How this instrument works
Rotating a point (x, y) by an angle θ around the origin follows directly from writing the point in polar form first. Any point sits at some distance r from the origin and some angle φ from the positive x-axis, so x = r cos φ and y = r sin φ. Rotating by θ simply adds to that angle, replacing φ with φ + θ, and expanding cos(φ + θ) and sin(φ + θ) with the angle-addition identities turns that single sentence into the two formulas this sheet evaluates: x' = x cos θ − y sin θ and y' = x sin θ + y cos θ, with neither r nor φ ever appearing explicitly.
The matrix behind those two lines, [[cos θ, −sin θ], [sin θ, cos θ]], has determinant cos²θ + sin²θ = 1 for every angle — a fixed identity, not a coincidence of the example below — which is the algebraic signature of a transformation that changes direction and nothing else. Distance from the origin survives untouched: a point 5 units out stays 5 units out no matter what θ is fed in, because r² = x² + y² is invariant under this exact pairing of cos and sin. That property is what makes the formula the standard tool for spinning a shape in computer graphics, robotics, and orbital mechanics — every vertex keeps its length, so the shape itself neither stretches nor tears.
Positive θ turns counterclockwise under this convention, the same orientation used throughout mathematics wherever angles are measured from the positive x-axis — a genuine source of mix-ups when porting a formula between a math notebook and a screen coordinate system with y pointing down, where the identical arithmetic visually spins the other way. A second edge case is purely numerical rather than mathematical: rotating by exactly 90° ought to send (3, 4) to a clean (−4, 3), but cos(π⁄2) is not stored as a perfect zero in floating-point arithmetic, so the sheet can report 3.0000000000000004 in place of 3 — off by less than one part in 10^15, a reminder that the formula is exact even when the printed decimals are not.
- Enter the point to rotate into Point: x and Point: y — any real numbers, including negatives or zero.
- Enter the turn amount into Rotation angle (counterclockwise); the default unit is degrees, with radians and turns available from the unit selector.
- Read Rotated point: x and Rotated point: y for the coordinates of the point after the spin.
- Use a negative angle, such as −90 instead of 90, to rotate clockwise — no other change to the inputs is needed.
- Set the angle to 0 as a sanity check: the rotated point should come back identical to the point you entered.
Worked example — spinning (3, 4) by 90°
Take the point (3, 4) — three units right and four up from the origin, distance 5 away by the Pythagorean relation 3² + 4² = 5² — and set Rotation angle (counterclockwise) to 90°, which the sheet converts internally to 1.5707963267948966 radians. Feeding that θ into x' = x cos θ − y sin θ gives x' = 3×0 − 4×1 = −4 exactly, since cos(90°) is 0 and sin(90°) is 1.
The second coordinate follows the same rule: y' = x sin θ + y cos θ = 3×1 + 4×0 = 3, though the sheet actually reports 3.0000000000000004 rather than a bare 3, because θ arrives as the double-precision value 1.5707963267948966 rather than a mathematically perfect π⁄2, leaving cos θ a hair above true zero. The rotated point (−4, 3) still sits exactly 5 units from the origin, matching (3, 4) — confirmation that the rotation only turned the point; it never moved it closer to or farther from the center.
Questions
What does the rotation formula actually compute?
It computes where a point (x, y) lands after spinning it by angle θ counterclockwise around the origin, using x' = x cos θ − y sin θ and y' = x sin θ + y cos θ — the same two lines whether θ is 1° or 359°. Both formulas fall out of writing (x, y) in polar form and adding θ to its angle, then expanding with the cosine and sine addition identities.
Why is a positive angle counterclockwise instead of clockwise?
Because the formula follows the standard mathematical convention, where angles are measured counterclockwise from the positive x-axis — the same orientation used for sine and cosine everywhere else in trigonometry. Enter a negative angle, such as −90 instead of 90, to rotate clockwise; the formula needs no other change, since cos(−θ) = cos θ and sin(−θ) = −sin θ handle the flip automatically.
Does rotating a point change its distance from the origin?
No — that distance is exactly preserved, which is the defining property of a rotation. For any θ, x'² + y'² works out identical to x² + y², because the rotation matrix's determinant is cos²θ + sin²θ = 1 for every angle. Rotating (3, 4), a point 5 units out, by any amount always returns a point exactly 5 units out — only the direction changes.
Why does the sheet sometimes show a value like 3.0000000000000004 instead of a clean 3?
That extra digit string is floating-point noise, not a math error. A 90° angle converts internally to the double-precision value 1.5707963267948966, extremely close to but not exactly π⁄2, so cos θ lands a hair above zero instead of exactly on it — and that hair carries through into the final coordinate. The true mathematical answer is exactly 3.
How is this different from converting a point to polar coordinates?
A cartesian-to-polar conversion re-describes one fixed point using distance and angle from the origin, r and θ, and changes nothing about the point itself. This calculator instead moves the point: it takes an existing (x, y), adds θ to its underlying angle, and hands back a genuinely different location, still the same distance from the origin but turned to face a new direction.
What happens when θ is a full 360° or a multiple of it?
The point returns to exactly where it started, since cos and sin both repeat every 360° (equivalently every 2π radians, or every full turn), so x' = x and y' = y once θ completes a full circle. Any angle beyond 360°, such as 450°, behaves identically to that angle minus 360° — here, the same result as a plain 90° rotation.