\documentclass[11pt]{article}

\usepackage{precalc}
\usepackage{graphicx}

\newcommand{\diagram}[2]{
  \begin{figure}[htp]
  \centering
  \includegraphics{diagrams/#1.eps}
  \caption{#2}
  \label{fig:#1}
  \end{figure}
}
  

\begin{document}

\assigntitle{14}{Polar coordinates}

\topic{yo yo yo and a bottle of gum} 
One sunny Saturday afternoon, you
are out hunting for buried treasure, as usual.  It so happens that at
this particular time, on this particular Saturday, you are standing
immediately above the treasure (consisting of \$15 trillion billion
dollars' worth of vintage yo-yos and vintage bubblegum in a vintage
coke bottle) but you don't know it, because your map is wrong.  Or
maybe you read the map wrong.  Or maybe the directions you got from
that nice man with the penguin were wrong.  Come to think of it, his
penguin \emph{did} seem a little confused.

In any event, your map says to go 50 feet east and 20 feet north, even
though ideally it should say ``dig here to become filthy rich.''

\begin{problem}
  If you follow your map's instructions, how far away will you be from
  the treasure?
\end{problem}

\begin{problem}
  Being the clever lads that you are, you realize that you could save
  some time by walking directly to the point 50 feet east and 20 feet
  north of you, instead of first walking 50 feet east and then turning
  and walking 20 feet north.  In what direction should you face before
  you begin walking?  Give your answer as an angle in radians
  counterclockwise from due east.  (For example, due north would be
  $\pi/2$.)
\end{problem}

\topic{oops}
(Unfortunately, it turns out that your map was cleverer than you, and
you end up walking directly into a gorse\footnote{any spiny shrub of
  the genus \emph{Ulex}, of the legume family, native to the Old
  World, especially \emph{U. europaeus}, having rudimentary leaves and
  yellow flowers and growing in waste places and sandy soil}-thicket
which you could have avoided by first walking 50 feet east and then
turning and walking 20 feet north. Oh well.)

\section{Two-dimensional coordinate systems}
\label{sec:coordinates}

There are two important and commonly used systems for thinking about
position in a two-dimensional world: \emph{Cartesian} (or
\emph{rectangular}) coordinates, and \emph{polar} coordinates.

\subsection{Cartesian coordinates}
\label{sec:cartesian}

\topic{Cartesian coordinates}
You are already familiar with the Cartesian coordinate system (also
sometimes called the \emph{rectangular} coordinate system): this is
the familiar $(x,y)$ system that you learned in
fifth\footnote{something like that} grade: points in the plane are
represented by $x$ and $y$ coordinates representing their
\emph{horizontal} and \emph{vertical} distance from a distinguished
point called the \emph{origin}.  Cartesian coordinates are what your
map used when it told you to go to a point 50 feet east and 20 feet
north of you: in other words, it told you to go to the point
$(50,20)$.

\diagram{cartesian}{Cartesian coordinates}

\topic{a really smart dude from the past}
The Cartesian coordinate system is named for Ren\'e Descartes
(1596--1650), a French philosopher, mathematician, and scientist who
is known as (among other things) the ``father of analytic geometry.''
\emph{Analytic geometry} simply refers to the idea of using a
numerical coordinate system to analyze geometrical; this may seem
obvious to you now (having been taught it in school), but in its time
it was a brilliant new idea that directly led to the development of
calculus by Newton and Leibniz.

\begin{problem}
  Can there be two pairs of $(x,y)$ coordinates that refer to the same
  point?  Why or why not?
\end{problem}

\subsection{Polar coordinates}
\label{sec:polar}

\topic{polar coordinates}
The second coordinate system is the \emph{polar} coordinate system.
In this system, points in the plane are represented by their
\emph{distance} from the origin, and the \emph{angle} that they make
with the positive $x$-axis.  (Actually, calling it the ``positive
$x$-axis'' is kind of silly, since $x$ specifically has to do with
Cartesian coordinates; in the polar coordinate system, we call it the
\emph{pole}.)  Polar coordinates are what you used when you computed a
shortcut through the gorse-thicket: you figured out the
\emph{distance} and \emph{angle} at which you had to travel to get to
the desired point.  In a polar coordinate system, the distance from
the origin to a point is called $r$ (R for Radius) and the angle from
the pole is called $\theta$.

