\documentclass[11pt]{article}
\usepackage{precalc}
\usepackage{graphicx}
\begin{document}
\assigntitle{9}{Fundamentals of Trigonometry, Part II}
In the last assignment, you learned general definitions of the sine
and cosine functions. This week, we will explore some of the special
properties of these functions (there are many)!
\topic{do not panic.}
WARNING: this is the point at which many people decide that ``trig
sucks.'' The reason is that they think (or are taught) that the way
to learn trig is to memorize a huge number of equations, formulas, and
procedures. But that is exactly the \emph{wrong} way to learn
trigonometry! What's really cool about trigonometry is how many
interesting equations, formulas, and procedures \emph{can be derived}
from just a few fundamental definitions, and how many interesting
interrelationships there are between all of these. The only things
you need to memorize are those few fundamental definitions, which
gives you a basis from which to understand (and enjoy) all the
ramifications, without stressing about memorizing a gazillion things.
\section{Fundamental fundamentals}
\label{sec:fund-fund}
So, what are those fundamental definitions? There are only two; you
learned them both last week. Here they are, repeated for your
convenience:
\begin{center}
\fbox{\parbox{0.9\textwidth}{
\begin{quote}
There are $2\pi$ radians in a full revolution.
\end{quote}
\hspace{\fill} and \hspace{\fill}
\begin{quote}
Draw $\theta$ as an angle in standard position, on top of a \emph{unit}
(radius $1$) circle. The point where $\theta$'s terminal ray
intersects the unit circle has coordinates $(\cos \theta, \sin \theta)$. (See
\pref{fig:sin-cos-defn}.)
\end{quote}
}}
\end{center}
\begin{figure}[htp]
\centering
\includegraphics{diagrams/sin-cos-defn.eps}
\caption{Definition of sine and cosine}
\label{fig:sin-cos-defn}
\end{figure}
That's it! That's \emph{all you need to know}. If you ever get
confused, you can just come back to these two definitions.
\begin{problem} \label{prob:write-fundamentals}
Write out the two fundamental fundamentals (complete with diagram
for the second) on a separate sheet of paper, \emph{without
peeking}! After writing them out, compare and see how you did. If
you got confused or forgot something, write them out again, repeating
until you can confidently write them down from memory and are
sure you understand them.
I am actually serious about this! For your answer
to~\pref{prob:write-fundamentals}, you should submit the statement
``I fully followed the instructions in this problem, cross my heart
and may my toes be eaten by ravenous tarantulas if I am not telling
the truth.'' Of course, you should only write it as your answer if
it is a true statement. (Because otherwise your toes will be eaten
by ravenous tarantulas.)
\end{problem}
\section{Special angles}
\label{sec:special-angles}
The definition of sine and cosine is very useful for \emph{reasoning}
about these functions, but in general it is not very useful for
actually \emph{computing} the sine or cosine of a particular
angle. (You would have to draw the angle using a protractor---not a
very precise operation---and then measure some distances in your
diagram with a ruler---also not very precise.)
In general, if you want to know the sine or cosine of any old angle,
you should use a calculator.
\footnote{This is a good time to mention that you should always be
very careful about which mode your graphing calculator is
in---degree mode or radian mode. When it is in degree mode, it
expects angles in degrees; when in radian mode, it expects angles in
radians. Obviously, if you type in an angle in radians, but your
calculator thinks it is in degrees (or vice versa), you will get
very strange (and wrong) answers.}
\footnote{Before calculators were invented, people (very tediously)
made huge books full of the sine and cosine of any angle (to five or
six or more decimal places), so if you wanted to know the sine or
cosine of an angle, you could just look up the angle in the
book. Handy!}
\footnote{There are better ways to calculate the sine and cosine of an
angle; in particular, your calculator does not actually calculate sine
and cosine using this definition, by drawing a really accurate
picture and so on. Both sine and cosine have elegant \term{infinite
series expansions} that allow you to calculate the sine or cosine of
an angle with as much accuracy as you want. But to understand these
methods you'll have to wait until calculus!}
However, there are certain special angles for which we know the
\emph{exact} sine and cosine, and it's very important to know them
(they tend to come up a lot---after all, they are special!).
\begin{problem} For each part, give your answer in exact form and
\emph{explain} the reasoning behind your answer. Use the
fundamental fundamentals!
\begin{subproblems}
\item What is $\sin 0$?
\item What is $\sin (\pi/2)$?
\item What is $\cos 0$?
\item What is $\cos (\pi/2)$?
\end{subproblems}
\end{problem}
\begin{problem}
Consider the diagram in~\pref{fig:45-45-90}. It shows an angle of
$\pi/4$ (otherwise known as $45^\circ$) drawn on a unit circle.
\begin{subproblems}
\item What is special about triangle $OPQ$? (\emph{Hint}: the
angles of any triangle add up to $\pi$\dots)
\item What is the length of segment $OP$?
