From: Andrew Lorimer <andrew@lorimer.id.au>
Date: Mon, 30 Sep 2019 03:10:21 +0000 (+1000)
Subject: [spec] formatting
X-Git-Tag: yr12~20
X-Git-Url: https://git.lorimer.id.au/notes.git/diff_plain/9de32b2705dd0129427870e7f869f694bd79013a

[spec] formatting
---

diff --git a/spec/spec-collated.pdf b/spec/spec-collated.pdf
index bb64725..21130bc 100644
Binary files a/spec/spec-collated.pdf and b/spec/spec-collated.pdf differ
diff --git a/spec/spec-collated.tex b/spec/spec-collated.tex
index 0b1d56e..785a7fd 100644
--- a/spec/spec-collated.tex
+++ b/spec/spec-collated.tex
@@ -85,7 +85,7 @@
 \newcolumntype{R}[1]{>{\hsize=#1\hsize\raggedleft\arraybackslash}X}%
 \newcolumntype{Y}{>{\centering\arraybackslash}X}
 
-\definecolor{cas}{HTML}{e6f0fe}
+\definecolor{cas}{HTML}{cde1fd}
 \definecolor{important}{HTML}{fc9871}
 \definecolor{dark-gray}{gray}{0.2}
 \definecolor{light-gray}{HTML}{cccccc}
@@ -103,17 +103,16 @@
 
 \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
@@ -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]