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.

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