5.3 Coordinates rotation

Coordinates rotation. If we indicate with $\alpha$ the angle of rotation between the origin cartesian system to the new cartesian system, and if we indicate with $P \equiv (\theta, d)$ the polar coordinates of point P in origin system, we can determine these relations (look the figure):

$\begin{cases}\theta = \theta' + \alpha \\ x = d \sin{\theta} \\ y = d \cos{\theta}\end{cases} ~~~\begin{cases}\theta' = \theta - \alpha \\ x' = d \sin{\theta'}\\y'=d \cos{\theta'}\end{cases}$

There is another (but most used) kind of way to change the coordinates:

$\begin{cases}x'=x\cos{\alpha} - y\sin{\alpha} \\ y'=y\cos{\alpha} + x\sin{alpha} \end{cases}~~~\begin{cases}x=x'\cos{\alpha} + y'\sin{\alpha} \\ y=y'\cos{\alpha} - x'\sin{\alpha} \end{cases}$