\diagram{polar}{Polar coordinates}

\begin{problem}
  In which quadrant is the point with polar coordinates $(10,4\pi/3)$?
\end{problem}

\begin{problem}
  Can the same point have multiple different polar coordinate pairs?
  If not, explain why; if yes, give an example.
\end{problem}

\section{Conversion}
\label{sec:conversion}

\topic{converting between Cartesian and polar}
Since we have two different ways to refer to points in the plane, it's
useful to know how to convert between the two representations.
Everything you need to know about the relationship between Cartesian
and polar coordinates is shown in the diagram in
\pref{fig:cartesian-polar-rel}. 

\diagram{cartesian-polar-rel}{The relationship between Cartesian and
  polar coordinates}

The point at the upper right corner of the triangle has Cartesian
coordinates $(x,y)$ and polar coordinates $(r,\theta)$.

\begin{problem}
  Use \pref{fig:cartesian-polar-rel}, your knowledge of
  trigonometry, and the Pythagorean Theorem to help answer the
  following questions:
  \begin{subproblems}
    \item What is $x$ in terms of $r$ and $\theta$?
    \item What is $y$ in terms of $r$ and $\theta$?
    \item What is $r$ in terms of $x$ and $y$?
    \item What is $\theta$ in terms of $x$ and $y$?
  \end{subproblems}
\end{problem}

\begin{problem}
  Convert from polar to Cartesian coordinates.
  \begin{subproblems}
    \item $(\sqrt{2}, \pi/4)$
    \item $(5, 4\pi/3)$
    \item $(2,\pi)$
    \item $(6,0)$
  \end{subproblems}
\end{problem}

\begin{problem}
  Convert $(-5, \pi/2)$ from polar to Cartesian coordinates.  Does
  this make sense?  What should negative values of $r$ mean?
\end{problem}

\begin{problem}
  Convert from Cartesian to polar coordinates.
  \begin{subproblems}
    \item $(-1,-1)$
    \item $(0,2)$
    \item $(50,20)$
  \end{subproblems}
\end{problem}

\section{Polar graphs}
\label{sec:graphs}

\topic{Cartesian vs polar graphs}
As you know, we can graph any equation involving $x$ and $y$ by drawing a
line through all the points whose $x$ and $y$ coordinates satisfy the
equation.  For example, the graph of $y = 2x + 1$ is a line; the graph
of $y = \cos x$ is a squiggly wave; and the graph of $x^2 + y^2 = 16$
is a circle with radius $4$.  We can do the same thing with equations
involving $r$ and $\theta$.

\begin{problem}
  What do you think the graph of $r = 4$ looks like?  (Which points
  have polar coordinates with $r = 4$?)
\end{problem}

\begin{problem}
  What do you think the graph of $r = \theta$ looks like?
\end{problem}

\topic{use the graphing calculator, Luke} Your graphing calculator can
probably make polar graphs!  Go to the ``mode'' screen and look for
something like ``Polar'' or ``Pol'' mode.  Now when you go to the
screen where you can enter equations to be graphed, it should say ``$r
=$'' instead of ``$y =$''!  You can now type equations in terms of
$\theta$ (which you can type with the same button you would otherwise
use for $x$).

\begin{problem}
  Try graphing each of the following polar equations using your
  graphing calculator, and describe what they look like.
  \begin{subproblems}
    \item $r = \cos \theta$
    \item $r = \sin \theta$
    \item $r = \cos (5 \theta)$
    \item $r = 1 + \cos \theta$
    \item $r = 1 + 2 \cos \theta$
  \end{subproblems}
\end{problem}

\begin{problem}
  Try graphing some polar equations of your own, and write down one of
  your favorites.
\end{problem}

\end{document}