Valerie Russell

Assignment 10

Parametric Equations

Investigating parametric equations in the form x = a cos(t) and y = b sin(t).

When a = b.

Notice the parametrization of a circle of radius a or b.

Parametric equations of the form:

x = a cos(t)

y = b sin(t)

a < b:

When a is less than b as in the first parametric equation, we no longer have a circle, An ellipse is formed whose major axis is vertical.

a > b:

When a is greater than b as in the second parametric equation, we also have an ellipse. The major axis is horizontal.

It seems to appear that the values of a and b affect the direction of the major axis of an ellipse.

 

 

Let's investigate parametric equations for various values of a & b for x = cos (at) and y = sin (bt). We will let n represent these values.

x = cos(nt)

y = sin(nt)

Notice what happens when the coefficients of t are the

same (a = b). The parametric equation still forms a circle

but the radius does not change.

x = cos(nt): -20 < n <20

y = sin(t)

As the coefficient of t changes for cosine, the parametric

equation forms an increasing vertical coil. Whether n is

smaller than 1 or larger than 1, we still have the same

vertical coil forming.

 

 

x = cos(t)

y = sin(nt) -20 < n < 20

As the coefficient of t changes for sine, the parametric

equation forms an increasing horizontal coil. The period I

used here is 0 < t < 6.28. When b is zero, you have a

horizontal line from -1 to 1. Whether n is smaller than 1

or greater than 1, the same horizontal coil is formed.

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