methods / circ-functions.texon commit [chem] start organic reactions summary document (5447e6c)
   1\section{Circular functions}
   2
   3\subsection*{Radians and degrees}
   4
   5\[1 \thinspace \operatorname{rad}={{180 \operatorname{deg}}\over \pi}\]
   6
   7\subsection*{Exact values}
   8\adjustbox{trim=0.7cm 0cm}{
   9  \begin{tikzpicture}
  10    \matrix{
  11      \begin{scope}[scale=0.8]
  12        \draw [orange, thick] (0,0) -- (3,3) node [black, pos=0.5, above left] {\(\sqrt{2}\)};
  13        \draw [orange, thick] (0,0) -- (3,0) node [black, below, pos=0.5] {\(1\)} node[black, above, pos=0.3] {\(\dfrac{\pi}{4}\)};
  14        \draw [orange, thick] (3,0) -- (3,3) node [black, right, pos=0.5] {1} node[black, left, pos=0.7] {\(\dfrac{\pi}{4}\)};
  15        \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};
  16      \end{scope}
  17      &
  18      \begin{scope}[scale=0.8]
  19        \draw [orange, thick] (0,3) -- (5.19,0) node [black, pos=0.5, above right] {2};
  20        \draw [orange, thick] (0,0) -- (5.19,0) node [black, below, pos=0.5] {\(\sqrt{3}\)} node[black, above, pos=0.7] {\(\dfrac{\pi}{6}\)};
  21        \draw [orange, thick] (0,0) -- (0,3) node [black, left, pos=0.5] {1} node [black, pos=0.8, right] {\(\dfrac{\pi}{3}\)};
  22        \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};
  23      \end{scope}
  24      \\
  25    };
  26  \end{tikzpicture}
  27}
  28
  29                  \subsection*{Compound angle formulas}
  30
  31
  32                  \begin{align*}
  33                    \cos(x \pm y) &= \cos x + \cos y \mp \sin x \sin y \\
  34                    \sin(x \pm y) &= \sin x \cos y \pm \cos x \sin y \\
  35                    \tan(x \pm y) &= {{\tan x \pm \tan y} \over {1 \mp \tan x \tan y}}
  36                  \end{align*}
  37
  38                  \subsection*{Double angle formulas}
  39
  40                  \begin{align*}
  41                    \cos 2x &= \cos^2 x - \sin^2 x \\
  42                    & = 1 - 2\sin^2 x \\
  43                    & = 2 \cos^2 x -1 \\ 
  44                    \sin 2x &= 2 \sin x \cos x \\
  45                    \tan 2x &= \dfrac{2 \tan x}{1 - \tan^2 x}
  46                  \end{align*}
  47
  48
  49
  50\subsection*{Symmetry}
  51
  52\begin{align*}
  53  \sin(\theta+\frac{\pi}{2}) &= \sin\theta \\
  54  \sin(\theta+\pi) &= -\sin\theta \\ \\
  55  \cos(\theta+\frac{\pi}{2}) &= -\cos\theta \\
  56  \cos(\theta+\pi) &= -\cos(\theta+\frac{3\pi}{2}) \\
  57  &= \cos(-\theta)
  58\end{align*}
  59
  60\subsection*{Complementary relationships}
  61
  62\begin{align*}
  63  \sin \theta &= \cos(\frac{\pi}{2} - \theta) \\
  64  &= -\cos(\theta+\frac{\pi}{2}) \\
  65  \cos\theta &= \sin(\frac{\pi}{2} - \theta) \\ 
  66  &= \sin(\theta+\frac{\pi}{2})
  67\end{align*}
  68
  69\subsection*{Pythagorean identity}
  70
  71\[\cos^2\theta+\sin^2\theta=1\]
  72
  73                  \subsection*{Inverse circular functions}
  74
  75                  \begin{tikzpicture}
  76                    \begin{axis}[ymin=-2, ymax=4, xmin=-1.1, xmax=1.1, ytick={-1.5708, 1.5708, 3.14159},yticklabels={$-\frac{\pi}{2}$, $\frac{\pi}{2}$, $\pi$}]
  77                      \addplot[color=red, smooth] gnuplot [domain=-2:2,unbounded coords=jump,samples=500] {asin(x)} node [pos=0.25, below right] {\(\sin^{-1}x\)};
  78                      \addplot[color=blue, smooth] gnuplot [domain=-2:2,unbounded coords=jump,samples=500] {acos(x)} node [pos=0.25, below left] {\(\cos^{-1}x\)};
