Department of Mathematics
J.Wilson, EMAT6680

Lesson # 11
by Jan White

An investigation of polar equations:     r = a + b cos(kt)

     When a and b are equal, and k is an integer, this is one textbook version of the " n-leaf  rose."
     Compare with r = a + b cos(kt) for various k.
     What if . . . cos( ) is replaced with sin( )?
 

First let's look at the graph when a =2 and b =2 and k = 2, 3, -4, -5.
 

When the absolute value of k is an integer greater than or equal to 2 then there will be k leaves. Notice that the length of the petals from the origin to the tip of each petal will be a+b, as long as a = b.

What happens when "cos" is replaced with "sin"?
There will be the same number of petals except the flower will be rotated 45 degrees.

Compare r = b cos (kt).

Let's compare when b = 2, and k = 2,3,-4, and -5

In this case the number of petals will be 2k petals when the absolute value of k is even and k petals when the absolute value of k is odd. Again, the length of the petals are dependent on the value of b in the equation.

What happens when "cos" is replaced with "sin"?

Again the graphs are the same except that there is a rotation of 45 degrees. By looking the coordinate graphs of sin x and cos x you can see why this rotation occurs.