+---
+geometry: margin=1.9cm
+columns: 2
+graphics: yes
+author: Andrew Lorimer
+---
+
# Circular functions
+Period of $a\sin(bx)$ is ${2\pi} \over b$
+
+Period of $a\tan(nx)$ is $\pi \over n$
+Asymptotes at $x={2k+1)\pi \over 2n} \> \vert \> k \in \mathbb{Z}$
+
## Reciprocal functions
### Cosecant
-$$\mathrm{cosec} \Theta = {1 \over \sin \Theta} \vert \sin \Theta \ne 0$$
+
-- **Domain** $= \mathbb{R} \ {n\pi : n \in \mathbb{Z}$
-- **Range** $= \mathbb{R} \ (-1, 1)$
-- **Turning points** at $\Theta = {{(2n + 1)\pi} \over 2} \vert n \in \mathbb{Z}$
-- **Asymptotes** at $\Theta = n\pi \vert n \in \math{Z}$
+$$\operatorname{cosec} \theta = {1 \over \sin \theta} \> \vert \> \sin \theta \ne 0$$
+- **Domain** $= \mathbb{R} \setminus {n\pi : n \in \mathbb{Z}}$
+- **Range** $= \mathbb{R} \setminus (-1, 1)$
+- **Turning points** at $\theta = {{(2n + 1)\pi} \over 2} \> \vert \> n \in \mathbb{Z}$
+- **Asymptotes** at $\theta = n\pi \> \vert \> n \in \mathbb{Z}$
### Secant
-$$\mathrm{sec} \Theta = {1 \over \cos \Theta} \vert \cos \Theta \ne =$$
+
-- **Domain** $= \mathbbb{R} \ \{{{(2n + 1) \pi} \over 2 } : n \in \mathbb{Z}\}$
-- **Range** $= \mathbb{R} \ (-1, 1)$
-- **Turning points** at $\Theta n \pi \vert n \in \mathbb{Z}$
-- **Asymptotes** at $\Theta = {{(2n + 1) \pi} \over 2} \vert n \in \mathb{Z}$
+$$\operatorname{sec} \theta = {1 \over \cos \theta} \> \vert \> \cos \theta \ne 0$$
+- **Domain** $= \mathbb{R} \setminus \{{{(2n + 1) \pi} \over 2 } : n \in \mathbb{Z}\}$
+- **Range** $= \mathbb{R} \setminus (-1, 1)$
+- **Turning points** at $\theta = n\pi \> \vert \> n \in \mathbb{Z}$
+- **Asymptotes** at $\theta = {{(2n + 1) \pi} \over 2} \> \vert \> n \in \mathbb{Z}$
### Cotangent
-$$\mathrm{cot} \Theta = {{\cos \Theta} \over {\sin \Theta}}$$
+
+
+$$\operatorname{cot} \theta = {{\cos \theta} \over {\sin \theta}} \> \vert \> \sin \theta \ne 0$$
+
+- **Domain** $= \mathbb{R} \setminus \{n \pi: n \in \mathbb{Z}\}$
+- **Range** $= \mathbb{R}$
+- **Asymptotes** at $\theta = n\pi \> \vert \> n \in \mathbb{Z}$
+
+### Symmetry properties
+
+\begin{equation}\begin{split}
+ \operatorname{sec} (\pi \pm x) & = -\operatorname{sec} x \\
+ \operatorname{sec} (-x) & = \operatorname{sec} x \\
+ \operatorname{cosec} (\pi \pm x) & = \mp \operatorname{cosec} x \\
+ \operatorname{cosec} (-x) & = - \operatorname{cosec} x \\
+ \operatorname{cot} (\pi \pm x) & = \pm \operatorname{cot} x \\
+ \operatorname{cot} (-x) & = - \operatorname{cot} x
+\end{split}\end{equation}
+
+### Complementary properties
+
+\begin{equation}\begin{split}
+ \operatorname{sec} \left({\pi \over 2} - x\right) & = \operatorname{cosec} x \\
+ \operatorname{cosec} \left({\pi \over 2} - x\right) & = \operatorname{sec} x \\
+ \operatorname{cot} \left({\pi \over 2} - x\right) & = \tan x \\
+ \tan \left({\pi \over 2} - x\right) & = \operatorname{cot} x
+\end{split}\end{equation}
+
+### Pythagorean identities
+
+\begin{equation}\begin{split}
+ 1 + \operatorname{cot}^2 x & = \operatorname{cosec}^2 x, \quad \text{where } \sin x \ne 0 \\
+ 1 + \tan^2 x & = \operatorname{sec}^2 x, \quad \text{where } \cos x \ne 0
+\end{split}\end{equation}
+
+## Compound angle formulas
+
+$$\cos(x \pm y) = \cos x + \cos y \mp \sin x \sin y$$
+$$\sin(x \pm y) = \sin x \cos y \pm \cos x \sin y$$
+$$\tan(x \pm y) = {{\tan x \pm \tan y} \over {1 \mp \tan x \tan y}}$$
+
+## Double angle formulas
+
+\begin{equation}\begin{split}
+ \cos 2x &= \cos^2 x - \sin^2 x \\
+ & = 1 - 2\sin^2 x \\
+ & = 2 \cos^2 x -1
+\end{split}\end{equation}
+
+$$\sin 2x = 2 \sin x \cos x$$
+
+$$\tan 2x = {{2 \tan x} \over {1 - \tan^2 x}}$$
+
+## Inverse circular functions
+
+Inverse functions: $f(f^{-1}(x)) = x, \quad f(f^{-1}(x)) = x$
+Must be 1:1 to find inverse (reflection in $y=x$
+
+Domain is restricted to make functions 1:1.
+
+### $\arcsin$
+
+$$\sin^{-1}: [-1, 1] \rightarrow \mathbb{R}, \quad \sin^{-1} x = y, \quad \text{where } \sin y = x \text{ and } y \in [{-\pi \over 2}, {\pi \over 2}]$$
+
+### $\arcos$
+
+$$\cos^{-1} \rightarrow \mathbb{R}, \quad \cos^{-1} x = y, \quad \text{where } \cos y = x \text{ and } y \in [0, \pi]$$
+
+### $\arctan$
+
+$$\tan^{-1}: \mathbb{R} \rightarrow \mathbb{R}, \quad \tan^{-1} x = y, \quad \text{where } \tan y = x \text{ and } y \in \left(-{\pi \over 2}, {\pi \over 2}\right)$$