Andrew's git - notes.git/blob

   1---
   2geometry: margin=1.9cm
   3columns: 2
   4graphics: yes
   5author: Andrew Lorimer
   6---
   7
   8# Circular functions
   9
  10Period of $a\sin(bx)$ is ${2\pi} \over b$
  11
  12Period of $a\tan(nx)$ is $\pi \over n$  
  13Asymptotes at $x={2k+1)\pi \over 2n} \> \vert \> k \in \mathbb{Z}$
  14
  15## Reciprocal functions
  16
  17### Cosecant
  18
  19![](graphics/csc.png)
  20
  21$$\operatorname{cosec} \theta = {1 \over \sin \theta} \> \vert \> \sin \theta \ne 0$$
  22
  23- **Domain** $= \mathbb{R} \setminus {n\pi : n \in \mathbb{Z}}$
  24- **Range** $= \mathbb{R} \setminus (-1, 1)$
  25- **Turning points** at $\theta = {{(2n + 1)\pi} \over 2} \> \vert \> n \in \mathbb{Z}$
  26- **Asymptotes** at $\theta = n\pi \> \vert \> n \in \mathbb{Z}$
  27
  28### Secant
  29
  30![](graphics/sec.png)
  31
  32$$\operatorname{sec} \theta = {1 \over \cos \theta} \> \vert \> \cos \theta \ne 0$$
  33
  34- **Domain** $= \mathbb{R} \setminus \{{{(2n + 1) \pi} \over 2 } : n \in \mathbb{Z}\}$
  35- **Range** $= \mathbb{R} \setminus (-1, 1)$
  36- **Turning points** at $\theta = n\pi \> \vert \> n \in \mathbb{Z}$
  37- **Asymptotes** at $\theta = {{(2n + 1) \pi} \over 2} \> \vert \> n \in \mathbb{Z}$
  38
  39### Cotangent
  40
  41![](graphics/cot.png)
  42
  43$$\operatorname{cot} \theta = {{\cos \theta} \over {\sin \theta}} \> \vert \> \sin \theta \ne 0$$
  44
  45- **Domain** $= \mathbb{R} \setminus \{n \pi: n \in \mathbb{Z}\}$
  46- **Range** $= \mathbb{R}$
  47- **Asymptotes** at $\theta = n\pi \> \vert \> n \in \mathbb{Z}$
  48
  49### Symmetry properties
  50
  51\begin{equation}\begin{split}
  52  \operatorname{sec} (\pi \pm x) & = -\operatorname{sec} x \\
  53  \operatorname{sec} (-x) & = \operatorname{sec} x \\
  54  \operatorname{cosec} (\pi \pm x) & = \mp \operatorname{cosec} x \\
  55  \operatorname{cosec} (-x) & = - \operatorname{cosec} x \\
  56  \operatorname{cot} (\pi \pm x) & = \pm \operatorname{cot} x \\
  57  \operatorname{cot} (-x) & = - \operatorname{cot} x
  58\end{split}\end{equation}
  59
  60### Complementary properties
  61
  62\begin{equation}\begin{split}
  63  \operatorname{sec} \left({\pi \over 2} - x\right) & = \operatorname{cosec} x \\
  64  \operatorname{cosec} \left({\pi \over 2} - x\right) & = \operatorname{sec} x \\
  65  \operatorname{cot} \left({\pi \over 2} - x\right) & = \tan x \\
  66  \tan \left({\pi \over 2} - x\right) & = \operatorname{cot} x
  67\end{split}\end{equation}
  68
  69### Pythagorean identities
  70
  71\begin{equation}\begin{split}
  72  1 + \operatorname{cot}^2 x & = \operatorname{cosec}^2 x, \quad \text{where } \sin x \ne 0 \\
  73  1 + \tan^2 x & = \operatorname{sec}^2 x, \quad \text{where } \cos x \ne 0
  74\end{split}\end{equation}
  75
  76## Compound angle formulas
  77
  78$$\cos(x \pm y) = \cos x + \cos y \mp \sin x \sin y$$
  79$$\sin(x \pm y) = \sin x \cos y \pm \cos x \sin y$$
  80$$\tan(x \pm y) = {{\tan x \pm \tan y} \over {1 \mp \tan x \tan y}}$$
  81
  82## Double angle formulas
  83
  84\begin{equation}\begin{split}
  85  \cos 2x &= \cos^2 x - \sin^2 x \\
  86  & = 1 - 2\sin^2 x \\
  87  & = 2 \cos^2 x -1
  88\end{split}\end{equation}
  89
  90$$\sin 2x = 2 \sin x \cos x$$
  91
  92$$\tan 2x = {{2 \tan x} \over {1 - \tan^2 x}}$$
  93
  94## Inverse circular functions
  95
  96Inverse functions: $f(f^{-1}(x)) = x, \quad f(f^{-1}(x)) = x$  
  97Must be 1:1 to find inverse (reflection in $y=x$
  98
  99Domain is restricted to make functions 1:1.
 100
 101### $\arcsin$
 102
 103$$\sin^{-1}: [-1, 1] \rightarrow \mathbb{R}, \quad \sin^{-1} x = y, \quad \text{where } \sin y = x \text{ and } y \in [{-\pi \over 2}, {\pi \over 2}]$$
 104
 105### $\arcos$
 106
 107$$\cos^{-1} \rightarrow \mathbb{R}, \quad \cos^{-1} x = y, \quad \text{where } \cos y = x \text{ and } y \in [0, \pi]$$
 108
 109### $\arctan$
 110
 111$$\tan^{-1}: \mathbb{R} \rightarrow \mathbb{R}, \quad \tan^{-1} x = y, \quad \text{where } \tan y = x \text{ and } y \in \left(-{\pi \over 2}, {\pi \over 2}\right)$$