Inverse Rotation of Axes

Question

I have rotated point coordinates of original coordinates. I am trying to find original coordinates from rotated coordinates..

For example;

Original Points: x=0, y=60000

Degree: -45

If i calculate with here: https://keisan.casio.com/exec/system/1223522781

I found the x=-42,426.406871193, y=42,426.406871193

When I look here, the result is correct.

Now i want to calculate,

If i have,

degree = -45
rotated_x = -42,426.406871193
rotated_y = 42,426.406871193

How can i find original coordinates with above values?

Result must be = (0, 60000)

Answer 1

In general, the 2 by 2 Rotation Matrix to rotate a point $\theta$ degrees counter clockwise, is given by $ \begin{bmatrix}cos\theta & -sin\theta\\sin\theta & cos\theta\end{bmatrix}$. The derivation of this matrix can be found elsewhere on the internet. So if I want to rotate $ \begin{bmatrix}-42426 \\42426 \end{bmatrix}$ "back" I can plug in $\theta=-45$ into the matrix and multiply $ \begin{bmatrix}cos(-45) & -sin(-45)\\sin(-45) & cos(-45)\end{bmatrix}$$ \begin{bmatrix}-42426 \\42426 \end{bmatrix}$ and this will give $ \begin{bmatrix}0 \\60000 \end{bmatrix}$ (Forgive my rounding)