# Sinje J. Butler

## Write-up 10

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.

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