The rules of rotation in mathematics, particularly in coordinate geometry, describe how to rotate a point or shape around the origin or a specific point on the coordinate plane. These rules depend on the angle and direction of rotation. Below are the common rules of rotation about the origin for angles of 90°, 180°, and 270°, typically in counterclockwise direction, but can also be stated for clockwise.

1. 90° Counterclockwise Rotation (or 270° Clockwise)

If a point $(x, y)$ is rotated 90° counterclockwise around the origin, the new coordinates become: $(x', y') = (-y, x)$

2. 180° Rotation (Counterclockwise or Clockwise, both are the same)

If a point $(x, y)$ is rotated 180° around the origin, the new coordinates become: $(x', y') = (-x, -y)$

3. 270° Counterclockwise Rotation (or 90° Clockwise)

If a point $(x, y)$ is rotated 270° counterclockwise (equivalent to 90° clockwise) around the origin, the new coordinates become: $(x', y') = (y, -x)$

4. 360° Rotation

A 360° rotation (either direction) brings the point back to its original position: $(x', y') = (x, y)$

5. Clockwise Rotation Rules

To rotate clockwise, you can apply the same rules as above but interpret the angles differently:

• 90° clockwise is equivalent to 270° counterclockwise.
• 180° clockwise is the same as 180° counterclockwise.
• 270° clockwise is equivalent to 90° counterclockwise.

General Formula for Rotation by Any Angle $\theta$

For a general counterclockwise rotation by an arbitrary angle $\theta$, the new coordinates $(x', y')$ of the point $(x, y)$ are given by:

x' &= x \cos(\theta) - y \sin(\theta) \\ y' &= x \sin(\theta) + y \cos(\theta) \end{aligned}$$ This formula works for any angle $$\theta$$ (in radians or degrees), and for clockwise rotation, you can replace $$\theta$$ with $$-\theta$$. --- Would you like to see a specific example of rotation applied, or explore more about transformations? Here are 5 related questions: 1. How do we apply these rotation rules to polygons in the plane? 2. What is the difference between rotation and reflection in transformations? 3. How does rotating around a point other than the origin affect the equations? 4. Can rotation rules be extended to 3D space, and if so, how? 5. How do sine and cosine in the general rotation formula relate to trigonometric functions? **Tip:** When rotating a point, always be mindful of the direction (clockwise or counterclockwise) and the angle, as small changes can drastically alter the result.