# Unit Circle - Sine & Cosine

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=
 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:

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