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