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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{Derivatives of \protect\( \sin (x)\protect \) and \protect\( \cos (x)\protect \).}

\begin{example}
Find the tangent line to the curve \( y=\sin (x) \) at \( (\pi /6,1/2) \),
and use that to find, approximately, \( \sin (\pi /6+0.1) \). 
\end{example}

\subsubsection*{Solution }

You may want to glance back at the section on linear approximation first.

\( f(x)=\sin (x) \), so \( f'(x)=\cos (x) \),  and \( f'(\pi /6)=\sqrt{3}/2=0.86602540378 \),
so the tangent line is  \( y=f(a)+f'(a)(x-a)=1/2+\sqrt{3}/2(x-\pi /6) \). The
approximate value of  \( \sin (\pi +0.1) \) is then: \begin{eqnarray*}
f(x) & \approx  & f(a)+f'(a)(x-a)\\
\sin (x) & \approx  & \frac{1}{2}+\sqrt{3}/2(x-\pi /6)\\
\sin (\pi /6+0.1) & \approx  & \frac{1}{2}+\sqrt{3}/2(0.1)\\
\sin (\pi /6+0.1) & \approx  & \frac{1}{2}+0.86602540378(0.1)=0.5866025403784438
\end{eqnarray*}
On the other hand, the machine finds its own (similarly determined) value of
\( \sin (\pi /6+0.1) \) as \( \sin (\pi /6+0.1)\approx 0.5839603578 \). Which
is closer? 
\end{document}
