-\PassOptionsToPackage{unicode=true}{hyperref} % options for packages loaded elsewhere
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-%
-\documentclass[]{article}
-\usepackage{lmodern}
-\usepackage{amssymb,amsmath}
-\usepackage{ifxetex,ifluatex}
-\usepackage{fixltx2e} % provides \textsubscript
-\ifnum 0\ifxetex 1\fi\ifluatex 1\fi=0 % if pdftex
- \usepackage[T1]{fontenc}
- \usepackage[utf8]{inputenc}
- \usepackage{textcomp} % provides euro and other symbols
-\else % if luatex or xelatex
- \usepackage{unicode-math}
- \defaultfontfeatures{Ligatures=TeX,Scale=MatchLowercase}
-\fi
-% use upquote if available, for straight quotes in verbatim environments
-\IfFileExists{upquote.sty}{\usepackage{upquote}}{}
-% use microtype if available
-\IfFileExists{microtype.sty}{%
-\usepackage[]{microtype}
-\UseMicrotypeSet[protrusion]{basicmath} % disable protrusion for tt fonts
-}{}
-\IfFileExists{parskip.sty}{%
-\usepackage{parskip}
-}{% else
-\setlength{\parindent}{0pt}
-\setlength{\parskip}{6pt plus 2pt minus 1pt}
-}
-\usepackage{hyperref}
-\hypersetup{
- pdfborder={0 0 0},
- breaklinks=true}
-\urlstyle{same} % don't use monospace font for urls
-\usepackage[margin=2cm]{geometry}
-\usepackage{multicol}
-\newcommand{\columnsbegin}{\begin{multicols}{2}}
-\newcommand{\columnsend}{\end{multicols}}
-\setlength\columnsep{20pt}
-\usepackage{graphicx,grffile}
-\makeatletter
-\def\maxwidth{\ifdim\Gin@nat@width>\linewidth\linewidth\else\Gin@nat@width\fi}
-\def\maxheight{\ifdim\Gin@nat@height>\textheight\textheight\else\Gin@nat@height\fi}
-\makeatother
-% Scale images if necessary, so that they will not overflow the page
-% margins by default, and it is still possible to overwrite the defaults
-% using explicit options in \includegraphics[width, height, ...]{}
-\setkeys{Gin}{width=\maxwidth,height=\maxheight,keepaspectratio}
-\setlength{\emergencystretch}{3em} % prevent overfull lines
-\providecommand{\tightlist}{%
- \setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}}
-\setcounter{secnumdepth}{0}
-% Redefines (sub)paragraphs to behave more like sections
-\ifx\paragraph\undefined\else
-\let\oldparagraph\paragraph
-\renewcommand{\paragraph}[1]{\oldparagraph{#1}\mbox{}}
-\fi
-\ifx\subparagraph\undefined\else
-\let\oldsubparagraph\subparagraph
-\renewcommand{\subparagraph}[1]{\oldsubparagraph{#1}\mbox{}}
-\fi
-
-% set default figure placement to htbp
-\makeatletter
-\def\fps@figure{htbp}
-\makeatother
-
-
-\date{}
-
-\begin{document}
-
-\columnsbegin
-\hypertarget{circular-functions}{%
-\section{Circular functions}\label{circular-functions}}
-
-\hypertarget{radians-and-degrees}{%
-\subsection{Radians and degrees}\label{radians-and-degrees}}
+\section{Circular functions}
-\[1 \thinspace \operatorname{rad}={{180 \operatorname{deg}}\over \pi}\]
-
-\hypertarget{exact-values}{%
-\subsection{Exact values}\label{exact-values}}
+\subsection*{Radians and degrees}
-\includegraphics[scale=0.5]{./graphics/exact-values-1.png}
+\[1 \thinspace \operatorname{rad}={{180 \operatorname{deg}}\over \pi}\]
-\hypertarget{sin-and-cos-graphs}{%
-\subsection{\texorpdfstring{\(\sin\) and \(\cos\)
-graphs}{\textbackslash{}sin and \textbackslash{}cos graphs}}\label{sin-and-cos-graphs}}
+\subsection*{Exact values}
-\[f(x)=a \sin(bx-c)+d\] \[f(x)=a \cos(bx-c)+d\]
-where
+ \begin{tikzpicture}[scale=0.75]
+ \draw [orange, thick] (0,0) -- (3,3) node [black, pos=0.5, above left] {\(\sqrt{2}\)};
+ \draw [orange, thick] (0,0) -- (3,0) node [black, below, pos=0.5] {\(1\)} node[black, above, pos=0.3] {\(\frac{\pi}{4}\)};
+ \draw [orange, thick] (3,0) -- (3,3) node [black, right, pos=0.5] {1} node[black, left, pos=0.7] {\(\frac{\pi}{4}\)};
