Rotation Formula — Definition, Formula & Examples

The rotation formula gives you the new coordinates of a point after it is rotated by a specified angle around the origin. It uses sine and cosine to compute where the point lands.

Given a point $(x, y)$ in the coordinate plane and an angle $\theta$ measured counterclockwise from the positive $x$ -axis, the image of the point under rotation about the origin is $(x\cos\theta - y\sin\theta,\; x\sin\theta + y\cos\theta)$ .

Key Formula

x' = x\cos\theta - y\sin\theta \qquad y' = x\sin\theta + y\cos\theta

Where:

$(x, y)$ = Coordinates of the original point
$\theta$ = Angle of rotation (counterclockwise positive)
$(x', y')$ = Coordinates of the rotated point

How It Works

To apply the rotation formula, identify the original coordinates

(x, y)

and the angle of rotation

\theta

. Substitute these values into the two equations to get the new

x'

and

y'

coordinates. A positive angle rotates the point counterclockwise, while a negative angle rotates it clockwise. For common angles like

90°

180°

, and

270°

, the sine and cosine values simplify to

0

1

, or

-1

, making the arithmetic straightforward.

Worked Example

Problem: Rotate the point (3, 4) by 90° counterclockwise about the origin.

Identify values: Here x = 3, y = 4, and θ = 90°. Recall that cos 90° = 0 and sin 90° = 1.

\cos 90° = 0, \quad \sin 90° = 1

Compute x': Substitute into the formula for x'.

x' = 3(0) - 4(1) = -4

Compute y': Substitute into the formula for y'.

y' = 3(1) + 4(0) = 3

Answer: The image of (3, 4) after a 90° counterclockwise rotation is (−4, 3).

Why It Matters

Rotation formulas appear throughout precalculus when deriving trigonometric identities and in physics when resolving force components in rotated coordinate systems. They are also foundational in computer graphics, where every on-screen rotation of a shape is computed using these equations.

Common Mistakes

Mistake: Using a clockwise angle as positive, which swaps the sign of the sine terms.

Correction: The standard convention treats counterclockwise as positive. If you rotate clockwise by θ, substitute −θ into the formula, which changes sin θ to −sin θ.