methods / circ-functions.texon commit [methods] collate notes for sac (9822645)
   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
   9
  10    \begin{tikzpicture}[scale=0.75]
  11      \draw [orange, thick] (0,0) -- (3,3) node [black, pos=0.5, above left] {\(\sqrt{2}\)};
  12      \draw [orange, thick] (0,0) -- (3,0) node [black, below, pos=0.5] {\(1\)} node[black, above, pos=0.3] {\(\frac{\pi}{4}\)};
  13      \draw [orange, thick] (3,0) -- (3,3) node [black, right, pos=0.5] {1} node[black, left, pos=0.7] {\(\frac{\pi}{4}\)};
  14      \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};
  15    \end{tikzpicture}
  16    \begin{tikzpicture}[scale=0.75]
  17      \draw [orange, thick] (0,3) -- (5.19,0) node [black, pos=0.5, above right] {2};
  18      \draw [orange, thick] (0,0) -- (5.19,0) node [black, below, pos=0.5] {\(\sqrt{3}\)} node[black, above, pos=0.7] {\(\frac{\pi}{6}\)};
  19      \draw [orange, thick] (0,0) -- (0,3) node [black, left, pos=0.5] {1} node [black, pos=0.8, right] {\(\frac{\pi}{3}\)};
  20      \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};
  21    \end{tikzpicture}
  22
  23                  \subsection*{Compound angle formulas}
  24
  25
  26                  \begin{align*}
  27                    \cos(x \pm y) &= \cos x + \cos y \mp \sin x \sin y \\
  28                    \sin(x \pm y) &= \sin x \cos y \pm \cos x \sin y \\
  29                    \tan(x \pm y) &= {{\tan x \pm \tan y} \over {1 \mp \tan x \tan y}}
  30                  \end{align*}
  31
  32                  \subsection*{Double angle formulas}
  33
  34                  \begin{align*}
  35                    \cos 2x &= \cos^2 x - \sin^2 x \\
  36                    & = 1 - 2\sin^2 x \\
  37                    & = 2 \cos^2 x -1 \\ 
  38                    \sin 2x &= 2 \sin x \cos x \\
  39                    \tan 2x &= \dfrac{2 \tan x}{1 - \tan^2 x}
  40                  \end{align*}
  41
  42
  43
  44\subsection*{Symmetry}
  45
  46\begin{align*}
  47  \sin(\theta+\frac{\pi}{2}) &= \sin\theta \\
  48  \sin(\theta+\pi) &= -\sin\theta \\ \\
  49  \cos(\theta+\frac{\pi}{2}) &= -\cos\theta \\
  50  \cos(\theta+\pi) &= -\cos(\theta+\frac{3\pi}{2}) \\
  51  &= \cos(-\theta)
  52\end{align*}
  53
  54\subsection*{Complementary relationships}
  55
  56\begin{align*}
  57  \sin \theta &= \cos(\frac{\pi}{2} - \theta) \\
  58  &= -\cos(\theta+\frac{\pi}{2}) \\
  59  \cos\theta &= \sin(\frac{\pi}{2} - \theta) \\ 
  60  &= \sin(\theta+\frac{\pi}{2})
  61\end{align*}
  62
  63\subsection*{Pythagorean identity}
  64
  65\[\cos^2\theta+\sin^2\theta=1\]
  66
  67                  \subsection*{Inverse circular functions}
  68
  69                  \begin{tikzpicture}
  70                    \begin{axis}[ymin=-2, ymax=4, xmin=-1.1, xmax=1.1, ytick={-1.5708, 1.5708, 3.14159},yticklabels={$-\frac{\pi}{2}$, $\frac{\pi}{2}$, $\pi$}]
  71                      \addplot[color=red, smooth] gnuplot [domain=-2:2,unbounded coords=jump,samples=500] {asin(x)} node [pos=0.25, below right] {\(\sin^{-1}x\)};
  72                      \addplot[color=blue, smooth] gnuplot [domain=-2:2,unbounded coords=jump,samples=500] {acos(x)} node [pos=0.25, below left] {\(\cos^{-1}x\)};
