[methods] determining equations for graphs

diff --git a/spec/circ.md b/spec/circ.md
index c3c5b173675e8db11d22477f347d86d814786223..da26542e6d4e3f4053b6f7c61105c76d3784bb9d 100644 (file)
--- a/spec/circ.md
+++ b/spec/circ.md
@@ -1,3 +1,10 @@
+---
+geometry: margin=1.9cm
+columns: 2
+graphics: yes
+author: Andrew Lorimer
+---
+
 # Circular functions
 
 Period of $a\sin(bx)$ is ${2\pi} \over b$
@@ -18,10 +25,9 @@ $$\operatorname{cosec} \theta = {1 \over \sin \theta} \> \vert \> \sin \theta \n
 - **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]
+![](graphics/sec.png)
 
 $$\operatorname{sec} \theta = {1 \over \cos \theta} \> \vert \> \cos \theta \ne 0$$
 
@@ -30,10 +36,9 @@ $$\operatorname{sec} \theta = {1 \over \cos \theta} \> \vert \> \cos \theta \ne
 - **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]
+![](graphics/cot.png)
 
 $$\operatorname{cot} \theta = {{\cos \theta} \over {\sin \theta}} \> \vert \> \sin \theta \ne 0$$
 
@@ -77,7 +82,9 @@ $$\tan(x \pm y) = {{\tan x \pm \tan y} \over {1 \mp \tan x \tan y}}$$
 ## Double angle formulas
 
 \begin{equation}\begin{split}
-  \cos 2x = \cos^2 x = \sin^2 x
+  \cos 2x &= \cos^2 x - \sin^2 x \\
+  & = 1 - 2\sin^2 x \\
+  & = 2 \cos^2 x -1
 \end{split}\end{equation}
 
 $$\sin 2x = 2 \sin x \cos x$$