%Polar Coordinates
%If you modify this file, please indicate here and in the footnote
%Math263B

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{\Large
Polar Coordinates\footnote{Copyright \copyright 2002 Steve Chapin.
All rights reserved. Please address comments to young@math.ohiou.edu.}}
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\begin{enumerate}
\item Enter the following sequence of commands:\\
      \verb&   syms t& \\
      \verb&   r = cos(4*t)& \dotfill Use \verb$t$ in place of
      $\theta$. \\ 
      \verb&   ezplot(r*cos(t), r*sin(t), [0,2*pi])& \\
This plots the polar equation $r = \cos{4\theta}$. Explain why.

\item  Using the pattern above, plot the polar equation $r = \sin(n\theta)$ for several 
positive integers $n$.  (Use the $\uparrow$ key.)  Find a formula 
for the number of loops.

\item  Plot the polar equation $r = \sin(p\theta/q)$ for various integers
$p$ and $q$, satisfying\\ 
$p$ \verb$>$ $q$ \verb$>$ $0$.  Write $p/q$ in lowest terms 
and plot over the interval [0, 2$\pi$q].   
Find a formula for the number of loops.

\item Plot the polar equation $r = \sin(\sqrt{2} \, \theta)$
on the interval $[0, 100\pi]$. Explain the resulting plot.

\item Plot the polar equation 
	$r = e^{\cos \theta} - 2\cos4\theta + \sin^5(\theta/12)$
for $0 \le \theta \le 24\pi$.  (This curve was discovered by Temple H. Fay.)
What does the graph resemble? (Type: \verb& exp(cos(t)) & for $e^{\cos \theta}$ and \verb& (sin(t/12))^5 &
for $\sin^5 (\theta/12)$.)

\item On a separate piece of paper, prepare a brief written report
describing what happened and answering all the questions.  Use complete
sentences and use standard mathematical notation.  Hand-in sketches of graphs or
computer plots as directed by your instructor.

\end{enumerate}

\vfill
\noindent
{\sf Polar equations can be plotted by transforming them into 
parametric equations.}

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