Math101 · Independent Ontario learning library
Math101Coterminal Angles
Coterminal angles share the same initial and terminal sides in standard position. They differ by whole rotations: in degrees $\theta+360^\circ k$, and in radians $\theta+2\pi k$, where $k$ is any integer.
Coterminal reduction makes periodic trig evaluation, rotation, and angular modelling efficient while preserving direction.
Intuition and core definition
Coterminal angles share the same initial and terminal sides in standard position. They differ by whole rotations: in degrees $\theta+360^\circ k$, and in radians $\theta+2\pi k$, where $k$ is any integer. Their sine, cosine, and tangent values are equal wherever defined.
Notation, language, and conditions
Positive angles rotate counterclockwise and negative angles clockwise. A principal interval is a chosen representative range, commonly $[0,360^\circ)$ or $[0,2\pi)$, but the requested interval controls which representative is reported.
Why this idea matters
Coterminal angles differ by full rotations and therefore share a terminal side while retaining different numerical measures.
A dependable method
- Identify whether the measure uses degrees or radians.
- Add or subtract full turns using integer multiples.
- Continue until the result lies in the requested interval.
- For all coterminal angles, append $+360^\circ k$ or $+2\pi k$.
- Check that the difference from the original is an exact whole-turn multiple.
