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\usepackage{tabu}

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\title{Polar time}
\date{November 23, 2020}

\begin{document}

\ 

\vspace{-.25in}

\maketitle

\vspace{-.15in}

The polar point $(r, \theta)$ is the point at a distance $r$ from the origin in the direction of angle $\theta$.
To convert from polar to Cartesian: \begin{align*}
x &= r\cos \theta \\
y &= r \sin \theta 
\end{align*}

\prob{ 
Plot the polar points and find their Cartesian coordinates:

\noindent \begin{minipage}[c]{.4\textwidth}
\begin{enumerate}[a)]
\item $\displaystyle \left(2, \frac{2 \pi}{3}\right)$

\vspace{.3in}

\item $\displaystyle \left(1, -\frac{\pi}{4}\right)$

\vspace{.3in}

\item $\displaystyle \left(-1, \frac{3 \pi}{4}\right)$

\vspace{.3in}

\end{enumerate}
\end{minipage}
\hfill
\begin{minipage}[c]{.4\textwidth}
\begin{tikzpicture}
\draw[step=0.5cm,color=lightgray,very thin] (-3,-3) grid (3,3);
\end{tikzpicture}
\end{minipage}
}

When converting from Cartesian to polar coordinates:
\begin{align*}
r^2 &= x^2 + y^2 \\
\tan \theta &= \frac{y}{x}
\end{align*}
Some values of $\tan \theta$:
{ \def\arraystretch{1.5}
\begin{tabular}{|r|ccccc|}
\hline 
$\theta$ & 0 & $\frac{\pi}{6}$ & $\frac{\pi}{4}$ & $\frac{\pi}{3}$ & $\frac{\pi}{2}$ \\
\hline
$\tan \theta$ & 0 & $\frac{1}{\sqrt{3}}$ & 1 & $\sqrt{3}$ & U \\
\hline
\end{tabular}
}

\prob{
Find the polar coordinates of the Cartesian points:
\begin{enumerate}[a)]
\item $(\sqrt{3}, 1)$
\item $(-1,1)$
\item $(2,-\sqrt{12})$
\end{enumerate}
}


\newpage

\prob{
Sketch a graph of the polar curve: $\displaystyle r=1 + 2 \cos \theta $.
Give coordinates for all axis intercepts.

\begin{tikzpicture}
\draw[step=0.5cm,color=lightgray,very thin] (-3,-3) grid (3,3);
\end{tikzpicture}
}

\vfill

\prob{
Sketch graphs of the polar curves. $\displaystyle r= \sin \theta $.
Label all axis intercepts.
\begin{enumerate}[a)]
\item $\displaystyle r= \sin \theta $.
\item $\displaystyle r= \sin (2\theta)$.
\item $\displaystyle r= \sin (3\theta)$.
\end{enumerate}

\begin{tikzpicture}
\draw[step=0.5cm,color=lightgray,very thin] (-3,-3) grid (3,3);
\end{tikzpicture}
\begin{tikzpicture}
\draw[step=0.5cm,color=lightgray,very thin] (-3,-3) grid (3,3);
\end{tikzpicture}
\begin{tikzpicture}
\draw[step=0.5cm,color=lightgray,very thin] (-3,-3) grid (3,3);
\end{tikzpicture}
}

\vfill



\end{document}