Assignment 11

Polar Equations

Amena Warrayat

 


Problem 1 states:
Investigate
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Note:

  • When a and b are equal, and k is an integer, this is one textbook version of the " n-leaf rose."
  • Compare with

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for various k. What if . . . cos( ) is replaced with sin( )?
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Let’s start with a =1, b =1, and k = 1
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Let’s try changing the value of k, but keeping a and b constant.
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From the graphs above we can deduce that k corresponds to n, where n represents the number of leaves on the n-leaf rose. Also notice that the x-intercepts of each graph is 2, and in some cases -2, which is the value of +/- (a+b).

If we let k= .5
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You must double your rotations to get a complete graph. We can see there are 2 inverted leaves.

When k = .25
We must increase the # of rotations by 4 to get a complete graph. Notice that we now have 4 leaves inverted in each other.
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Now let’s look at graphs of when the values of k and b are varied and a is the constant.
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Now we have a double leaf rose. Notice k = n for small leaves as well as k = n for larger leaves.
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Notice that the x intercepts are still equated to +/- (a+b), and as b increases the length of the second set of leaves elongates.
What if cos() is replaced with sin()?
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The shape of the graph stays the same but when the translated from cosine to sine the graph is rotated 90°. It can also be noted that when a = b, the range of r is from 0 to 2a. On the other hand, when a ≠ b, r takes on negative values and hence the second set of leaves is produced.

Now, let a = 0 and we can compare our previous results using the equation with a ≠  b and k a real number.

 

It is interesting to see that by using the equation , having a ≠ b does not guarantee the second set of leaves being produced!

 

Now, let's explore with k = 7.5. Is it a 7.5 leaf rose?

It is to be noted that when k is not an integer, it does not always form a k-leaf rose. Here, upon choosing the specific value of k = 7.5, we have three cases:

Case 1: a ≠ b, a = 0: two 7.5 leaf roses are disfiguredly formed.

Case 2: a ≠ b, a ≠ 0, and a < b: a second set of leaves are formed.

Case 3: a ≠ b, a ≠ 0, and a >b: a 7.5 leaf rose is not fully formed because the vertex of the rose is not at the origin.

An exciting topic!

 


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