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\assigntitle{12}{Trigonometric Identities}
\topic{many fascinating trig relationships}
As I hinted before, there are many fascinating relationships among
trigonometric functions, far too many to cover in even several
assignments! (For example, see
\url{http://en.wikipedia.org/wiki/List_of_trigonometric_identities}.)
This week you'll learn just a few of the most important
relationships---those that will come in handy over and over when
manipulating trigonometric functions.
\section{Symmetric and cofunction identities}
\label{sec:periodic}
\topic{periodic identities}
There are several identities which you have already discovered in
previous assignments, but are worth repeating here so they are all in
one place. These identities are directly related to the definitions
of sine and cosine.
First, you may recall that the cosine of a negative angle is always
the same as the cosine of the corresponding positive angle:
\begin{equation}
\label{eq:cos-even}
\cos (-\theta) = \cos (\theta)
\end{equation}
We call any function $f(x)$ with this property (that $f(-x) = f(x)$)
an \term{even} function, so the above equation could be stated as
``cosine is even.''
You may also recall that the sine of a negative angle is the negative
of the sine of the corresponding positive angle:
\begin{equation}
\label{eq:sin-odd}
\sin (-\theta) = -\sin (\theta)
\end{equation}
We call any function with this property \term{odd}, so the above
equation could also be stated as ``sine is odd.''
\begin{problem}
In a previous assignment, you discovered that the graphs of cosine
and sine are identical, except for the fact that the graph of sine
is shifted $\pi/2$ radians to the right of cosine. Can you express
this fact with an equation relating $\sin$ and $\cos$?
\end{problem}
\section{Pythagorean identities}
\label{sec:pythagorean}
\topic{a Pythagorean identity}
Consider \pref{fig:pythagorean}, which shows an angle of $\theta$ in
standard position on a unit circle. Note, $\theta$ could be
\emph{any} angle; it may look close to $\pi/4$ radians, but you should
ignore that.
\begin{figure}[htp]
\centering
\includegraphics{diagrams/pythagorean.eps}
\caption{An angle $\theta$ in standard position on a unit circle.}
\label{fig:pythagorean}
\end{figure}
\begin{problem}
What is the length of segment $OQ$ in terms of $\theta$?
(\emph{Hint}: remember your fundamental fundamentals!)
\end{problem}
\begin{problem}
What is the length of segment $PQ$ in terms of $\theta$?
\end{problem}
\begin{problem} \label{prob:pythagorean}
I won't insult your intelligence by asking you what the length of
segment $OP$ is. Since triangle $OPQ$ is a right triangle and you
now know the lengths of its three sides, what can you conclude by
the Pythagorean Theorem?
\end{problem}
Your answer to \pref{prob:pythagorean} is known as a \term{Pythagorean
identity} (for hopefully obvious reasons), and is an extremely
fundamental trigonometric identity that comes up all over the place!
\topic{Notes on notation}
You should note that $\sin^2 \theta$ and $\cos^2 \theta$ are
commonly-used abbreviations for $(\sin \theta)^2$ and $(\cos
\theta)^2$, respectively. You will often see the former written down,
but you have to type the latter into your calculator (if you try
typing $\sin^2 x$ into your calculator, it will get mad at you).
\begin{problem}
Are $\sin^2 (x)$ and $\sin (x^2)$ the same? If not, what is the
difference?
\end{problem}
\begin{problem}
When is the sine of an angle the same as its cosine? This problem
will walk you through using the Pythagorean identity to solve this
equation: \[ \sin \theta = \cos \theta. \]
\begin{subproblems}
\item First, square both sides, and write down the equation which
results. Note that this may result in some extra ``solutions''
which are not actually valid solutions, so we will have to check
the solutions we get at the end. \label{prob:pyth-app-1}
\item Rearrange the Pythagorean identity you found in
\pref{prob:pythagorean} to solve for $\sin^2(\theta)$ in terms
of $\cos^2 (\theta)$, and substitute into the equation from part
\ref{prob:pyth-app-1}. Write down the equation which results.
\item Simplify and solve for $\theta$ using arccosine.
\item Which solutions are valid for the original equation?
