\input{decls.tex}
\title{Tutorial 2023-10-10}
\begin{document}
\maketitle
\section{}
\section{}
They're rotated by 90 degrees
\section{}
\begin{tabular}{ c c c c }
$\theta$   & $\cos\theta$   & $\sin\theta$   & $\tan\theta$    \\ \hline
$0$        & $ 1 $          & $ 0 $          & $ 0 $           \\ \hline
$\pi/6$    & $\sqrt{3}/2$   & $1/2$          & $\sqrt{3}/3$    \\ \hline
$\pi/4$    & $\sqrt{2}/2$   & $\sqrt{2}/2$   & $1$             \\ \hline
$\pi/3$    & $1/2$          & $\sqrt{3}/2$   & $\sqrt{3}$      \\ \hline
$\pi/2$    & $0$            & $1$            & undefined       \\ \hline
$-\pi/6$   & $\sqrt{3}/2$   & $-1/2$         & $-\sqrt{3}/3$   \\ \hline
$-\pi/4$   & $\sqrt{2}/2$   & $-\sqrt{2}/2$  & -1              \\ \hline
$-\pi/3$   & $1/2$          & $-\sqrt{3}/2$  & $-\sqrt{3}$     \\ \hline
$-\pi/2$   & $0$            & $-1$           & undefined       \\ \hline
\end{tabular}
\section{}
\subsection{}
\begin{align*}
& +1 = \polar{1}{0} & {-1} = \polar{1}{\pi} \\
& +i = \polar{1}{\frac{\pi}{2}} & {-i} = \polar{1}{-\frac{\pi}{2}}
\end{align*}
\subsection{}
\[1 - i = \polar{\sqrt{2}}{\frac{-\pi}{4}}\]
\[1 + i = \polar{\sqrt{2}}{\frac{\pi}{4}} \]
\subsection{}
\[-1 + i\sqrt{3} = \polar{2}{\frac{-\pi}{3}}\]
\[-1 - i\sqrt{3} = \polar{2}{\frac{+\pi}{3}}\]
\subsection{}
\[\]
\section{}
\subsection{}
\[i\]
\subsection{}
\[\sqrt{2} + \sqrt{2}i\]
\subsection{}

\end{document}