Parametric Equations
A parametric curve in the plane is a pair of functions where
the two continuous functions define ordered pairs (x,
y). The extent of the
curve will depend on the range of t.
The following is the graph of the parametric equations
for
.
where 6.28 is approximately 2π.
Notice that the unit circle is the curve defined by the parametric equations
.
Why?
Recall that for the acute angle
which we will call
in the right triangle below
and
.
In the unit circle which is defined by the equation
and
.
Recall the graphs of the sine and cosine functions.
Remember that the graphs are periodic and that they repeat themselves at ≈ every 6.28 units.
. Thus, that is
why
in the parametric
equations
.
Notice the following table. Keep in mind that some of the below values are
approximations.
t |
x=cos(t) |
y=sin(t) |
0 |
1 |
0 |
0.52 |
0.87 |
0.5 |
0.78 |
0.71 |
0.71 |
1.05 |
0.5 |
0.87 |
1.57 |
0 |
1 |
3.14 |
-1 |
0 |
4.71 |
0 |
-1 |
6.28 |
1 |
0 |
For each value of t in the table above
because in the unit circle the cosine and sine are defined as
and
.
Thus, the parametric equations for
for
define the unit circle.