\end{subproblems}
\end{problem}
\topic{Other Pythagorean identities}
We can also use the basic Pythagorean identity you found in
\pref{prob:pythagorean} to derive a few other related identities.
\begin{problem}
Starting from the Pythagorean identity that you found in
\pref{prob:pythagorean}, divide both sides by $\sin^2(\theta)$. Can
you simplify the resulting equation? (\emph{Hint}: think about the
other trigonometric functions you learned about---$\tan$, $\csc$,
$\sec$, and $\cot$.)
\end{problem}
\begin{problem}
Now divide both sides by $\cos^2(\theta)$ instead. Can you simplify
the resulting equation?
\end{problem}
All three of these equations are known as Pythagorean identities,
since they arise from the Pythagorean theorem, and are in some sense
all equivalent. The first one involving sine and cosine is by far the
most useful, but it's good to be aware of the other two as well.
\section{Sum and difference identities}
\label{sec:addition}
Suppose we have two angles, $\theta$ and $\phi$. What can we say
about the sine or cosine of their sum or difference, $\theta \pm
\phi$?
\topic{Deriving cosine sum identity}
Consider the diagram in \pref{fig:sum-difference-proof}. $\theta$ and
$\phi$ could be any arbitrary angles.
\begin{figure}[htp]
\centering
\includegraphics{diagrams/sum-difference-proof.eps}
\caption{Proving an angle sum identity for cosine.}
\label{fig:sum-difference-proof}
\end{figure}
\begin{problem}
First, let's figure out coordinates of points in the diagram.
\begin{subproblems}
\item What are the coordinates of point $P$?
\item What are the coordinates of point $Q$ in terms of $\theta$?
\item What are the coordinates of point $S$ in terms of $\phi$?
(Keep in mind that the angle from $P$ to $S$ is a
\emph{negative} angle.)
\item What are the coordinates of point $R$ in terms of $\theta$
and $\phi$? (\emph{Hint}: the angle from $P$ to $R$ is $\theta
+ \phi$.)
\end{subproblems}
\end{problem}
\begin{problem} \label{prob:sum-ident}
\begin{subproblems}
\item Use the distance formula to write down an expression for the
length of segment $RP$ (shown by a dotted line). \label{pt:rp}
\item Use the distance formula to write down an expression for
the length of segment $QS$. \label{pt:qs}
\item Segments $RP$ and $QS$ are the same length, since the angle
they subtend is the same. Set the expressions from parts
\ref{pt:rp} and \ref{pt:qs} equal, solve for $\cos(\theta +
\phi)$, and use the symmetric and Pythagorean identities to help
you simplify!
\end{subproblems}
\end{problem}
In \pref{prob:sum-ident}, you should have come up with the fact that
\begin{equation}
\label{eq:cos-sum}
\cos(\theta + \phi) = \cos(\theta) \cos(\phi) - \sin(\theta) \sin(\phi).
\end{equation}
\topic{Other sum and difference identities}
From this, using the symmetric and cofunction identities, it's not
hard to derive other laws for addition and subtraction of angles:
\begin{gather}
\cos(\theta - \phi) = \cos(\theta) \cos(\phi) + \sin(\theta)
\sin(\phi) \label{eq:cos-diff} \\
\sin(\theta + \phi) = \sin(\theta) \cos(\phi) + \cos(\theta)
\sin(\phi) \\
\sin(\theta - \phi) = \sin(\theta) \cos(\phi) - \cos(\theta)
\sin(\phi)
\end{gather}
although I won't make you do so. (It's not hard, though: for example,
to derive \eqref{eq:cos-diff}, substitute $\theta$ and $-\phi$ into
\eqref{eq:cos-sum}, and simplify using the symmetric identities.)
\begin{problem}
What is the exact value of $\sin(5\pi/12)$? (\emph{Hint}: $5/12 =
1/4 + 1/6$.)
\end{problem}
\section{Other identities}
\label{sec:other}
\topic{Other identities}
Other identities can also be derived, such as half- and double-angle
identities, power-reduction identities, and product-to-sum or
sum-to-product identities. See
\url{http://staff.jccc.net/swilson/trig/anglesumidentities.htm} for
more information.
\begin{problem}
Visit the above website, and pick one of the identities that we
haven't discussed on this assignment. Write it down and explain how
it can be derived.
\end{problem}
\end{document}