Coterminal Angle Calculator (positive & negative coterminal angles, reference angle)

Understanding the coterminal angle calculator — concept and principle

This calculator finds angles that share a terminal side with a given angle: the smallest positive one, the largest negative one, and the neighbours one turn away. It also gives the number of whole turns, the reference angle and the quadrant. Use it for trigonometry homework and to simplify large or negative angles before evaluating sine and cosine.

How it is calculated

Coterminal angles: θ + 360° × k (or θ + 2π × k in radians), for any whole number k

Smallest positive = remainder of θ after removing whole turns, 0° ≤ angle < 360°

Largest negative = smallest positive − 360°

Reference angle: θ in Quadrant I; 180° − θ in II; θ − 180° in III; 360° − θ in IV

• Angles on an axis (multiples of 90°) are in no quadrant and have no acute reference angle.

The principle

An angle in standard position starts on the positive x-axis and turns counterclockwise for positive values, clockwise for negative ones. A full turn of 360° brings the terminal side back to the same place, so adding or subtracting whole turns does not change where the angle ends. Since trigonometric functions depend only on the terminal side, coterminal angles have identical sine, cosine and tangent. The reference angle is the acute angle to the x-axis, which gives the size of those values; the quadrant gives their signs.

Worked example

Default inputs: θ = −415°, degrees.

1. Add two full turns: −415 + 720 = 305°, the smallest positive coterminal angle; whole turns k = −2 (since −415 = 305 − 2 × 360).
2. Largest negative coterminal angle = 305 − 360 = −55°.
3. One turn less / one turn more: −775°, −55°.
4. 305° lies between 270° and 360°: Quadrant IV.
5. Reference angle = 360 − 305 = 55; principal angle 305.

How to read the result

To evaluate a trigonometric function of −415°, use 305°, or the reference angle 55° with the sign for Quadrant IV: cos(−415°) = cos 55°, positive, and sin(−415°) = −sin 55°, negative. Coterminal angles also describe the same direction for rotating objects, such as a wheel that has turned past a full revolution.

Common mistakes

• Adding 180° instead of 360° to find a coterminal angle.
• Giving 0° and 360° as different answers for the smallest positive angle (the page uses 0° ≤ angle < 360°).
• Measuring the reference angle from the y-axis.
• Mixing degrees and radians.

Frequently asked questions

How do you find a coterminal angle?

Add or subtract 360° (or 2π) as many times as needed.

What is the reference angle of 305°?

55°, the acute angle between the terminal side and the x-axis.

Do coterminal angles have the same sine and cosine?

Yes, because they end on the same terminal side.

Sources: OpenStax, Precalculus 2e (2021), §5.1 Angles and §5.2 Unit Circle