[chem] start organic reactions summary document
[notes.git] / methods / circ-functions.tex
index 7870df145d038410cbca68c05664c4ab48de8347..2b615cd7f8f3011cd58e04d22b8317841b4d93e9 100644 (file)
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-\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
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-\renewcommand{\subparagraph}[1]{\oldsubparagraph{#1}\mbox{}}
-\fi
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-\makeatletter
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-\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}}
-
-\[1 \thinspace \operatorname{rad}={{180 \operatorname{deg}}\over \pi}\]
-
-\hypertarget{exact-values}{%
-\subsection{Exact values}\label{exact-values}}
-
-\includegraphics[scale=0.5]{./graphics/exact-values-1.png}
-
-\hypertarget{sin-and-cos-graphs}{%
-\subsection{\texorpdfstring{\(\sin\) and \(\cos\)
-graphs}{\textbackslash{}sin and \textbackslash{}cos graphs}}\label{sin-and-cos-graphs}}
+\section{Circular functions}
 
-\[f(x)=a \sin(bx-c)+d\] \[f(x)=a \cos(bx-c)+d\]
+\subsection*{Radians and degrees}
 
-where
+\[1 \thinspace \operatorname{rad}={{180 \operatorname{deg}}\over \pi}\]
 
-\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*{Exact values}
+\adjustbox{trim=0.7cm 0cm}{
+  \begin{tikzpicture}
+    \matrix{
+      \begin{scope}[scale=0.8]
+        \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] {\(\dfrac{\pi}{4}\)};
+        \draw [orange, thick] (3,0) -- (3,3) node [black, right, pos=0.5] {1} node[black, left, pos=0.7] {\(\dfrac{\pi}{4}\)};
+        \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{scope}
+      &
+      \begin{scope}[scale=0.8]
+        \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] {\(\dfrac{\pi}{6}\)};
+        \draw [orange, thick] (0,0) -- (0,3) node [black, left, pos=0.5] {1} node [black, pos=0.8, right] {\(\dfrac{\pi}{3}\)};
+        \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{scope}
+      \\
+    };
+  \end{tikzpicture}
+}
 
-Domain is \(\mathbb{R}\)
+                  \subsection*{Compound angle formulas}
 
-Range is \([-b+c, b+c]\);
 
-Graph of \(\cos(x)\) starts at \((0,1)\). Graph of \(\sin(x)\) starts at
-\((0,0)\).
+                  \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*}
 
-\textbf{Mean / equilibrium:} line that the graph oscillates around
-(\(y=d\))
+                  \subsection*{Double angle formulas}
 
-\hypertarget{amplitude}{%
-\subsubsection{Amplitude}\label{amplitude}}
+                  \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*}
 
-Amplitude of \(a\) means graph oscillates between \(+a\) and \(-a\) in
-\(y\)-axis
 
-\(a=0\) produces straight line
 
-\(a < 0\) inverts the phase (\(\sin\) becomes \(\cos\), vice vera)
+\subsection*{Symmetry}
 
-\hypertarget{period}{%
-\subsubsection{Period}\label{period}}
+\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*}
 
-Period \(T\) is \({2 \pi}\over b\)
+\subsection*{Complementary relationships}
 
-\(b=0\) produces straight line
+\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*}
 
-\(b<0\) inverts the phase
+\subsection*{Pythagorean identity}
 
-\hypertarget{phase}{%
-\subsubsection{Phase}\label{phase}}
+\[\cos^2\theta+\sin^2\theta=1\]
 
-\(c\) moves the graph left-right in the \(x\) axis.
+                  \subsection*{Inverse circular functions}
 
-If \(c=T={{2\pi}\over b}\), the graph has no actual phase shift.
+                  \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}\\
 
-\hypertarget{symmetry}{%
-\subsection{Symmetry}\label{symmetry}}
+                  Inverse functions: \(f(f^{-1}(x)) = x\) (restrict domain)
 
-\[\sin(\theta+{\pi\over 2})=\sin\theta\]
-\[\sin(\theta+\pi)=-\sin\theta\]
+                  \[\sin^{-1}: [-1, 1] \rightarrow \mathbb{R}, \quad \sin^{-1} x = y\]
+                  \hfill where \(\sin y = x, \> y \in [{-\pi \over 2}, {\pi \over 2}]\)
 
-\[\cos(\theta+{\pi \over 2})=-\cos\theta\]
-\[\cos(\theta+\pi)=-cos(\theta+{3\pi \over 2})=\cos(-\theta)\]
+                  \[\cos^{-1}: [-1,1] \rightarrow \mathbb{R}, \quad \cos^{-1} x = y\]
+                  \hfill where \(\cos y = x, \> y \in [0, \pi]\)
 
-\hypertarget{pythagorean-identity}{%
-\subsection{Pythagorean identity}\label{pythagorean-identity}}
+                  \[\tan^{-1}: \mathbb{R} \rightarrow \mathbb{R}, \quad \tan^{-1} x = y\]
+                  \hfill where \(\tan y = x, \> y \in \left(-{\pi \over 2}, {\pi \over 2}\right)\)
 
-\[\cos^2\theta+\sin^2\theta=1\]
+                  \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} node [black, font=\footnotesize, below right, pos=0] {\(y=\frac{\pi}{2}\)};
+                      \addplot[gray, dotted, thick, domain=-35:35] {-1.5708} node [black, font=\footnotesize, above left, pos=1] {\(y=-\frac{\pi}{2}\)};
+                    \end{axis}
+                  \end{tikzpicture}
 
-\hypertarget{complementary-relationships}{%
-\subsection{Complementary
-relationships}\label{complementary-relationships}}
+\subsection*{\(\sin\) and \(\cos\) graphs}
 
-\[\sin({\pi \over 2} - \theta)=\cos\theta\]
-\[\cos({\pi \over 2} - \theta)=\sin\theta\]
+\[ f(x)=a\sin(bx-c)+d \]
 
-\[\sin\theta=-\cos(\theta+{\pi \over 2})\]
-\[\cos\theta=\sin(\theta+{\pi \over 2})\]
+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}
 
-\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}
+\textbf{Asymptotes should always have equations}
 
-\hypertarget{solving-trig-equations}{%
-\subsection{Solving trig equations}\label{solving-trig-equations}}
+\subsection*{Solving trig equations}
 
 \begin{enumerate}
 \def\labelenumi{\arabic{enumi}.}
@@ -211,5 +150,3 @@ where
 \(2\theta={\pi\over 3}, {2\pi \over 3}, {7\pi \over 3}, {8\pi \over 3}\)
 
 \(\therefore \theta = {\pi \over 6}, {\pi \over 3}, {7 \pi \over 6}, {4\pi \over 3}\)
-\columnsend
-\end{document}