\documentclass[letterpaper,12pt]{article}
\usepackage[utf8x]{inputenc}
\usepackage{amsmath,amssymb}
\usepackage[letterpaper,margin=1in,includehead=true]{geometry}
\usepackage{comment}
\usepackage{graphicx}
\usepackage{fancyhdr}
\usepackage{enumerate}
\pagestyle{fancy}
\usepackage{color}
\usepackage{wrapfig}
%\usepackage{enumitem}
\usepackage{tikz, pgfplots}
%\usetikzlibrary{calc}
%\usepackage{titling}
%\setlength{\droptitle}{-1.5in}

%opening
\title{\vspace*{-1.5in}\textbf{Polar Coordinates (Appendix H1)}\\[3pt]\normalsize Thanks to Faan Tone Liu}
\date{}
\author{}

\begin{document}

\maketitle
%\underline{Math 1300-400: Quiz 1}
\lhead{Math 2300--007: Calculus II}
\chead{April 25, 2018}
\rhead{Polar Coordinates}
\vspace{-.7in}
\noindent
\textbf{Key Points:}
\begin{itemize}
\item Every location in the plane can be described by $(r,\theta)$, where
\begin{align*}
r &= \text{distance from the origin}\\
\theta &= \text{angle from the positive $x$-axis}.
\end{align*}
\item Consider the following picture:\vskip1.5in
\item Converting from polar to rectangular coordinates:
\begin{align*}
x &= \phantom{alksjflasdkjflakdjflasdkjf}\\
y &=
\end{align*}
\item Converting from rectangular to polar coordinates:
\begin{align*}
r^2 &= \phantom{alksjflasdkjflakdjflasdkjf}\\
\tan\theta &=
\end{align*}
\end{itemize}
\noindent
\textbf{Examples:}\vskip12pt
\begin{enumerate}
\item Plot the following points:
\begin{enumerate}[A.]
\item $(r,\theta) = (2, \frac{2\pi}{3})$
\item $(r,\theta) = (4, \frac{3\pi}{2})$
\item $(r,\theta) = (-3, \frac{3\pi}{4})$
\item $(r,\theta) = (0, \frac{11\pi}{6})$
\end{enumerate}
\item Convert $(2,\frac{2\pi}{3})$ into rectangular coordinates.
\newpage
\item Convert $(-5, -5\sqrt{3})$ into polar coords.\vskip0.5in
\item Convert $r=2$ to rectangular coords.\vskip.5in
\item Convert $r=3\cos\theta$ to rectangular coords.\vskip1.4in
\item Graph the following polar curves:
\begin{center}
\begin{tabular}{c@{\qquad\qquad}c@{\qquad\qquad}c}
$r =1$ & $r=\theta$ & $\theta = \frac{5\pi}{6}$\\ 
\begin{tikzpicture}[x=1.8cm,y=1.8cm]
\draw (-1,0)--(1,0);
\draw (0,-1)--(0,1);
\end{tikzpicture}
&
\begin{tikzpicture}[x=1.8cm,y=1.8cm]
\draw (-1,0)--(1,0);
\draw (0,-1)--(0,1);
\end{tikzpicture}
&
\begin{tikzpicture}[x=1.8cm,y=1.8cm]
\draw (-1,0)--(1,0);
\draw (0,-1)--(0,1);
\end{tikzpicture}
\end{tabular}
\end{center}
\item Graph the following polar curves (Hint: first graph in rect. coords):
\begin{center}
\begin{tabular}{c@{\hspace{2in}}c}
$r = 3 + 3\sin\theta$ & $r=3\cos\theta$ \\ 
\begin{tikzpicture}[x=1.5cm,y=1.2cm]
\draw (-.2,0)--(3,0);
\draw (0,-1)--(0,1);
\end{tikzpicture}
&
\begin{tikzpicture}[x=1.5cm,y=1.2cm]
\draw (-.2,0)--(3,0);
\draw (0,-1)--(0,1);
\end{tikzpicture}\\\\[6pt]
\begin{tikzpicture}[x=1.8cm,y=1.8cm]
\draw (-1,0)--(1,0);
\draw (0,-1)--(0,1);
\end{tikzpicture}
&
\begin{tikzpicture}[x=1.8cm,y=1.8cm]
\draw (-1,0)--(1,0);
\draw (0,-1)--(0,1);
\end{tikzpicture}
\end{tabular}
\end{center}


\end{enumerate}
\end{document}
