Parametric Curves and Trigonometry
By Sharon K. OÕKelley
Introduction
Consider the equationsÉ
Notice that squaring both
equations yieldsÉ
Using the Pythagorean Trigonometric
Identity, we know thatÉ
Using substitution, it can
be established thatÉ
which is the equation for the Unit Circle.
The parameter of t, therefore, can be viewed as the angle of rotation
as the terminal ray containing point (x, y) moves counterclockwise around the
unit circle. (See figure 1.). The ordered pair (x,y) can also be regarded as
(cos(t), sin(t)).
Figure 1
Varying ÒaÓ and/or ÒbÓ
What happens if a and/or b are
varied in the following equations?
LetÕs experiment. What if
ÒaÓ is two and ÒbÓ is one? This would yield the following valuesÉ.
t (degrees) 
t (radians) 
x 
y 


2 
0 


0 
1 


2 
0 


0 
1 
If these equations were
graphed, it would look like the followingÉ.
(Note that if ÒaÓ
were negative, it would still yield the same graph because the values would
switch – e.g. t = 0 would yield (2,0) and t=180 would yield (2,0)).
Notice that this yields a
horizontallyoriented ellipse whose major axis has a measure of 4 units and
whose minor axis has a measure of 2 units. In essence, the unit circle has been
horizontally dilated by a factor of 2 to yield the ellipse.
What if the following
equations were given?
A prediction can be made
that a graph of these equations would be a verticallyoriented ellipse with points
(1/2, 0), (1/2,0), (0, 3) and (0,3).