This write up explores polar equations using graphing calculator software.
Here we investigate r = a + b cos(kq) and explores for different values of a, b and k and also compares with r = a + b sin(kq) and explains the observation made.
First lets look at r = a + b cos(kq), with a = b = 2(for any a = b) and for different integer values of k.(we see that positive or negative makes no difference, since cos(q) = cos(-q). For different integer values of k, you get that many “k-leaf rose”, with leaves of magnitude b + b = 2b.
For the function, r = b cos(kq), we observe that the leafs are of magnitude b.
Consider the function r = a + b sin((kq) with a = b and k being an integer values.
We observe that the function results in a “k-leaf rose” with the difference from the cosine function being a phase shift. And the phase shift is 90o/k. For example, 2 leaf rose has phase shift of 45o.
And in case of r = bsin(kq), if k is negative, then the graph is flipped since
sin(-q) = -sin(q). Graphs are shown below for comparison.