+ \draw [black] (0,0) coordinate (A) (3,0) coordinate (B) (3,3) coordinate (C) pic [draw,black,angle radius=2mm] {right angle = A--B--C};
+ \end{tikzpicture}
+ \begin{tikzpicture}[scale=0.75]
+ \draw [orange, thick] (0,3) -- (5.19,0) node [black, pos=0.5, above right] {2};
+ \draw [orange, thick] (0,0) -- (5.19,0) node [black, below, pos=0.5] {\(\sqrt{3}\)} node[black, above, pos=0.7] {\(\frac{\pi}{6}\)};
+ \draw [orange, thick] (0,0) -- (0,3) node [black, left, pos=0.5] {1} node [black, pos=0.8, right] {\(\frac{\pi}{3}\)};
+ \draw [black] (5.19,0) coordinate (A) (0,0) coordinate (B) (0,3) coordinate (C) pic [draw,black,angle radius=2mm] {right angle = A--B--C};
+ \end{tikzpicture}
-\begin{itemize}
-\tightlist
-\item
- \(a\) is the \(y\)-dilation (amplitude)
-\item
- \(b\) is the \(x\)-dilation (period)
-\item
- \(c\) is the \(x\)-shift (phase)
-\item
- \(d\) is the \(y\)-shift (equilibrium position)
-\end{itemize}
+ \subsection*{Compound angle formulas}
-Domain is \(\mathbb{R}\)
-Range is \([-b+c, b+c]\);
+ \begin{align*}
+ \cos(x \pm y) &= \cos x + \cos y \mp \sin x \sin y \\
+ \sin(x \pm y) &= \sin x \cos y \pm \cos x \sin y \\
+ \tan(x \pm y) &= {{\tan x \pm \tan y} \over {1 \mp \tan x \tan y}}
+ \end{align*}
-Graph of \(\cos(x)\) starts at \((0,1)\). Graph of \(\sin(x)\) starts at
-\((0,0)\).
+ \subsection*{Double angle formulas}
-\textbf{Mean / equilibrium:} line that the graph oscillates around
-(\(y=d\))
+ \begin{align*}
+ \cos 2x &= \cos^2 x - \sin^2 x \\
+ & = 1 - 2\sin^2 x \\
+ & = 2 \cos^2 x -1 \\
+ \sin 2x &= 2 \sin x \cos x \\
+ \tan 2x &= \dfrac{2 \tan x}{1 - \tan^2 x}
+ \end{align*}
-\hypertarget{amplitude}{%
-\subsubsection{Amplitude}\label{amplitude}}
-Amplitude of \(a\) means graph oscillates between \(+a\) and \(-a\) in
-\(y\)-axis
-\(a=0\) produces straight line
+\subsection*{Symmetry}
-\(a < 0\) inverts the phase (\(\sin\) becomes \(\cos\), vice vera)
+\begin{align*}
+ \sin(\theta+\frac{\pi}{2}) &= \sin\theta \\
+ \sin(\theta+\pi) &= -\sin\theta \\ \\
+ \cos(\theta+\frac{\pi}{2}) &= -\cos\theta \\
+ \cos(\theta+\pi) &= -\cos(\theta+\frac{3\pi}{2}) \\
+ &= \cos(-\theta)
+\end{align*}
-\hypertarget{period}{%
-\subsubsection{Period}\label{period}}
+\subsection*{Complementary relationships}
-Period \(T\) is \({2 \pi}\over b\)
+\begin{align*}
+ \sin \theta &= \cos(\frac{\pi}{2} - \theta) \\
+ &= -\cos(\theta+\frac{\pi}{2}) \\
+ \cos\theta &= \sin(\frac{\pi}{2} - \theta) \\
+ &= \sin(\theta+\frac{\pi}{2})
+\end{align*}
-\(b=0\) produces straight line
+\subsection*{Pythagorean identity}
-\(b<0\) inverts the phase
+\[\cos^2\theta+\sin^2\theta=1\]
-\hypertarget{phase}{%
-\subsubsection{Phase}\label{phase}}
+ \subsection*{Inverse circular functions}
-\(c\) moves the graph left-right in the \(x\) axis.
+ \begin{tikzpicture}
+ \begin{axis}[ymin=-2, ymax=4, xmin=-1.1, xmax=1.1, ytick={-1.5708, 1.5708, 3.14159},yticklabels={$-\frac{\pi}{2}$, $\frac{\pi}{2}$, $\pi$}]
+ \addplot[color=red, smooth] gnuplot [domain=-2:2,unbounded coords=jump,samples=500] {asin(x)} node [pos=0.25, below right] {\(\sin^{-1}x\)};
+ \addplot[color=blue, smooth] gnuplot [domain=-2:2,unbounded coords=jump,samples=500] {acos(x)} node [pos=0.25, below left] {\(\cos^{-1}x\)};
+ \addplot[mark=*, red] coordinates {(-1,-1.5708)} node[right, font=\footnotesize]{\((-1,-\frac{\pi}{2})\)} ;
+ \addplot[mark=*, red] coordinates {(1,1.5708)} node[left, font=\footnotesize]{\((1,\frac{\pi}{2})\)} ;
+ \addplot[mark=*, blue] coordinates {(1,0)};
+ \addplot[mark=*, blue] coordinates {(-1,3.1415)} node[right, font=\footnotesize]{\((-1,\pi)\)} ;
+ \end{axis}
+ \end{tikzpicture}\\
-If \(c=T={{2\pi}\over b}\), the graph has no actual phase shift.
