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\begin{document}


\title{On-line Math 21}


\email{david.johnson@lehigh.edu}


\urladdr{http://www.lehigh.edu/\textasciitilde{}dlj0}

\maketitle

\subsection{Trigonometric functions}


\subsubsection{Limits of trig functions}

In order to understand the calculus of \( \sin (x) \), or any trigonometric
function, we first need to find some limits involving trigonometric functions.
First, we deal with the limit  \[
\lim _{x\rightarrow 0}\frac{\sin (x)}{x}.\]


\vspace{0.3cm}
{\par\centering \includegraphics{squeeze-triangles.ps} \par}
\vspace{0.3cm}


\subsection*{So What? }

Those three regions are arranged from the smallest to the largest by: \begin{eqnarray*}
Area(\triangle OAB) & \leq Area(Sector(OAB)) & \leq Area(\triangle OBD),\\
\frac{1}{2}\sin (\theta ) & \leq \frac{1}{2}\theta  & \leq \frac{1}{2}\tan (\theta )
\end{eqnarray*}


Multiply through the second line by 2 and divide by \( \sin (\theta ) \), and
you get: \[
1\leq \frac{\theta }{\sin (\theta )}\leq \frac{1}{\cos \theta }.\]


This is kind of upside down, so flip these fractions over. That reverses the
 inequalities: \[
1\geq \frac{\sin (\theta )}{\theta }\geq \cos \theta .\]


Now as \( \theta  \) heads down to 0, both the outside functions go to 1 (one
is there all the time), so the  squeeze principle implies that the center one
does also. 

Don't worry about the fact that this only works for positive values of  \( \theta  \),
if a negative value for \( \theta  \) is put in, it just changes the sign of
both the top and the bottom. So, you can substitute \( x \) for \( \theta  \)
above, and not worry about whether it's positive or negative. The limit then
has to be 1: \[
\lim _{x\rightarrow 0}\frac{\sin (x)}{x}=1.\]


All other trigonometric limits we will need follow from this one. 
\end{document}
