[spec] compound & double angle formulas

diff --git a/spec/circ.md b/spec/circ.md
index a119a4af35c22fc7724b9b2040a20b0634c3cabf..c3c5b173675e8db11d22477f347d86d814786223 100644 (file)
--- a/spec/circ.md
+++ b/spec/circ.md
@@ -1,27 +1,86 @@
 # 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
 
-$$\mathrm{cosec} \Theta = {1 \over \sin \Theta} \vert \sin \Theta \ne 0$$
+![](graphics/csc.png)
+
+$$\operatorname{cosec} \theta = {1 \over \sin \theta} \> \vert \> \sin \theta \ne 0$$
 
-- **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 $\Theta = n\pi \vert n \in \math{Z}$
+- **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
 
-$$\mathrm{sec} \Theta = {1 \over \cos \Theta} \vert \cos \Theta \ne =$$
+!()[graphics/sec.png]
 
-- **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
 
-$$\mathrm{cot} \Theta = {{\cos \Theta} \over {\sin \Theta}}$$
+!()[graphics/cot.png]
+
+$$\operatorname{cot} \theta = {{\cos \theta} \over {\sin \theta}} \> \vert \> \sin \theta \ne 0$$
+
+- **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
+\end{split}\end{equation}
+
+$$\sin 2x = 2 \sin x \cos x$$
+
+$$\tan 2x = {{2 \tan x} \over {1 - \tan^2 x}}$$
+