+ Inverse functions: \(f(f^{-1}(x)) = x\) (restrict domain)
-\hypertarget{symmetry}{%
-\subsection{Symmetry}\label{symmetry}}
+ \[\sin^{-1}: [-1, 1] \rightarrow \mathbb{R}, \quad \sin^{-1} x = y\]
+ \hfill where \(\sin y = x, \> y \in [{-\pi \over 2}, {\pi \over 2}]\)
-\[\sin(\theta+{\pi\over 2})=\sin\theta\]
-\[\sin(\theta+\pi)=-\sin\theta\]
+ \[\cos^{-1}: [-1,1] \rightarrow \mathbb{R}, \quad \cos^{-1} x = y\]
+ \hfill where \(\cos y = x, \> y \in [0, \pi]\)
-\[\cos(\theta+{\pi \over 2})=-\cos\theta\]
-\[\cos(\theta+\pi)=-cos(\theta+{3\pi \over 2})=\cos(-\theta)\]
+ \[\tan^{-1}: \mathbb{R} \rightarrow \mathbb{R}, \quad \tan^{-1} x = y\]
+ \hfill where \(\tan y = x, \> y \in \left(-{\pi \over 2}, {\pi \over 2}\right)\)
-\hypertarget{pythagorean-identity}{%
-\subsection{Pythagorean identity}\label{pythagorean-identity}}
+ \begin{tikzpicture}
+ \begin{axis}[yticklabel style={yshift=1.0pt, anchor=north east},x=0.1cm, y=1cm, ymax=2, ymin=-2, xticklabels={}, ytick={-1.5708,1.5708},yticklabels={\(-\frac{\pi}{2}\),\(\frac{\pi}{2}\)}]
+ \addplot[color=orange, smooth] gnuplot [domain=-35:35, unbounded coords=jump,samples=350] {atan(x)} node [pos=0.5, above left] {\(\tan^{-1}x\)};
+ \addplot[->, gray, dotted, thick, domain=-35:35] {1.5708};
+ \addplot[->, gray, dotted, thick, domain=-35:35] {-1.5708};
+ \end{axis}
+ \end{tikzpicture}
-\[\cos^2\theta+\sin^2\theta=1\]
+\subsection*{\(\sin\) and \(\cos\) graphs}
-\hypertarget{complementary-relationships}{%
-\subsection{Complementary
-relationships}\label{complementary-relationships}}
+\[ f(x)=a\sin(bx-c)+d \]
-\[\sin({\pi \over 2} - \theta)=\cos\theta\]
-\[\cos({\pi \over 2} - \theta)=\sin\theta\]
+where:
+\begin{description}
+ \item Period \(=\frac{2\pi}{n}\)
+ \item dom \(= \mathbb{R}\)
+ \item ran \(= [-b+c, b+c]\);
+ \item \(\cos(x)\) starts at \((0,1)\), \(\sin(x)\) starts at \((0,0)\)
+ \item 0 amplitidue \(\implies\) straight line
+ \item \(a<0\) or \(b<0\) inverts phase (swap \(\sin\) and \(\cos\))
+ \item \(c=T={{2\pi}\over b} \implies\) no net phase shift
+\end{description}
-\[\sin\theta=-\cos(\theta+{\pi \over 2})\]
-\[\cos\theta=\sin(\theta+{\pi \over 2})\]
-
-\hypertarget{tan-graph}{%
-\subsection{\texorpdfstring{\(\tan\)
-graph}{\textbackslash{}tan graph}}\label{tan-graph}}
+\subsection*{\(\tan\) graphs}
\[y=a\tan(nx)\]
-where
-
-\begin{itemize}
-\tightlist
-\item
- \(a\) is \(x\)-dilation (period)
-\item
- \(n\) is \(y\)-dilation (\(\equiv\) amplitude)
-\item
- period \(T\) is \(\pi \over n\)
-\item
- range is \(R\)
-\item
- roots at \(x={k\pi \over n}\)
-\item
- asymptotes at \(x={{(2k+1)\pi}\over 2n},\quad k \in \mathbb{Z}\)
-\end{itemize}
+\begin{description}
+ \item Period \(= \dfrac{\pi}{n}\)
+ \item Range is \(\mathbb{R}\)
+ \item Roots at \(x={\dfrac{k\pi}{n}}\) where \(k \in \mathbb{Z}\)
+ \item Asymptotes at \(x=\frac{(2k+1)\pi}{2n}\)
+\end{description}
\textbf{Asymptotes should always have equations and arrow pointing up}
-\hypertarget{solving-trig-equations}{%
-\subsection{Solving trig equations}\label{solving-trig-equations}}
+\subsection*{Solving trig equations}
\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\(2\theta={\pi\over 3}, {2\pi \over 3}, {7\pi \over 3}, {8\pi \over 3}\)
\(\therefore \theta = {\pi \over 6}, {\pi \over 3}, {7 \pi \over 6}, {4\pi \over 3}\)
-\columnsend
-\end{document}