2D Rotation Calculator - Rotate a Point

A two-dimensional rotation changes a point's direction from the origin while preserving its distance from the origin. The transformation is rigid: lengths, angles, area, and shape do not change. Only orientation and coordinates change. This calculator rotates a Cartesian point around the origin using either degrees or radians and returns the transformed x and y coordinates. For an original point (x, y) and angle theta, the new x-coordinate equals x times cosine theta minus y times sine theta. The new y-coordinate equals x times sine theta plus y times cosine theta. These equations are the expanded form of multiplying the point by the standard two-dimensional rotation matrix. The calculator evaluates sine and cosine at full floating-point precision before rounding the displayed coordinates. By mathematical convention, positive angles rotate counterclockwise and negative angles rotate clockwise. A 90-degree counterclockwise rotation sends (x, y) to (-y, x). A 180-degree rotation sends it to (-x, -y), and a 270-degree rotation sends it to (y, -x). A full 360-degree turn returns the point to its starting location. Equivalent radian measures are pi divided by two, pi, three pi divided by two, and two pi. Degrees divide one full revolution into 360 equal parts and are convenient for geometry and everyday measurement. Radians measure angle by arc length relative to radius and are the natural unit for calculus, physics, and most programming language trigonometric functions. The calculator converts degree input to radians internally. When radians are selected, enter a decimal approximation such as 1.5707963268 for pi divided by two. Rotation preserves the point's radius because cosine squared plus sine squared equals one. You can verify a result by comparing x squared plus y squared before and after transformation. Small differences near zero can occur because trigonometric values such as cosine of 90 degrees are represented approximately by computers. The display normalizes tiny residuals to zero so common quarter-turn results remain readable. This origin-based rotation is useful in analytic geometry, computer graphics, robotics, game development, mapping, and physics. Rotating around another center requires translating that center to the origin, applying the rotation, and translating back. Coordinate systems in screen graphics often place positive y downward, which can make the visual direction appear reversed. Confirm the axis convention used by your application before applying the result.