## find new coordinates in a rotated axes

We could just rotate the axes clockwise by $\theta$ to revert to our original axes. Simply apply the rotation matrix $\begin{bmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{bmatrix}$ on $\begin{bmatrix} x\\y \end{bmatrix}$ to get $\begin{bmatrix} x'=x\cos\theta + y\sin\theta \\y'=y\cos\theta - x\sin\theta \end{bmatrix}$.

Alternatively, we can see that

$x'=r\cos\alpha = r\cos(\theta+\alpha-\theta) = r(\cos(\theta+\alpha)\cos\theta+\sin(\theta+\alpha)\sin\theta) = r(\frac{x}{r}\cos\theta + \frac{y}{r}\sin\theta)$

and y’ similarly.