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Summer Tuggle

Assignment 11

Polar Equations

Investigate r= a + b cos(kΘ)

Compare with r= b cos(kΘ) for various k.

What if cos( ) is replaced with sin( )?

First let’s investigate what effect each variable has on the graph.  Let’s look at what happens when b and k are 1 and we vary “a”.

On the graph below we will look at a few cases:

r = 1 + 1 cos(1Θ)        r = 2 + 1 cos(1Θ)                r = 3 + 1 cos(1Θ)

As “a” becomes bigger, the graph becomes more and more like a circle.

It is interesting to note that when is “a” is 0 the graphs appears to be a circle with an origin at (1/2, 0)

If we keep a and k constant and alter “b” we get another pattern:

r = 1 + 1 cos(1Θ)        r = 1 + 2 cos(1Θ)                r = 1 + 3 cos(1Θ)

r = 1 - 3 cos(1Θ)

When a and b are constant at 1 and k is varied you get the basic n-leaf rose where “k” gives the number of petals.

r = 1 + 1 cos(2Θ)                r = 1 + 1 cos(3Θ)                r = 1 + 1 cos(4Θ)

Changing the size of a and b but keeping them equal to each other gives you longer petals.

r = 5 + 5 cos(3Θ)       r = 3 + 3 cos(4Θ)       r = 4 + 4 cos(5Θ)

When a and b are no longer equal, our rose petals no longer meet at the origin:

r= 3 + 1 cos(2Θ)        r = 2 + 1 cos(3Θ)               r = 3 + 2 cos(4Θ)

Now let’s compare r= a + b cos(kΘ) with r= b cos(kΘ)

r= 2 + 2 cos(4Θ)                r = 2 cos(4Θ)

The purple flower has 4 stretched out petals as would be expected based on the previous explorations. The red flower has 8 petals not 4 as would be expected with a k value of 4. Upon further investigation, I found that in the graph of r= b cos(kΘ) the number of petals follows an interesting pattern.  For all even k, the number of petals is double the k value while for all odd k the number of petals is k.

r=  2 cos(2Θ)    r=  2 cos(3Θ)    r=  2 cos(4Θ)    r=  2 cos(5Θ)

When you compare the graphs of r= a + b cos(kΘ) and r= b cos(kΘ) with r= a + b sin(kΘ) and r= b sin(kΘ) you see that the graph is basically the same except that the petals have rotated.

r=  3 + 3 sin(3Θ)                r=  3 + 3 sin(3Θ)

r=  3 + 3 cos(3Θ)       r=  3 cos(3Θ)