methods / circ-functions.texon commit [spec] start vector calculus (42fe40b)
   1\PassOptionsToPackage{unicode=true}{hyperref} % options for packages loaded elsewhere
   2\PassOptionsToPackage{hyphens}{url}
   3%
   4\documentclass[]{article}
   5\usepackage{lmodern}
   6\usepackage{amssymb,amsmath}
   7\usepackage{ifxetex,ifluatex}
   8\usepackage{fixltx2e} % provides \textsubscript
   9\ifnum 0\ifxetex 1\fi\ifluatex 1\fi=0 % if pdftex
  10  \usepackage[T1]{fontenc}
  11  \usepackage[utf8]{inputenc}
  12  \usepackage{textcomp} % provides euro and other symbols
  13\else % if luatex or xelatex
  14  \usepackage{unicode-math}
  15  \defaultfontfeatures{Ligatures=TeX,Scale=MatchLowercase}
  16\fi
  17% use upquote if available, for straight quotes in verbatim environments
  18\IfFileExists{upquote.sty}{\usepackage{upquote}}{}
  19% use microtype if available
  20\IfFileExists{microtype.sty}{%
  21\usepackage[]{microtype}
  22\UseMicrotypeSet[protrusion]{basicmath} % disable protrusion for tt fonts
  23}{}
  24\IfFileExists{parskip.sty}{%
  25\usepackage{parskip}
  26}{% else
  27\setlength{\parindent}{0pt}
  28\setlength{\parskip}{6pt plus 2pt minus 1pt}
  29}
  30\usepackage{hyperref}
  31\hypersetup{
  32            pdfborder={0 0 0},
  33            breaklinks=true}
  34\urlstyle{same}  % don't use monospace font for urls
  35\usepackage[margin=2cm]{geometry}
  36\usepackage{multicol}
  37\newcommand{\columnsbegin}{\begin{multicols}{2}}
  38\newcommand{\columnsend}{\end{multicols}}
  39\setlength\columnsep{20pt}
  40\usepackage{graphicx,grffile}
  41\makeatletter
  42\def\maxwidth{\ifdim\Gin@nat@width>\linewidth\linewidth\else\Gin@nat@width\fi}
  43\def\maxheight{\ifdim\Gin@nat@height>\textheight\textheight\else\Gin@nat@height\fi}
  44\makeatother
  45% Scale images if necessary, so that they will not overflow the page
  46% margins by default, and it is still possible to overwrite the defaults
  47% using explicit options in \includegraphics[width, height, ...]{}
  48\setkeys{Gin}{width=\maxwidth,height=\maxheight,keepaspectratio}
  49\setlength{\emergencystretch}{3em}  % prevent overfull lines
  50\providecommand{\tightlist}{%
  51  \setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}}
  52\setcounter{secnumdepth}{0}
  53% Redefines (sub)paragraphs to behave more like sections
  54\ifx\paragraph\undefined\else
  55\let\oldparagraph\paragraph
  56\renewcommand{\paragraph}[1]{\oldparagraph{#1}\mbox{}}
  57\fi
  58\ifx\subparagraph\undefined\else
  59\let\oldsubparagraph\subparagraph
  60\renewcommand{\subparagraph}[1]{\oldsubparagraph{#1}\mbox{}}
  61\fi
  62
  63% set default figure placement to htbp
  64\makeatletter
  65\def\fps@figure{htbp}
  66\makeatother
  67
  68
  69\date{}
  70
  71\begin{document}
  72
  73\columnsbegin
  74\hypertarget{circular-functions}{%
  75\section{Circular functions}\label{circular-functions}}
  76
  77\hypertarget{radians-and-degrees}{%
  78\subsection{Radians and degrees}\label{radians-and-degrees}}
  79
  80\[1 \thinspace \operatorname{rad}={{180 \operatorname{deg}}\over \pi}\]
  81
  82\hypertarget{exact-values}{%
  83\subsection{Exact values}\label{exact-values}}
  84
  85\includegraphics[scale=0.5]{./graphics/exact-values-1.png}
  86
  87\hypertarget{sin-and-cos-graphs}{%
  88\subsection{\texorpdfstring{\(\sin\) and \(\cos\)
  89graphs}{\textbackslash{}sin and \textbackslash{}cos graphs}}\label{sin-and-cos-graphs}}
  90
  91\[f(x)=a \sin(bx-c)+d\] \[f(x)=a \cos(bx-c)+d\]
  92
  93where
  94
  95\begin{itemize}
  96\tightlist
  97\item
  98  \(a\) is the \(y\)-dilation (amplitude)
  99\item
 100  \(b\) is the \(x\)-dilation (period)
 101\item
 102  \(c\) is the \(x\)-shift (phase)
 103\item
 104  \(d\) is the \(y\)-shift (equilibrium position)
 105\end{itemize}
 106
 107Domain is \(\mathbb{R}\)
 108
 109Range is \([-b+c, b+c]\);
 110
 111Graph of \(\cos(x)\) starts at \((0,1)\). Graph of \(\sin(x)\) starts at
 112\((0,0)\).
