Polar Curves

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Textbook Reference: Section 8.4 Polar Coordinates: Graphs

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When you have a new toy, you just kind of want to play with it and see what it can do.

Here, the "toy" is our new way of graphing points. And our way of playing with it is graphing curves where \(r\) depends on \(\theta\) — where \(r\) is a function of \(\theta\).

And the best way to try out a new toy is to just jump right into it.

Preview Activity 26

Answer these questions and submit your answers as a document on Moodle. (Please submit as .docx or .pdf if possible.)

1. One example of a polar curve is a cardioid. You may have heard of a "cardioid microphone", which picks up sound in front of the microphone and blocks everything else; its region of sensitivity is shaped like the curve you're about to graph.

   Consider the cardioid whose equation is \(r=3-3\sin(\theta)\).

   1. Fill in the following table of values for \(r\).
      

      \(\theta\)

      \(0\)

      \(\tfrac\pi6\)

      \(\tfrac\pi3\)

      \(\tfrac\pi2\)

      \(\tfrac{2\pi}3\)

      \(\tfrac{5\pi}6\)

      \(\pi\)

      \(\tfrac{7\pi}6\)

      \(\tfrac{4\pi}3\)

      \(\tfrac{3\pi}2\)

      \(\tfrac{5\pi}3\)

      \(\tfrac{11\pi}6\)

      \(2\pi\)

      \(r\)

   2. Carefully graph the points on a sheet of polar graph paper (click the link). Then connect the dots. (Don't worry, this is the only time I'm ever going to ask you to painstakingly graph a polar equation point-by-point. Once you've done it once, that's enough.)
   3. Desmos can graph in polar coordinates too! Open Desmos, change the setting to a polar grid, and then type r = 3 - 3 cos theta to graph the cardioid. Double-check that it's the same shape you got in part (b).
   4. Why do you think the shape is called a "cardioid"?
   5. In Desmos, also graph \(r=3+3\sin\theta\). Compare and contrast this cardioid to the first one.
2. Cardioids are actually part of a special class of curves called limaçons, which comes from Latin for "little snail." There are two general forms of a limaçon: \[r=a+b\cos(\theta)\qquad r=a+b\sin(\theta)\] In Desmos, graph each of these and use sliders for \(a\) and \(b\). In your own words, compare and contrast the features of the different graphs for various values of \(a\) and \(b\). When does the limaçon look like a cardioid? When does the limaçon have an inner loop?

3. One more interesting type of curve we can graph in polar coordinates is the rose curve, which has one of the following equations: \[r=a\cos(n\theta)\qquad r=a\sin(n\theta)\] Note that \(n\) must be an integer here.

   In Desmos, graph each of these and use sliders for \(a\) and \(n\). (You might want to set the "step" for \(n\) to \(1\) to make sure it can't be a decimal.) In your own words, compare and contrast the features of the different graphs for various values of \(a\) and \(n\). How can you tell how many "petals" a rose curve has from the equation alone?

4. Answer AT LEAST one of the following questions:
   1. What was something you found interesting about this activity?
   2. What was an "a-ha" moment you had while doing this activity?
   3. What was the muddiest point of this activity for you?
   4. What question(s) do you have about anything you've done?

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Posted by Bill Shillito