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Math101learn.math101.caCoterminal 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.
Worked example
Representations and interpretation
On a unit circle, multiple rotations trace the same path repeatedly and end at the same terminal ray. Reducing modulo one full turn records only the endpoint direction, not the number of revolutions.
Reasoning about variations
Coterminal does not mean complementary or supplementary. Angles $30^\circ$ and $390^\circ$ are coterminal; $30^\circ$ and $150^\circ$ are supplementary but end on different rays.
Common mistakes
How to check your work
- Subtract the measures and divide by a full turn; the quotient must be an integer.
- Plot both terminal rays.
- Compare sine and cosine values.
Practice
- Find the coterminal angle in $[0,360^\circ)$ for $470^\circ$.
- Give all angles coterminal with $-\pi/4$.
- Are $20^\circ$ and $200^\circ$ coterminal?
Answers and brief solutions
Show answers
- $110^\circ$ $470-360=110$.
- $-\pi/4+2\pi k$, $k\in\mathbb Z$ Add any integer number of full radian turns.
- No They differ by $180^\circ$, not a multiple of $360^\circ$.
Synthesis and transfer
A rotating wheel can finish at the same spoke position after positive or negative turns; adding an integer multiple of $360^\circ$ or $2\pi$ records that equivalence.
A wheel turned through $-250^\circ$ and one turned through $110^\circ$ finish with the same terminal spoke because the measures differ by $360^\circ$. To find the representative in $[0,360^\circ)$, add or subtract full turns without altering the remainder direction. In radians, the corresponding step uses multiples of $2\pi$. Coterminal does not mean equal as real numbers or identical as rotations: one route may include several extra revolutions and therefore different elapsed motion. Trigonometric functions repeat at these angles, but a physical context may still care about total turns, direction, or time. The requested interval determines which representative is acceptable.
Related topics
Teaching and accessibility note
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Find the coterminal angle in $[0,360^\circ)$ for $470^\circ$.
- $470-360=110$.
End of lesson
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