Axis Rotation Calculator

Find the new coordinates of a point when the coordinate axes are rotated by an angle. Enter the original point and rotation angle to get the new coordinates with a diagram.

Input

Enter a point and the rotation angle of the axes to get the new coordinates in the rotated system.

Point x

Point y

Axis angle (deg)

Result

New coordinates

(3.098076, -0.633975)

Original point

(3, 1)

Rotation angle

30 deg

Distance from origin

3.162278

x prime = x cos theta + y sin theta, y prime = minus x sin theta + y cos theta. The distance from the origin is unchanged by axis rotation.

How it works

When the coordinate axes are rotated counterclockwise by an angle theta, the new coordinates of the same point are x prime = x cos theta + y sin theta and y prime = minus x sin theta + y cos theta.
This is the opposite sign of rotating the point itself by theta. Point rotation moves the point, while axis rotation keeps the point fixed and turns the coordinate system. Rotating the axes by plus theta gives the same result as rotating the point by minus theta.
The distance r from the origin does not change when the axes are rotated. Only the coordinate values change to match the new axes.
Enter the angle in degrees. Counterclockwise is taken as positive.

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