Coterminal Angle Calculator
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Coterminal angles share the same terminal side, so they differ by whole turns of 360° (2π radians). For 765°: 765° − 2(360°) = 45°, so 45°, 405° and −315° are all coterminal with 765°.
Degrees like −120, or radians like 13pi/4.
The terminal side (green) and the rotation from the positive x-axis (blue spiral).
| Angle in standard position (0 to 360°) | \(45^{\circ}\) |
| Smallest positive coterminal angle | \(45^{\circ}\) |
| Largest negative coterminal angle | \(-315^{\circ}\) |
| Location | Quadrant I |
| Reference angle | \(45^{\circ}\) |
| In the other unit | \(\frac{17\pi}{4}\) rad |
| More coterminal angles | \(45^{\circ},\ 405^{\circ},\ 1125^{\circ},\ 1485^{\circ},\ 1845^{\circ}\) |
Show steps
- Coterminal angles differ by whole turns: \(765^{\circ} + k \cdot 360^{\circ}\) for any integer k.
- Subtract 2 full turns: \(765^{\circ} – 2 \cdot 360^{\circ} = 45^{\circ}\)
- Add or subtract one more turn for other coterminal angles: \(45^{\circ} + 360^{\circ} = 405^{\circ}\) and \(45^{\circ} – 360^{\circ} = -315^{\circ}\).
How to use the coterminal angle calculator
- Type the angle. For radians you can use pi, like 7pi/3.
- Choose degrees or radians and press Find.
- You get the angle between 0 and one full turn, the smallest positive and largest negative coterminal angles, the quadrant and the reference angle.
Formula
Coterminal angles: \(\theta + 360^{\circ}k\) in degrees, or \(\theta + 2\pi k\) in radians, for any integer k.
To find the angle between 0° and 360°, add or subtract 360° until it lands in that range.
Worked examples
- −120° + 360° = 240° (Quadrant III). Another negative one: −120° − 360° = −480°.
- \(\frac{13\pi}{4} – 2\pi = \frac{5\pi}{4}\), and \(\frac{5\pi}{4} – 2\pi = -\frac{3\pi}{4}\).
Quadrants and reference angles
| Angle (0°–360°) | Quadrant | Reference angle |
|---|---|---|
| 0°–90° | I | θ |
| 90°–180° | II | 180° − θ |
| 180°–270° | III | θ − 180° |
| 270°–360° | IV | 360° − θ |
Coterminal angles practice quiz: 10 questions
Test yourself with 10 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.
Quiz complete
Frequently asked questions
What are coterminal angles?
Angles in standard position that share the same terminal side. They differ by a whole number of full rotations.
How do you find a coterminal angle?
Add or subtract 360° (or 2π radians) as many times as you need.
How do I find the positive coterminal angle of a negative angle?
Add 360° (or 2π) until the result is positive.
Do coterminal angles have the same trig values?
Yes. Because they end at the same point on the unit circle, their sine, cosine and tangent are equal.
Can I enter radians with π?
Yes. Type pi, for example 13pi/4 or -5pi/6, and choose Radians.
Sources
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