Investigation of

To start this investigation, again,  we will look at the graph of the above equation when a = 1 and k = 1.

Looking at this graph we see that equation is symmetric around the x-axis and that it appears that the circle still has a center of 1 on the polar scale.

To investigate further we will see what happens to the graph when we vary the value of a...therefore let's explore the graph of the equation when a = 3 and k = 1.

It appears that the circle has expanded in this case as well...it now appears that the center is equal to 3 on the polar scale.

Now let's see what happens when we vary the value of k...so let's look at what happens if a = 1 and k = 2

It seems that once again if we allow k > 1 we get a completely different type of graph...instead of a circle we have pedals.

Since the graph has 4 pedals...we could possible assume that k times 2 would equal the number of pedals.

Let's check to see if our theory for the sine equation is the same as the cosine equation.  Let's look at another graph to check...how about when k = 5 and a is still equal to 1.

Here it looks like the number of pedals equals the value of k

Click on the links to see the graphs when k = 4 and k = 7.

Yes, it appears that our theory for sine is the same for cosine

After looking at these graphs we can assume that when k is an even number, the number of pedals will be twice the value of k.  However, when k is an odd number, the number of pedals is equal to the value of k.

So what is the difference in the equations...let's look at the following graph for both the sine equation and the cosine equation.

Now what happens when we vary both the values of "a" and "k"?

Lets look when a = 3 and k = 5...if our assumptions stand true we should get a graph with 5 pedals and the center of the pedal being equal to 3.

Yep, not only do we have a graph with 5 pedals where the center of the pedal appears to be 3 but looking at the graphs overlapped we can also see that the cosine equations rotates the graph to the right