Rotating a 2D Coordinate System About The Origin Counterclockwise

In a 2D rectangular coordinate system, if you rotate the axes about the origin to get two new coordinate axes, you can covert points from one axis to the other. To convert the original axis to the transformed, you have to use the following relations where x' and y' represent the points in the new systems while x and y represent the point in the original axes.

x' = x cos(θ) + y sin(θ)

y' = - x sin(θ) + y cos(θ)

Conversely, if you have the point in the new coordinate system and you want to know what it would be in the original system, you can either put the angle in reverse in these equations or directly you can use the following relation which is made by shifting the signs for sin(θ).

x = x' cos(θ) - y' sin(θ)

y = x' sin(θ) + y' cos(θ)

This is different from rotating a point in the same coordinate system which you can find here.