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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 @@ -147,12 +146,11 @@ \[k\left(r \operatorname{cis}\theta\right)=kr \operatorname{cis}\left(\begin{cases}\theta - \pi & |0<\operatorname{Arg}(z)\le \pi \\ \theta + \pi & |-\pi<\operatorname{Arg}(z)\le 0\end{cases}\right)\] \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} @@ -196,14 +194,11 @@ \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{\(\operatorname{cis}\pi=-1,\qquad \operatorname{cis}0=1\)} @@ -215,7 +210,9 @@ \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} @@ -1128,9 +1125,11 @@ \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] -- 2.47.1