  79                      \addplot[mark=*, red] coordinates {(-1,-1.5708)} node[right, font=\footnotesize]{\((-1,-\frac{\pi}{2})\)} ;
  80                      \addplot[mark=*, red] coordinates {(1,1.5708)} node[left, font=\footnotesize]{\((1,\frac{\pi}{2})\)} ;
  81                      \addplot[mark=*, blue] coordinates {(1,0)};
  82                      \addplot[mark=*, blue] coordinates {(-1,3.1415)} node[right, font=\footnotesize]{\((-1,\pi)\)} ;
  83                    \end{axis}
  84                  \end{tikzpicture}\\
  85
  86                  Inverse functions: \(f(f^{-1}(x)) = x\) (restrict domain)
  87
  88                  \[\sin^{-1}: [-1, 1] \rightarrow \mathbb{R}, \quad \sin^{-1} x = y\]
  89                  \hfill where \(\sin y = x, \> y \in [{-\pi \over 2}, {\pi \over 2}]\)
  90
  91                  \[\cos^{-1}: [-1,1] \rightarrow \mathbb{R}, \quad \cos^{-1} x = y\]
  92                  \hfill where \(\cos y = x, \> y \in [0, \pi]\)
  93
  94                  \[\tan^{-1}: \mathbb{R} \rightarrow \mathbb{R}, \quad \tan^{-1} x = y\]
  95                  \hfill where \(\tan y = x, \> y \in \left(-{\pi \over 2}, {\pi \over 2}\right)\)
  96
  97                  \begin{tikzpicture}
  98                    \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={\(-\frac{\pi}{2}\),\(\frac{\pi}{2}\)}]
  99                      \addplot[color=orange, smooth] gnuplot [domain=-35:35, unbounded coords=jump,samples=350] {atan(x)} node [pos=0.5, above left] {\(\tan^{-1}x\)};
 100                      \addplot[gray, dotted, thick, domain=-35:35] {1.5708} node [black, font=\footnotesize, below right, pos=0] {\(y=\frac{\pi}{2}\)};
 101                      \addplot[gray, dotted, thick, domain=-35:35] {-1.5708} node [black, font=\footnotesize, above left, pos=1] {\(y=-\frac{\pi}{2}\)};
 102                    \end{axis}
 103                  \end{tikzpicture}
 104
 105\subsection*{\(\sin\) and \(\cos\) graphs}
 106
 107\[ f(x)=a\sin(bx-c)+d \]
 108
 109where:
 110\begin{description}
 111  \item Period \(=\frac{2\pi}{n}\)
 112  \item dom \(= \mathbb{R}\)
 113  \item ran \(= [-b+c, b+c]\);
 114  \item \(\cos(x)\) starts at \((0,1)\), \(\sin(x)\) starts at \((0,0)\)
 115  \item 0 amplitidue \(\implies\) straight line
 116  \item \(a<0\) or \(b<0\) inverts phase (swap \(\sin\) and \(\cos\))
 117  \item \(c=T={{2\pi}\over b} \implies\) no net phase shift
 118\end{description}
 119
 120\subsection*{\(\tan\) graphs}
 121
 122\[y=a\tan(nx)\]
 123
 124\begin{description}
 125  \item Period \(= \dfrac{\pi}{n}\)
 126  \item Range is \(\mathbb{R}\)
 127  \item Roots at \(x={\dfrac{k\pi}{n}}\) where \(k \in \mathbb{Z}\)
 128  \item Asymptotes at \(x=\frac{(2k+1)\pi}{2n}\)
 129\end{description}
 130
 131\textbf{Asymptotes should always have equations}
 132
 133\subsection*{Solving trig equations}
 134
 135\begin{enumerate}
 136\def\labelenumi{\arabic{enumi}.}
 137\tightlist
 138\item
 139  Solve domain for \(n\theta\)
 140\item
 141  Find solutions for \(n\theta\)
 142\item
 143  Divide solutions by \(n\)
 144\end{enumerate}
 145
 146\(\sin2\theta={\sqrt{3}\over2}, \quad \theta \in[0, 2\pi] \quad(\therefore 2\theta \in [0,4\pi])\)
 147
 148\(2\theta=\sin^{-1}{\sqrt{3} \over 2}\)
 149
 150\(2\theta={\pi\over 3}, {2\pi \over 3}, {7\pi \over 3}, {8\pi \over 3}\)
 151
 152\(\therefore \theta = {\pi \over 6}, {\pi \over 3}, {7 \pi \over 6}, {4\pi \over 3}\)