Rotations

Rotation is a Linear Transformations

Consider a rotation by angle $θ$
due to trig and unit circle, etc, $$
\begin{align}
T(\vec{e_{1}}) & =\left[ \begin{matrix}
\cos\theta \
\sin\theta
\end{matrix} \right] \
T(\vec{e_{2}}) & =\left[ \begin{matrix}
-\sin\theta \
\cos\theta
\end{matrix} \right] \
[T] & =\begin{bmatrix}
\cos\theta & -\sin\theta \
\sin\theta & \cos\theta
\end{bmatrix}
\end

- N o w t r y t w o r o t a t i o n s b y $ α $ a n d $ β $ r e s p e c t i v e l y, w h i c h i s a c o m p o s i t i o n o f f u n c t i o n s w r i t t e n a s $ r o t_{α} (r o t_{β} \vec{x}) $ - W e k n o w t h e m a t r i x o f a r o t a t i o n, a n d d u e t o t h e p r o p e r t i e s o f c o m p o s i t i o n o f l i n e a r t r a n s f o r m a t i o n s,

\begin{align}
[rot_{\alpha}(rot_{\beta}\vec{x})] & =([rot\alpha][rot\beta])\vec{x} \
& =\begin{pmatrix}
\cos\alpha \cos\beta-\sin\alpha \sin\beta & -\cos\alpha \sin\beta-\sin\alpha \cos\beta \
\sin\alpha \cos\beta+\cos\alpha \sin\beta & -\sin\alpha \sin\beta+\cos\alpha \cos\beta
\end{pmatrix}
\end

, w h i c h g i v e s [[F 0 - G l o s s a r y / C o n c r e t e C o n c e p t s / T r i g o n o m e t r i c I d e n t i t i e s ∥ T r i g o n o m e t r i c I d e n t i t i e s]]!