The angle of rotation is the angle through which a figure or point is rotated about a fixed point, typically the origin. It is measured in degrees or radians.
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The angle of rotation is positive if the rotation is counterclockwise and negative if clockwise.
In analytic geometry, rotating the axes by an angle of rotation can simplify equations of conic sections.
To rotate a point $(x, y)$ by an angle $\theta$, use the transformation formulas: $x' = x\cos(\theta) - y\sin(\theta)$ and $y' = x\sin(\theta) + y\cos(\theta)$.
Rotations preserve distances and angles between points, meaning they are rigid transformations.
The angle of rotation is often used to derive new coordinates in transformed coordinate systems.
Review Questions
What happens to a point $(x, y)$ when it is rotated by an angle $\theta$?
How does the sign of the angle of rotation affect the direction of rotation?
Why is rotating the axes useful for simplifying equations in analytic geometry?
Related terms
Rotation Matrix: A matrix used to perform a rotation in Euclidean space. For a two-dimensional space, it takes the form \( \begin{pmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{pmatrix} \).
Rigid Transformation: A transformation that preserves distances and angles, such as rotations, translations, and reflections.
Conic Sections: Curves obtained by intersecting a cone with a plane: ellipses, parabolas, and hyperbolas.