\documentclass[11pt]{article}

\usepackage{precalc}
\usepackage{graphicx}

\begin{document}

\assigntitle{10}{Reference angles}

\section{Sine and cosine, redux}
\label{sec:sin-cos-redux}

\topic{special angles recap}
Let's take a minute to remember the things we've learned about the
sine and cosine functions.  Last week, recall that you filled in a
table which looked something like \pref{tab:special}.

\begin{table}[htp]
  \centering
  \begin{tabular}{c|cc}
    $\theta$ & $\sin \theta$ & $\cos \theta$ \\
    \hline
    $0$     & $0$          & $1$          \\
    $\pi/6$ & $1/2$        & $\sqrt{3}/2$ \\
    $\pi/4$ & $\sqrt{2}/2$ & $\sqrt{2}/2$ \\
    $\pi/3$ & $\sqrt{3}/2$ & $1/2$        \\
    $\pi/2$ & $1$          & $0$          \\
  \end{tabular}
  \caption{All you will ever need to know!}
  \label{tab:special}
\end{table}

Actually, it looked slightly different, but it was equivalent:

\begin{problem}
  Show that $\sqrt{1/2} = \sqrt{2}/2$.  (\emph{Hint}: multiply by a
  something suitably equal to $1$\dots)
\end{problem}

\topic{special angles, more specialler}
\pref{tab:special} shows the way that this table is most commonly
presented.  However, there is another way to write the same table
which makes it easier to remember: see \pref{tab:special2} (you should
convince yourself that this is the same).

\begin{table}[htp]
  \centering
  \begin{tabular}{c|cc}
    $\theta$ & $\sin \theta$ & $\cos \theta$ \\
    \hline \\
    $0$     & $\sqrt{0/4}$ & $\sqrt{4/4}$ \\
    $\pi/6$ & $\sqrt{1/4}$ & $\sqrt{3/4}$ \\
    $\pi/4$ & $\sqrt{2/4}$ & $\sqrt{2/4}$ \\
    $\pi/3$ & $\sqrt{3/4}$ & $\sqrt{1/4}$ \\
    $\pi/2$ & $\sqrt{4/4}$ & $\sqrt{0/4}$ \\
  \end{tabular}
  \caption{All you will ever need to know, now even easier to remember!}
  \label{tab:special2}
\end{table}

\topic{special angles, the movie}
Nifty, eh?  And here's a nice diagram showing the same thing
(\pref{fig:special3}).

\begin{figure}[htp]
  \centering
  \includegraphics{diagrams/sin-cos-special-angles.eps}
  \caption{All you will ever need to know, in picture form!}
  \label{fig:special3}
\end{figure}

Why am I making such a big deal out of this? Well, as the captions
hint, this is pretty much all you will ever need to memorize when it
comes to sine and cosine, beyond the basic definition.  Technically,
if you know the definition of sine and cosine, you don't need to
memorize this table; after all, you figured out this table from the
basic definition in last week's assignment.  In practice, however,
these special angles come up so often that it's very helpful to know
these cold. You don't want to spend fifteen minutes drawing pictures
of triangles and working through the Pythagorean theorem every time
you need to evaluate $\sin (\pi/3)$.\footnote{Trust me on this.}

\begin{problem} \label{prob:memorize}
  Your mission, should you choose to accept it,\footnote{The foregoing
    learning-enhancement task (the ``ASSIGNMENT'') shall remain the
    sole responsibility and charge (the ``MISSION'') of any learners
    who are party to said ASSIGNMENT (the ``STUDENTS''), regardless of
    otherwise extenuating circumstances, such as, but not limited to,
    non-acceptance of the MISSION by the STUDENTS; prandial activities
    of any canine, feline, bovine, ursine, or other non-\emph{homo
      sapiens} companion organisms, even when said prandial activities
    are conducted in reference to the ASSIGNMENT; alignment or
    non-alignment of heavenly bodies; or destructive behavior by
    egregiously sized, genetically anomolous amphibious organisms.
    The author of the ASSIGNMENT (the ``TEACHER'') shall under no
    circumstances assume any liability for any adverse affects which
    may or may not arise as a result of completing the MISSION,
    including, but not limited to, hypertension, sleep apnea,
    halitosis, somnambulance, rheumatism, gnosticism, or anything else
    ending in -ism.} is to take as long as you need to memorize one of
  the above tables and/or picture (whichever is easiest for you to
  remember), and then evaluate each of the expressions found on the
  last page of the assignment, \emph{without peeking}.
\end{problem}