 113
 114\textbf{Mean / equilibrium:} line that the graph oscillates around
 115(\(y=d\))
 116
 117\hypertarget{amplitude}{%
 118\subsubsection{Amplitude}\label{amplitude}}
 119
 120Amplitude of \(a\) means graph oscillates between \(+a\) and \(-a\) in
 121\(y\)-axis
 122
 123\(a=0\) produces straight line
 124
 125\(a < 0\) inverts the phase (\(\sin\) becomes \(\cos\), vice vera)
 126
 127\hypertarget{period}{%
 128\subsubsection{Period}\label{period}}
 129
 130Period \(T\) is \({2 \pi}\over b\)
 131
 132\(b=0\) produces straight line
 133
 134\(b<0\) inverts the phase
 135
 136\hypertarget{phase}{%
 137\subsubsection{Phase}\label{phase}}
 138
 139\(c\) moves the graph left-right in the \(x\) axis.
 140
 141If \(c=T={{2\pi}\over b}\), the graph has no actual phase shift.
 142
 143\hypertarget{symmetry}{%
 144\subsection{Symmetry}\label{symmetry}}
 145
 146\[\sin(\theta+{\pi\over 2})=\sin\theta\]
 147\[\sin(\theta+\pi)=-\sin\theta\]
 148
 149\[\cos(\theta+{\pi \over 2})=-\cos\theta\]
 150\[\cos(\theta+\pi)=-cos(\theta+{3\pi \over 2})=\cos(-\theta)\]
 151
 152\hypertarget{pythagorean-identity}{%
 153\subsection{Pythagorean identity}\label{pythagorean-identity}}
 154
 155\[\cos^2\theta+\sin^2\theta=1\]
 156
 157\hypertarget{complementary-relationships}{%
 158\subsection{Complementary
 159relationships}\label{complementary-relationships}}
 160
 161\[\sin({\pi \over 2} - \theta)=\cos\theta\]
 162\[\cos({\pi \over 2} - \theta)=\sin\theta\]
 163
 164\[\sin\theta=-\cos(\theta+{\pi \over 2})\]
 165\[\cos\theta=\sin(\theta+{\pi \over 2})\]
 166
 167\hypertarget{tan-graph}{%
 168\subsection{\texorpdfstring{\(\tan\)
 169graph}{\textbackslash{}tan graph}}\label{tan-graph}}
 170
 171\[y=a\tan(nx)\]
 172
 173where
 174
 175\begin{itemize}
 176\tightlist
 177\item
 178  \(a\) is \(x\)-dilation (period)
 179\item
 180  \(n\) is \(y\)-dilation (\(\equiv\) amplitude)
 181\item
 182  period \(T\) is \(\pi \over n\)
 183\item
 184  range is \(R\)
 185\item
 186  roots at \(x={k\pi \over n}\)
 187\item
 188  asymptotes at \(x={{(2k+1)\pi}\over 2n},\quad k \in \mathbb{Z}\)
 189\end{itemize}
 190
 191\textbf{Asymptotes should always have equations and arrow pointing up}
 192
 193\hypertarget{solving-trig-equations}{%
 194\subsection{Solving trig equations}\label{solving-trig-equations}}
 195
 196\begin{enumerate}
 197\def\labelenumi{\arabic{enumi}.}
 198\tightlist
 199\item
 200  Solve domain for \(n\theta\)
 201\item
 202  Find solutions for \(n\theta\)
 203\item
 204  Divide solutions by \(n\)
 205\end{enumerate}
 206
 207\(\sin2\theta={\sqrt{3}\over2}, \quad \theta \in[0, 2\pi] \quad(\therefore 2\theta \in [0,4\pi])\)
 208
 209\(2\theta=\sin^{-1}{\sqrt{3} \over 2}\)
 210
 211\(2\theta={\pi\over 3}, {2\pi \over 3}, {7\pi \over 3}, {8\pi \over 3}\)
 212
 213\(\therefore \theta = {\pi \over 6}, {\pi \over 3}, {7 \pi \over 6}, {4\pi \over 3}\)
 214\columnsend
 215\end{document}