To quickly transform coordinates from one coordinate system to another that is translated and rotated, follow these steps:

1. Start with the coordinates (x,y) in the original system.
2. Rotate the coordinates using the formula:
   x′=xcos(θ)−ysin(θ), y′=xsin(θ)+ycos(θ) where θ is the counterclockwise angle between the two coordinate systems.
3. Apply the translation by adding the distance between the origins of the two systems, denoted as (𝑎,𝑏).
4. The final transformed coordinates are:
   x′′=xcos(θ)−ysin(θ)+a, y′′=xsin(θ)+ycos(θ)+b