This write up is for problems #2 in Assignment 11.

Investigating Polar Equations

by

Investigation of

In the following graphs we will overlay the cosine equation in red and the sine equation investigated above in blue.

When a =1 and k = 1, the polar equations will produce the following graphs.

At this point, we can observe that the circular shape has a center at 1 on the polar scale. The cosine graph (red) is symmetric about the x-axis. This is similar to the previous sine problem (blue) except the circular graph has been rotated to the right. If we first allow "a" to vary we can observe some of the following situations.

When a = 2 and k = 1, the circular graph has expanded.

Varying "a" in this situation has the same affect as when we varied "a" in the sine problem.

Now, we will vary "k" and hold the "a" constant at 1.

When k > 1, the graph no longer has a circular shape. It now begins to appear to have pedals. Such as the following graph with k = 2.

As one can observe from the graphs, the cosine graph appears to pass through the origin and is rotated to the right.

Does our theory about the number of pedals hold true for the cosine problem?

When k =5, the following graph is produced.

Yes, the same theory with "k" from the sine problem holds true with the cosine problem.

Finally, we will vary both "a" and "k".

When a = 2 and k = 5, we would expect the graph to have 5 pedals, to be expanded more than the graph above, and to be rotated to the right. (See graph below)

We have a graph with 5 pedal-like curves, expanded from the original graph, and rotated to the right when compared to the sine graph.

Return