Angle of rotation

Definition

The angle of rotation is the measure of the angle by which a figure is rotated about a fixed point, often the origin. It is typically measured in degrees or radians.

5 Must Know Facts For Your Next Test

1. A positive angle of rotation indicates a counterclockwise rotation, while a negative angle indicates a clockwise rotation.
2. The coordinates $(x, y)$ can be transformed to $(x', y')$ using the formulas $x' = x \cos(\theta) - y \sin(\theta)$ and $y' = x \sin(\theta) + y \cos(\theta)$ where $\theta$ is the angle of rotation.
3. Rotations preserve the shape and size of geometric figures, but change their orientation.
4. A complete rotation (360 degrees or $2\pi$ radians) brings a figure back to its original position.
5. In analytic geometry, rotating an axis changes the coordinates of points but not their relative distances.

Review Questions

• What happens to the coordinates $(3, 4)$ if they are rotated by 90 degrees counterclockwise around the origin?
• Explain how you can determine if an angle of rotation is positive or negative.
• How do you derive new coordinates after rotating a point using trigonometric functions?

Related terms

Rotation Matrix: A matrix used to perform a rotation in Euclidean space; for a two-dimensional space, it is given by $$R(\theta) = \begin{pmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{pmatrix}$$.

Radians: A unit of angular measure in which angles are represented as the length of arc divided by radius; there are $2 \pi$ radians in a full circle.

Trigonometric Functions: $\cos$, $\sin$, and $tan$ functions that relate angles to ratios of sides in right triangles and are used extensively in rotations.