+---
+geometry: margin=1.9cm
+columns: 2
+graphics: yes
+author: Andrew Lorimer
+---
+
# Circular functions
Period of $a\sin(bx)$ is ${2\pi} \over b$
- **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
-!()[graphics/sec.png]
+
$$\operatorname{sec} \theta = {1 \over \cos \theta} \> \vert \> \cos \theta \ne 0$$
- **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
-!()[graphics/cot.png]
+
$$\operatorname{cot} \theta = {{\cos \theta} \over {\sin \theta}} \> \vert \> \sin \theta \ne 0$$
## Double angle formulas
\begin{equation}\begin{split}
- \cos 2x = \cos^2 x = \sin^2 x
+ \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.
+
+### $\arcsin$
+
+$$\sin^{-1}: [-1, 1] \rightarrow \mathbb{R}, \quad \sin^{-1} x = y, \quad \text{where } \sin y = x \text{ and } y \in [{-\pi \over 2}, {\pi \over 2}]$$
+
+### $\arcos$
+
+$$\cos^{-1} \rightarrow \mathbb{R}, \quad \cos^{-1} x = y, \quad \text{where } \cos y = x \text{ and } y \in [0, \pi]$$
+
+### $\arctan$
+
+$$\tan^{-1}: \mathbb{R} \rightarrow \mathbb{R}, \quad \tan^{-1} x = y, \quad \text{where } \tan y = x \text{ and } y \in \left(-{\pi \over 2}, {\pi \over 2}\right)$$