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   1# Circular functions
   2
   3Period of $a\sin(bx)$ is ${2\pi} \over b$
   4
   5Period of $a\tan(nx)$ is $\pi \over n$  
   6Asymptotes at $x={2k+1)\pi \over 2n} \> \vert \> k \in \mathbb{Z}$
   7
   8## Reciprocal functions
   9
  10### Cosecant
  11
  12![](graphics/csc.png)
  13
  14$$\operatorname{cosec} \theta = {1 \over \sin \theta} \> \vert \> \sin \theta \ne 0$$
  15
  16- **Domain** $= \mathbb{R} \setminus {n\pi : n \in \mathbb{Z}}$
  17- **Range** $= \mathbb{R} \setminus (-1, 1)$
  18- **Turning points** at $\theta = {{(2n + 1)\pi} \over 2} \> \vert \> n \in \mathbb{Z}$
  19- **Asymptotes** at $\theta = n\pi \> \vert \> n \in \mathbb{Z}$
  20
  21
  22### Secant
  23
  24!()[graphics/sec.png]
  25
  26$$\operatorname{sec} \theta = {1 \over \cos \theta} \> \vert \> \cos \theta \ne 0$$
  27
  28- **Domain** $= \mathbb{R} \setminus \{{{(2n + 1) \pi} \over 2 } : n \in \mathbb{Z}\}$
  29- **Range** $= \mathbb{R} \setminus (-1, 1)$
  30- **Turning points** at $\theta = n\pi \> \vert \> n \in \mathbb{Z}$
  31- **Asymptotes** at $\theta = {{(2n + 1) \pi} \over 2} \> \vert \> n \in \mathbb{Z}$
  32
  33
  34### Cotangent
  35
  36!()[graphics/cot.png]
  37
  38$$\operatorname{cot} \theta = {{\cos \theta} \over {\sin \theta}} \> \vert \> \sin \theta \ne 0$$
  39
  40- **Domain** $= \mathbb{R} \setminus \{n \pi: n \in \mathbb{Z}\}$
  41- **Range** $= \mathbb{R}$
  42- **Asymptotes** at $\theta = n\pi \> \vert \> n \in \mathbb{Z}$
  43
  44### Symmetry properties
  45
  46\begin{equation}\begin{split}
  47  \operatorname{sec} (\pi \pm x) & = -\operatorname{sec} x \\
  48  \operatorname{sec} (-x) & = \operatorname{sec} x \\
  49  \operatorname{cosec} (\pi \pm x) & = \mp \operatorname{cosec} x \\
  50  \operatorname{cosec} (-x) & = - \operatorname{cosec} x \\
  51  \operatorname{cot} (\pi \pm x) & = \pm \operatorname{cot} x \\
  52  \operatorname{cot} (-x) & = - \operatorname{cot} x
  53\end{split}\end{equation}
  54
  55### Complementary properties
  56
  57\begin{equation}\begin{split}
  58  \operatorname{sec} \left({\pi \over 2} - x\right) & = \operatorname{cosec} x \\
  59  \operatorname{cosec} \left({\pi \over 2} - x\right) & = \operatorname{sec} x \\
  60  \operatorname{cot} \left({\pi \over 2} - x\right) & = \tan x \\
  61  \tan \left({\pi \over 2} - x\right) & = \operatorname{cot} x
  62\end{split}\end{equation}
  63
  64### Pythagorean identities
  65
  66\begin{equation}\begin{split}
  67  1 + \operatorname{cot}^2 x & = \operatorname{cosec}^2 x, \quad \text{where } \sin x \ne 0 \\
  68  1 + \tan^2 x & = \operatorname{sec}^2 x, \quad \text{where } \cos x \ne 0
  69\end{split}\end{equation}