (x,y) .
  • This means the graph is still 2D, that is 2 dimensional!
  • Cool thing about parameters - even though you don't plot them, they usually "mean something".
    • Here θ is the angle!
    • So here we can actually "see the parameter" on the graph.
InterActivity 1: Is the graph a unit circle?
*We make a table of values -

all our points should land on the circle!

\theta (x,y)=(cosθ,sinθ) Check!
t1=0° A1=(cos(t1),sin(t1)) A1=(1,0)
t2=90° A2=(cos(t2),sin(t2)) A2=(0,1)
t3=180° A3=(cos(t3),sin(t3)) A3=
t4=270° A4=(cos(t4),... A4=
t5=30° A5=(cos(t5),... A5=
t6=45° A6=... A6=
t7=60° A7=... A7=
t8=XXX° A8=... A8=
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More coming ...
  • We can use this circle to remember (or to learn) that: sin^2\theta+cos^2\theta=1 .
  • We can now draw the unit circle on our graphing calculator. Click on the mathcast below you want to watch.

Using a graphing calculator: Using the freeware GeoGebra:


ACT FU 33-36


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Page last modified on September 28, 2008, at 09:10 AM