  73                      \addplot[mark=*, red] coordinates {(-1,-1.5708)} node[right, font=\footnotesize]{\((-1,-\frac{\pi}{2})\)} ;
  74                      \addplot[mark=*, red] coordinates {(1,1.5708)} node[left, font=\footnotesize]{\((1,\frac{\pi}{2})\)} ;
  75                      \addplot[mark=*, blue] coordinates {(1,0)};
  76                      \addplot[mark=*, blue] coordinates {(-1,3.1415)} node[right, font=\footnotesize]{\((-1,\pi)\)} ;
  77                    \end{axis}
  78                  \end{tikzpicture}\\
  79
  80                  Inverse functions: \(f(f^{-1}(x)) = x\) (restrict domain)
  81
  82                  \[\sin^{-1}: [-1, 1] \rightarrow \mathbb{R}, \quad \sin^{-1} x = y\]
  83                  \hfill where \(\sin y = x, \> y \in [{-\pi \over 2}, {\pi \over 2}]\)
  84
  85                  \[\cos^{-1}: [-1,1] \rightarrow \mathbb{R}, \quad \cos^{-1} x = y\]
  86                  \hfill where \(\cos y = x, \> y \in [0, \pi]\)
  87
  88                  \[\tan^{-1}: \mathbb{R} \rightarrow \mathbb{R}, \quad \tan^{-1} x = y\]
  89                  \hfill where \(\tan y = x, \> y \in \left(-{\pi \over 2}, {\pi \over 2}\right)\)
  90
  91                  \begin{tikzpicture}
  92                    \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={\(-\frac{\pi}{2}\),\(\frac{\pi}{2}\)}]
  93                      \addplot[color=orange, smooth] gnuplot [domain=-35:35, unbounded coords=jump,samples=350] {atan(x)} node [pos=0.5, above left] {\(\tan^{-1}x\)};
  94                      \addplot[->, gray, dotted, thick, domain=-35:35] {1.5708};
  95                      \addplot[->, gray, dotted, thick, domain=-35:35] {-1.5708};
  96                    \end{axis}
  97                  \end{tikzpicture}
  98
  99\subsection*{\(\sin\) and \(\cos\) graphs}
 100
 101\[ f(x)=a\sin(bx-c)+d \]
 102
 103where:
 104\begin{description}
 105  \item Period \(=\frac{2\pi}{n}\)
 106  \item dom \(= \mathbb{R}\)
 107  \item ran \(= [-b+c, b+c]\);
 108  \item \(\cos(x)\) starts at \((0,1)\), \(\sin(x)\) starts at \((0,0)\)
 109  \item 0 amplitidue \(\implies\) straight line
 110  \item \(a<0\) or \(b<0\) inverts phase (swap \(\sin\) and \(\cos\))
 111  \item \(c=T={{2\pi}\over b} \implies\) no net phase shift
 112\end{description}
 113
 114\subsection*{\(\tan\) graphs}
 115
 116\[y=a\tan(nx)\]
 117
 118\begin{description}
 119  \item Period \(= \dfrac{\pi}{n}\)
 120  \item Range is \(\mathbb{R}\)
 121  \item Roots at \(x={\dfrac{k\pi}{n}}\) where \(k \in \mathbb{Z}\)
 122  \item Asymptotes at \(x=\frac{(2k+1)\pi}{2n}\)
 123\end{description}
 124
 125\textbf{Asymptotes should always have equations and arrow pointing up}
 126
 127\subsection*{Solving trig equations}
 128
 129\begin{enumerate}
 130\def\labelenumi{\arabic{enumi}.}
 131\tightlist
 132\item
 133  Solve domain for \(n\theta\)
 134\item
 135  Find solutions for \(n\theta\)
 136\item
 137  Divide solutions by \(n\)
 138\end{enumerate}
 139
 140\(\sin2\theta={\sqrt{3}\over2}, \quad \theta \in[0, 2\pi] \quad(\therefore 2\theta \in [0,4\pi])\)
 141
 142\(2\theta=\sin^{-1}{\sqrt{3} \over 2}\)
 143
 144\(2\theta={\pi\over 3}, {2\pi \over 3}, {7\pi \over 3}, {8\pi \over 3}\)
 145
 146\(\therefore \theta = {\pi \over 6}, {\pi \over 3}, {7 \pi \over 6}, {4\pi \over 3}\)