Key takeaway: Coterminal angles are angles in standard position that end on the same terminal side, differing by a whole number of full turns, 360° or 2π radians each.
Grade 11 – Algebra 2F-TF.A.1Trigonometry
Definition of Coterminal Angles
Coterminal angles are different angle measures that point the same way. Start both at the positive x-axis, rotate, and they stop on exactly the same ray. One may be positive and the other negative, and one may have wound around several extra times .
Since one complete rotation is 360°, adding or subtracting any number of full rotations gives a coterminal partner. Because they stop on the same ray, coterminal angles have identical sine, cosine and tangent values. This lets you shrink a huge angle like 1,000° to a friendly one between 0° and 360°.
Formula and notation
If \(\theta\) is an angle, every coterminal angle has the form
\[\theta+360^{\circ}k \quad\text{or}\quad \theta+2\pi k,\qquad k\in\mathbb{Z},\]
depending on whether degrees or radians are used. Here \(k\) counts full turns: positive \(k\) turns counterclockwise, negative \(k\) clockwise.
Key facts
- Two angles are coterminal exactly when their difference is a multiple of a full turn.
- Every angle has infinitely many partners, but only one between 0° and 360°.
- Trig function values are the same for all coterminal angles.
- A negative angle turns clockwise; 280° and -80° end on the same ray.
Coterminal Angles example
Find the angle between \(0^{\circ}\) and \(360^{\circ}\) that is coterminal with \(1000^{\circ}\), and one negative coterminal angle.
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Count full turns in 1,000°: \(2\times 360^{\circ}=720^{\circ}\).
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Subtract: \(1000^{\circ}-720^{\circ}=280^{\circ}\), which lies in the required range.
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For a negative partner subtract one more turn: \(280^{\circ}-360^{\circ}=-80^{\circ}\).
The angle in range is 280°, and \(-80^{\circ}\) is a negative coterminal partner.
Common mistake
Watch out
Bella subtracts just one turn from 1,000° and announces that 640° is the answer in range.
Why it is wrong
640° still exceeds 360°, so one more turn must go: \(640^{\circ}-360^{\circ}=280^{\circ}\). Keep subtracting full turns until the result falls between 0° and 360°.
Related terms and lessons
- Algebra 2 · Radians and the Unit CircleLessonWorksheetQuiz
- Precalculus · Angles, Radians and the Unit CircleLessonWorksheetQuiz
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