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+\definecolor{cas}{HTML}{cde1fd}
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\begin{document}
-\title{\vspace{-23mm}Year 12 Specialist\vspace{-5mm}}
+\title{\vspace{-22mm}Year 12 Specialist\vspace{-4mm}}
\author{Andrew Lorimer}
\date{}
\maketitle
-\vspace{-10mm}
+\vspace{-9mm}
\begin{multicols}{2}
\section{Complex numbers}
\[\mathbb{C}=\{a+bi:a,b\in\mathbb{R}\}\]
-
\begin{align*}
\text{Cartesian form: } & a+bi\\
\text{Polar form: } & r\operatorname{cis}\theta
\[k\left(r \operatorname{cis}\theta\right)=kr \operatorname{cis}\left(\begin{cases}\theta - \pi & |0<\operatorname{Arg}(z)\le \pi \\ \theta + \pi & |-\pi<\operatorname{Arg}(z)\le 0\end{cases}\right)\]
\subsection*{Conjugate}
-
+ \vspace{-7mm} \hfill \colorbox{cas}{\texttt{conjg(a+bi)}}
\begin{align*}
\overline{z} &= a \mp bi\\
&= r \operatorname{cis}(-\theta)
\end{align*}
- \noindent \colorbox{cas}{On CAS: \texttt{conjg(a+bi)}}
\subsubsection*{Properties}
\subsection*{Polar form}
- \begin{align*}
- z&=r\operatorname{cis}\theta\\
- &=r(\cos \theta + i \sin \theta)
- \end{align*}
+ \[ r \operatorname{cis} \theta = r\left( \cos \theta + i \sin \theta \right) \]
\begin{itemize}
\item{\(r=|z|=\sqrt{\operatorname{Re}(z)^2 + \operatorname{Im}(z)^2}\)}
- \item{\(\theta = \operatorname{arg}(z)\) \quad \colorbox{cas}{On CAS: \texttt{arg(a+bi)}}}
+ \item{\(\theta = \operatorname{arg}(z)\) \hfill \colorbox{cas}{\texttt{arg(a+bi)}}}
\item{\(\operatorname{Arg}(z) \in (-\pi,\pi)\) \quad \bf{(principal argument)}}
\item{Multiple representations:\\\(r\operatorname{cis}\theta=r\operatorname{cis}(\theta+2n\pi)\) with \(n \in \mathbb{Z}\) revolutions}
\item{\(\operatorname{cis}\pi=-1,\qquad \operatorname{cis}0=1\)}
\subsection*{de Moivres' theorem}
- \[(r \operatorname{cis} \theta)^n = r^n \operatorname{cis}(n\theta) \text{ where } n \in \mathbb{Z}\]
+ \begin{theorembox}{}
+ \[(r \operatorname{cis} \theta)^n = r^n \operatorname{cis}(n\theta) \text{ where } n \in \mathbb{Z}\]
+ \end{theorembox}
\subsection*{Complex polynomials}
\subsection*{Length of a curve}
- \[L = \int^b_a \sqrt{1 + ({\frac{dy}{dx}})^2} \> dx \quad \text{(Cartesian)}\]
-
- \[L = \int^b_a \sqrt{{\frac{dx}{dt}} + ({\frac{dy}{dt}})^2} \> dt \quad \text{(parametric)}\]
+ For length of \(f(x)\) from \(x=a \rightarrow x=b\):
+ \begin{align*}
+ &\text{Cartesian} \> & L &= \int^b_a \sqrt{1 + \left(\dfrac{dy}{dx}\right)^2} \> dx \\
+ &\text{Parametric} \> & L & = \int^b_a \sqrt{\left(\dfrac{dx}{dt}\right)^2 + \left(\dfrac{dy}{dt}\right)^2} \> dt
+ \end{align*}
\begin{cas}
\begin{enumerate}[label=\alph*), leftmargin=5mm]