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:

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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}   }

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