\item What is the length of segment $OQ$? (\emph{Hint}: there's a
Theorem named after some dead Greek guy that may be able to help
you\dots)
\item What is the length of segment $PQ$?
\item What can you conclude about $\sin (\pi/4)$? Give your
answer in exact form, and check it using a calculator.
\item What can you conclude about $\cos (\pi/4)$?
\end{subproblems}
\end{problem}
\begin{figure}[htp]
\centering
\includegraphics{diagrams/45-45-90.eps}
\caption{Determining $\sin (\pi/4)$ and $\cos (\pi/4)$}
\label{fig:45-45-90}
\end{figure}
\begin{problem}
Now take a look at the diagram in~\pref{fig:30-60-90}. It shows an
equilateral triangle $ABC$ with sides of lenth $2$. As you may remember
from geometry, an equilateral triangle is one which has three equal
sides; all of the angles in an equilateral triangle are also the
same (in particular, they are all $\pi/3$ since the angles of any
triangle add up to $\pi$).
Triangle $ABC$ has been cut in half by
altitude $AD$, with length $h$. Therefore segment $DC$ has length
$1$ (half of segment $BC$) and $\angle DAC$ is $\pi/6$ (half of
$\angle BAC$).
\begin{subproblems}
\item Find $h$.
\item What is $\sin (\pi/3)$?
\item What is $\cos (\pi/3)$?
\item What is $\sin (\pi/6)$?
\item What is $\cos (\pi/6)$?
\end{subproblems}
\end{problem}
\begin{figure}[htp]
\centering
\includegraphics[scale=1.2]{diagrams/30-60-90.eps}
\caption{Determining the sine and cosine of $\pi/3$ and $\pi/6$}
\label{fig:30-60-90}
\end{figure}
\begin{problem}
Let's summarize what we've learned so far. Fill in
\pref{tab:special}.
\begin{table}[htp]
\centering
\begin{tabular}{c|cc}
$\theta$ & $\sin \theta$ & $\cos \theta$ \\
\hline
$0$ & & \\
$\pi/6$ & & \\
$\pi/4$ & & \\
$\pi/3$ & & \\
$\pi/2$ & & \\
\end{tabular}
\caption{Sine and cosine of special angles}
\label{tab:special}
\end{table}
\end{problem}
\section{Properties of sine and cosine}
\label{sec:properties}
\begin{problem}
Using your calculator, evaluate the sine of $4\pi/5$, $14\pi/5$,
$24\pi/5$, and $34\pi/5$. What do you notice? Can you explain why
this happens?
\end{problem}
\begin{problem}
Use a graphing calculator to look at graphs of $\sin x$ and $\cos
x$. For best results, select the ``Zoom Trig'' zoom level.
\begin{subproblems}
\item Both graphs appear to have a repeating pattern. Functions
with a repeating pattern like this are called \term{periodic}.
How often do the graphs repeat? Can you explain why?
\item Notice that the graphs of $\sin x$ and $\cos x$ look very
similar. What is the difference between them? (It may help to
graph both functions at the same time.)
\end{subproblems}
\end{problem}
\begin{problem}
Consider \pref{fig:negative}, which shows an angle $\theta$ and its
negative both in standard position. Note that $\theta$ could be
anything; \pref{fig:negative} just shows one particular example.
The terminal rays of $\theta$ and $-\theta$ will always be mirror
images of each other across the $x$-axis, no matter what $\theta$ is.
\begin{figure}[htp]
\centering
\includegraphics{diagrams/negative.eps}
\caption{$\theta$ and $-\theta$}
\label{fig:negative}
\end{figure}
\begin{subproblems}
\item How are the $x$-coordinates of $P$ and $Q$ related? What
can you conclude about $\cos \theta$ and $\cos (-\theta)$?
\item How are the $y$-coordinates of $P$ and $Q$ related? What
can you conclude about $\sin \theta$ and $\sin (-\theta)$?
\end{subproblems}
\end{problem}
\section{Other trigonometric functions}
\label{sec:other-funcs}
There are several other trigonometric functions in common use;
however, they can all be defined in terms of sine and cosine. Tangent
($\tan$) is sine divided by cosine; cosecant ($\csc$) and secant
($\sec$) are the
reciprocals of sine and cosine, respectively; and cotangent ($\cot$)
is the reciprocal of tangent, that is, cosine divided by sine.
\begin{gather*}
\tan \theta = \frac{\sin \theta}{\cos \theta} \\
\csc \theta = \frac{1}{\sin \theta} \\
\sec \theta = \frac{1}{\cos \theta} \\
\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}
\end{gather*}
\begin{problem}
Evaluate. Give your answers in exact form.
\begin{subproblems}
\item $\tan (\pi/4)$
\item $\csc (\pi/2)$
\item $\sec (\pi/6)$
\item $\cot (\pi/3)$
\item Try typing $\tan (\pi/2)$ into your calculator. What
happens? Why?
\end{subproblems}
\end{problem}
\end{document}