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*What is
a polar equation?*

Simply stated: A polar equation is an equation
which is represented in polar form. *What does this statement
mean?* Well, suppose you have a point (2,3) and want to know
the equivalent polar coordinates of this point. In polar form,
this point looks like: (r, theta). This r is the distance from
the origin to the point and the second coordinate, theta, is the
angle created by the vector from the origin beginning at the positive
x-axis to the point. The following are conversion rules:

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Let's begin our investigation with the following polar equation:

What happens when a=b=1, and k varies?

To view the animation where k = 1, 2, ..., 100, click here.

*What pattern do you observe?* Imagine this image is a flower, so we see that when
k = 2, we have two petals, when k = 3, we have three petals, etc.
This example is a textbook version of the "n-leaf rose".

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What happens when a > b and k = 1:

For the best visual, click here and here. In both cases a > b; however, a varies then b varies, respectively. One can imagine these images would alter when k varies, as well. Recall that the absolute value of k dictates how many "petals" are observed.

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What happens when a < b and k = 1:

Again, for the best visual, click here and here. In both cases a < b; however, a varies then b varies, respectively. Likewise, these images would change when k varies where the number of "petals" would match the absolute value of k.

Now, what happens when we remove "a" from the equation?

Suppose k varies:

To observe where k = 1, 2, ..., 50, click here.

Lastly, what happens when cosine is replaced with sine in our original polar equation?

Suppose a = b = 1, and k varies:

Like our previous example, click here to see how the image changes for k = 1, 2, ..., 100. When compared to our original cosine equation, these sine images have shifted by 90 degrees where k = 1, 45 degrees where k = 2, etc.

These investigations only begin to explore polar equations. However, this introduction provides a basis for further exploration in comparing Cartesian equations with polar equations.

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