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%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Thu Feb 13 17:15:07 1997}
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\begin{document}


\section{Complex Numbers 
\index{Complex numbers@Complex numbers}\label{DeMoivre's Theorem}}

For a review of the arithmetic of complex numbers, click 
\hyperref{here}{}{}{DM2-5.tex#Complex numbers}.

\strut

\subsection{\ DeMoivre's Theorem}

Any pair $\left( a,b\right) $ of real numbers can be represented in polar
coordinates with $a=r\cos \theta $ and $b=r\sin \theta $ where $r=\sqrt{%
a^{2}+b^{2}}$ is the distance from the point $\left( a,b\right) $ to the
origin and $\theta $ is an angle satisfying $\tan \theta =\frac{a}{b}$. Thus
any complex number can be written in the form 
\[
z=r\left( \cos \theta +i\sin \theta \right) 
\]
where $r=\left| \,z\right| $.

\strut

\begin{quotation}
\emph{DeMoivre's Theorem} says that if $z=r\left( \cos \theta +i\sin \theta
\right) $ and $n$ is a positive integer, then 
\[
z^{n}=\left( r\left( \cos t+i\sin t\right) \right) ^{n}=r^{n}\left( \cos
nt+i\sin nt\right) 
\]
\end{quotation}

\strut

With \textsl{Scientific Notebook}, you can obtain this result for small 
\index{DeMoivre's Theorem@DeMoivre's Theorem}values of $n$ by the sequence
of operations \textsf{Expand} followed by\textsf{\ \QTR{MenuDialog}{Combine
+ Trig Functions} }and then\textsf{\ Factor}.

\begin{quote}
$\blacktriangleright $ \textsf{Expand, Combine + Trig Functions, Factor}
\end{quote}

\begin{quotation}
$\left( r\left( \cos t+i\sin t\right) \right) ^{3}\allowbreak =\,\,r^{3}\cos
^{3}t+3ir^{3}\cos ^{2}t\sin t-3r^{3}\cos t\sin ^{2}t-ir^{3}\sin
^{3}t\medskip \allowbreak =\,r^{3}\cos 3t+ir^{3}\sin 3t\allowbreak
=\,r^{3}\left( \cos 3t+i\sin 3t\right) $
\end{quotation}

\strut

Or, you can use \textsf{Simplify} followed by \textsf{Combine + Trig
Functions\ }and \textsf{Factor}.

\begin{quote}
$\blacktriangleright $ \textsf{Simplify, Combine + Trig Functions, Factor}
\end{quote}

\begin{quotation}
$\left( r\left( \cos t+i\sin t\right) \right) ^{3}\allowbreak =\,4r^{3}\cos
^{3}t+4ir^{3}\cos ^{2}t\sin t-3r^{3}\cos t-ir^{3}\sin t\allowbreak \medskip
=\,r^{3}\cos 3t+ir^{3}\sin 3t\allowbreak =\,r^{3}\left( \cos 3t+i\sin
3t\right) 
\vspace{6pt}$
\end{quotation}

\strut

You can get the same results in complete generality by working with $re^{it}$%
, since 
\[
\left( re^{it}\right) ^{n}=\,r^{n}e^{itn} 
\]

You can get the identity 
\[
re^{it}=r\left( \cos t+i\sin t\right) 
\]
with \textsf{Evaluate}. (You will find that \textsc{ctrl +e} has no effect
on the expression $re^{it}$. This is one circumstance where \textsf{Evaluate}
and \textsc{ctrl +e} produce a different result.)

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate, Factor}
\end{quote}

\begin{quotation}
$re^{it}=\allowbreak r\cos t+ir\sin t=\allowbreak r\left( \cos t+i\sin
t\right) \vspace{6pt}$
\end{quotation}

\strut

\begin{example}
The following sequence of operations will change the complex number $5+6i$
to trigonometric form.

\strut

\begin{enumerate}
\item[1.]  Take the absolute value $\left| 5+6i\right| =\,\sqrt{61}$ to find
the magnitude $r$.
\end{enumerate}

The trigonometric form of $5+6i$ is then 
\[
5+6i=\,\sqrt{61}\left( \frac{5}{\sqrt{61}}+\frac{6}{\sqrt{61}}i\right) 
\]

\begin{enumerate}
\item[2.]  Apply \textsf{Evaluate Numerically} to $\arccos \left( \frac{5}{%
\sqrt{61}}\right) $ (or $\arcsin \left( \frac{6}{\sqrt{61}}\right) $) to get
the angle$\,.87606$.
\end{enumerate}

Thus, 
\[
5+6i\approx \sqrt{61}\left( \cos .87606+i\sin .87606\right) . 
\]

\begin{enumerate}
\item[3.]  Apply \textsf{Evaluate Numerically} to verify this result.
\end{enumerate}

This produces 
\[
\sqrt{61}\left( \cos .87606+i\sin .87606\right) =5.0+6.0i 
\]
\end{example}

\strut

\subsection{\ Related topics}

\begin{itemize}
\item  \hyperref{Complex numbers}{}{}{DM2-5.tex#Complex numbers}

\item  \hyperref{Exercises and Solutions}{}{}{DM4-3.tex#Exercises}
\end{itemize}

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"XNPEU";}}}{}{}{DMIndex.tex#Math Index}\quad {\small Index entries: 
\hyperref{Complex numbers}{}{}{DMIndex.tex#Complex numbers}, \hyperref{%
DeMoivre's theorem}{}{}{DMIndex.tex#DeMoivre's theorem}}

\end{document}