The reason this is all you need to know has to do with a \emph{special
  amazing fact}\dots

\section{Reference angles}
\label{sec:ref-angles}

\topic{reference angles}
Every angle $\theta$ has a corresponding \term{reference angle}
$\theta_{ref}$, which is the smallest positive angle between
$\theta$'s terminal ray and the $x$-axis.  \pref{fig:reference-angles}
shows some examples.

\begin{figure}[htp]
  \centering
  \includegraphics{diagrams/reference-angles.eps}
  \caption{Reference angles}
  \label{fig:reference-angles}
\end{figure}

\begin{problem}
  Let's figure out how to compute reference angles.  Note, drawing
  pictures can help a lot!
  \begin{subproblems}
    \item Suppose $\theta \in [0,\pi/2]$.  What is $\theta_{ref}$?

    \item Suppose $\theta \in (\pi/2,\pi]$.  How can you determine
      $\theta_{ref}$?  For example, what is the reference angle for
      $\theta = 2\pi/3$?

    \item Now suppose $\theta \in (\pi, 3\pi/2]$.  What is
      $\theta_{ref}$?

    \item What is $\theta_{ref}$ when $\theta \in (3\pi/2, 2\pi)$?

    \item How would you determine the reference angle for an angle
      bigger than $2\pi$?  For example, what is $\theta_{ref}$ for
      $\theta = 98\pi/3$?
  \end{subproblems}
\end{problem}

\begin{problem}
  Find $\theta_{ref}$ for each of the following angles.
  \begin{subproblems}
    \item $9000^\circ$
    \item $11\pi/3$
    \item $-17\pi/9$
    \item $-\pi/20$
  \end{subproblems}
\end{problem}

\topic{special amazing fact!}
And now, for the \emph{special amazing fact}:
\begin{gather*}
  \sin \theta = \pm \sin (\theta_{ref}) \\
  \cos \theta = \pm \cos (\theta_{ref}) 
\end{gather*}

That is, in order to find the sine or cosine of an angle $\theta$, all
you have to do is find the reference angle $\theta_{ref}$, find the
sine or cosine of that, and then decide whether the answer should be
positive or negative!  This is why we only ever need to be able to
compute the sine or cosine of angles between $0$ and $\pi/2$.

\begin{problem}
  In your own words, explain why the \emph{special amazing fact} is
  true.  (It is pretty special, but it actually isn't all that amazing
  if you think about the definitions of sine, cosine, and reference
  angles.)
\end{problem}

\topic{determining signs}
Now, the one remaining important question: how to decide whether the
answer should be positive or negative?  Well, that's not too hard.

\begin{problem}
  Complete \pref{tab:cos-sin-signs}, which shows where cosine and sine
  have positive and negative results. Recall that the quadrants are
  labelled I--IV counterclockwise starting from the upper right.
  (Remember the fundamental fundamentals\dots)

  \begin{table}[htp]
  \centering
  \begin{tabular}{c|c|c}
    In quadrant\dots & $\sin \theta$ is\dots & $\cos \theta$ is\dots \\
    \hline
    I & + & + \\
    II & & \\
    III & & \\
    IV & &
  \end{tabular}
  \caption{Signs of sine and cosine} \label{tab:cos-sin-signs}
  \end{table}
\end{problem}

\begin{problem}
  Compute by finding the reference angle.  Don't use a calculator.
  \begin{subproblems}
    \item $\cos(90000\pi)$
    \item $\sin(77\pi/6)$
    \item $\cos(-983\pi/4)$
    \item $\sin(93\pi/2)$
  \end{subproblems}
\end{problem}

\newpage
\textbf{\pref{prob:memorize}, continued.} Evaluate each of the following,
\emph{without peeking}.  You can peek at last week's assignment to
remind yourself of the definitions of the other trigonometric
functions if you need to.
\begin{subproblems}
  \item $\sin (\pi/3)$
  \item $\cos (\pi/2)$
  \item $\cos (\pi/6)$
  \item $\sin (\pi/6)$
  \item $\cos 0$
  \item $\sin 0$
  \item $\cos (\pi/4)$
  \item $\tan (\pi/2)$
  \item $\sec (\pi/3)$
  \item $\csc (\pi/6)$
\end{subproblems}


\end{document}