** Write Up #11**:

__Part A__: Basic
Investigations of Polar Coordinates

Polar coordinates involve a pole and two rays, an initial ray, typically along the positive x-axis, and a terminal ray that sweeps counter-clockwise around a pole (the origin) and is measured by some angle theta, and is measured from the pole by a distance r (radius). The typical polar equation involves sine and cosine in the form

We wish to investigate the following curve

for various values of a, b, and multiples of theta.

__Case 1__: For a=1,
b=0, and __theta = 0__, the equation is the degenerate case
of a circle about the origin:

__Case 2__: For a=1,
b=0, and __odd__, integer values of k, we get a "k-pedal"
graph where the number of pedals __is equal to__ the value
of k: k = 1 yields one pedal (purple); k = 3 yields three pedals
(red); and k = 5 yields five pedals (green). Note that since a
is one, the coefficient of the sine function is 2 and the length
of the pedals is two.

__Case 3__: For a=1,
b=0, and __even__, integer values of k, we still get a "k-pedal"
graph, but the number of pedals __is twice__ the value of k:
k = 2 yields four pedals (red); k = 4 yields eight pedals (blue);
and k = 6 yields twelve pedals (not shown, because it is a mess).

__Case 4__: For a=1,
b=0, and using the __sine function__ instead of cosine for
odd values of n, we note the graphs are rotated:

__Case 5__: Returning
to the __cosine function__ for continued investigations, we
investigate values of a (when b=0 and theta multiple k is five
for 5-pedal figure), for values of 1, 2, and 3 of the following
equation:

Notice that a is a multiplier of the figures as noted below:

__Case 6__: Continuing
with the same cosine function ,

we will hold a=1 and study b = 1, 2, 3, 4,
5 for various values of k. If ** k = 1**, then we note
that b < the coefficient of cosine, and the figure is a lemiscon
(purple). If k = 2 (b = coefficient of cosine), then the figure
is a cardioid (red). For k = 3, 4, 5, and higher (b > coefficient
of cosine), then the cardioid figure looses its shape and approaches
a circle as k increases (green, light-blue, yellow, respectively):

__Case 7__: With
the same cosine function ,

we will hold a=1 and study b = 1, 2, 3, 4,
5 for ** k = 2**, then we notice the same things happen,
but the figures are different. With b < the coefficient of
cosine, and the figure is a 4-pedal figure (purple). If k = 2
(b = coefficient of cosine), then the figure loses shape and only
has 2 large pedals (red). For k = 3, 4, 5, and higher (b >
coefficient of cosine), then the 2-pedals continue to lose shape,
but increase in size as k increases (green and light-blue, respectively):

__Case 8__: Continuing
with the same cosine function ,

we will hold a=1 and study b = 1, 2, 3, 4,
5 for ** k = 3**, then we notice that the figures are
different, but the same things happen. With b < the coefficient
of cosine, and the figure is a 3-large pedal figure with an internal
pedals (purple). If k = 2 (b = coefficient of cosine), then the
figure loses its internal pedals (red). For k = 3, 4, 5, and higher
(b > coefficient of cosine), then the pedals themselves continue
to lose shape, but increase in size as k increases (green and
light-blue, yellow, and gray, respectively):

__Part B__: An
Interesting Form of Polar Coordinates

LINKHEREfor Ken Hayakaya's investigation.

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