Math Girls Manga

http://www.amazon.com/gp/aw/d/0983951349/ref=pd_aw_cart_recs_1?pi=SL500_SY115

Chapter 3 on Rotation is excellent ! He combines Analytic Geometry, Trigonometry, Linear Algebra (Matrix), and Physics (Rotation) into “one same thing” to show the beauty of Mathematics:

http://en.m.wikipedia.org/wiki/Rotation_matrix

The following matrix represents a rotation $\rho (\theta)$ by an angle $\theta$:
$\boxed { \begin{pmatrix} \cos {\theta} & -\sin {\theta}\\ \sin {\theta} & \cos {\theta} \end{pmatrix} = \rho (\theta) }$

Rotate by $2\theta$ will be:
$\begin{pmatrix} \cos {2\theta} & -\sin {2\theta}\\ \sin {2\theta} & \cos {2\theta} \end{pmatrix}$
which is equivalent to 2 successive rotations (same direction) of angle $\theta$:
$\rho (\theta) .\rho (\theta) = \rho^2 (\theta)$:

$\begin{pmatrix} \cos {\theta} & -\sin {\theta}\\ \sin {\theta} & \cos {\theta} \end{pmatrix}^2$
$= \begin{pmatrix} \cos {\theta} & -\sin {\theta}\\ \sin {\theta} & \cos {\theta} \end{pmatrix}$ $\begin{pmatrix} \cos {\theta} & -\sin {\theta}\\ \sin {\theta} & \cos {\theta} \end{pmatrix}$
$= \begin{pmatrix} \cos ^2 {\theta} - \sin ^2 {\theta } & -2\sin {\theta} \cos {\theta} \\ 2\sin {\theta} \cos {\theta} & \cos^2 {\theta}- \sin^2 {\theta} \end{pmatrix}$
$= \begin{pmatrix} \cos {2\theta} & -\sin {2\theta} \\ \sin {2\theta} & \cos {2\theta} \end{pmatrix}$

$\boxed { \cos 2 {\theta} = \cos ^{2} {\theta} - \sin ^{2} {\theta} }$

$\boxed { \sin 2 {\theta} = 2 \sin {\theta} \cos {\theta} }$