\docname{sess7.1}
\doctitle{Sinusoidal Functions}

\mylhead
\myrhead

\title{\doctitle}
\maketitle

\section{Introduction}

\hbcom{This paragraph highly modified; I'm trying to make this sound
  like an interesting part of the course.}
Euler's formula describes how sinusoidal functions go hand in hand
with complex numbers.  In addition, we will often use sinusoidal
functions as input to our systems.  This session lays out the terms
and formulas we will need when working with these functions.

\section{Definitions}

A {\bf sinusoidal function} (or {\bf sinusoidal oscillation} or {\bf
  signal}) is one that can be wrtten in the form
\begin{align}
\label{twentyfour}f(t)&=A\cos(\omega t-\phi).
\end{align}

The function $f(t)$ is a cosine function which has been {\bf
  amplified} by $A$, {\bf shifted} by $\phi$, and {\bf compressed} by $\omega$.
\begin{itemize}

\item $|A|$ is its {\bf amplitude}: how high its graph rises over the $t$-axis at its maximum points;

\item $\phi$ is its {\bf phase lag}: the value of $\omega t$ for which the graph is at its maximum (if $\phi=0$, the graph has the position of $\cos \omega t$; if $\phi=\pi/2$, it has the position of $\sin \omega t$);

\item $\phi/\omega$ is its {\bf time delay} or {\bf time lag}: how far to the right on the $t$-axis the graph of $\cos\omega t$ has been moved to make the graph of (\ref{twentyfour}); (to see this, write $A\cos(\omega t-\phi)=A\cos(\omega(t-\phi/\omega))$)

\item $\omega$ is its {\bf angular frequency}: the number of complete oscillations it makes in a time interval of length $2\pi$; that is, the number of radians per unit time;

\item $\omega/2\pi$ (usually written $\nu$) is its {\bf frequency}: the number of complete oscillations the graph makes in a time interval of length 1; that is, the number of cycles per unit time;

\item $P=2\pi/\omega=1/\nu$ is its {\bf period}, the $t$-interval
  required for one complete oscillation.
\end{itemize}

\section{Discussion}

Here are the instructions for building the graph of (\ref{twentyfour}) from the graph of $\cos t$. First \emph{amplify}, or vertically stretch, the graph by a factor of $A$; then \emph{shift} the result to the right by $\phi$ units; and finally \emph{compress} it horizontally by a factor of $\omega$.

\TODO
\begin{figure}[h]
\centering
%  \includegraphics[width=.5\textwidth]{figname}\\
  \caption{Features of the graph of a sinusoid.}\label{sinusoidfeatures}
\end{figure}

One can also write (\ref{twentyfour}) as 
\[f(t)=A\cos\left(\omega(t-t_0)\right),\]
where $wt_0=\phi$, or
\begin{align}
\label{two} t_0=\frac{\phi}{2\pi}P
\end{align}

$t_0$ is the {\em time lag}. It is measured in the same units as $t$, and represents the amount of time $f(t)$ lags behind the compressed cosine signal $\cos \omega t$. Equation \ref{two} expresses the fact that $t_0$ makes up the same fraction of the period $P$ as the phase lag $\phi$ does of the period of the cosine function.
\hbcom{This is our second definition of time lag.  I think we should
  delete this paragraph.}

\hbcom{Below is my feeble attempt at a concluding sentence.}  We can
completely describe any sinusoidal function by giving the values of
just three parameters.

