Andrew's git - notes.git/blob - methods/circ-functions.tex

   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*{\(\sin\) and \(\cos\) graphs}
  74
  75\[ f(x)=a\sin(bx-c)+d \]
  76
  77where:
  78\begin{description}
  79  \item Period \(=\frac{2\pi}{n}\)
  80  \item dom \(= \mathbb{R}\)
  81  \item ran \(= [-b+c, b+c]\);
  82  \item \(\cos(x)\) starts at \((0,1)\), \(\sin(x)\) starts at \((0,0)\)
  83  \item 0 amplitidue \(\implies\) straight line
  84  \item \(a<0\) or \(b<0\) inverts phase (swap \(\sin\) and \(\cos\))
  85  \item \(c=T={{2\pi}\over b} \implies\) no net phase shift
  86\end{description}
  87
  88\subsection*{\(\tan\) graphs}
  89
  90\[y=a\tan(nx)\]
  91
  92\begin{description}
  93  \item Period \(= \dfrac{\pi}{n}\)
  94  \item Range is \(\mathbb{R}\)
  95  \item Roots at \(x={\dfrac{k\pi}{n}}\) where \(k \in \mathbb{Z}\)
  96  \item Asymptotes at \(x=\frac{(2k+1)\pi}{2n}\)
  97\end{description}
  98
  99\textbf{Asymptotes should always have equations}
 100
 101\subsection*{Solving trig equations}
 102
 103\begin{enumerate}
 104\def\labelenumi{\arabic{enumi}.}
 105\tightlist
 106\item
 107  Solve domain for \(n\theta\)
 108\item
 109  Find solutions for \(n\theta\)
 110\item
 111  Divide solutions by \(n\)
 112\end{enumerate}
 113
 114\(\sin2\theta={\sqrt{3}\over2}, \quad \theta \in[0, 2\pi] \quad(\therefore 2\theta \in [0,4\pi])\)
 115
 116\(2\theta=\sin^{-1}{\sqrt{3} \over 2}\)
 117
 118\(2\theta={\pi\over 3}, {2\pi \over 3}, {7\pi \over 3}, {8\pi \over 3}\)
 119
 120\(\therefore \theta = {\pi \over 6}, {\pi \over 3}, {7 \pi \over 6}, {4\pi \over 3}\)