Coterminal Angle Calculator

The Coterminal Angle Calculator finds all coterminal angles for any given angle — angles that share the same terminal side on the unit circle despite having different degree measures. This calculator provides the smallest positive coterminal angle, the largest negative coterminal angle, and multiple additional coterminal angles in both directions.

What Are Coterminal Angles?

Two angles are coterminal if they share the same initial side and the same terminal side when drawn in standard position. This happens when the angles differ by a multiple of 360° (one full rotation):

$$\theta_1 \text{ and } \theta_2 \text{ are coterminal if } \theta_2 = \theta_1 + 360°n \text{ for some integer } n$$

For example, 45°, 405°, and −315° are all coterminal because they all end at the same position on the unit circle.

Finding the Standard Coterminal Angle

The smallest positive coterminal angle is the unique value in the range $$[0°, 360°)$$ that is coterminal with the given angle. It is found using the modular arithmetic formula:

$$\theta' = ((\theta \bmod 360) + 360) \bmod 360$$

The largest negative coterminal angle is the unique value in the range $$(-360°, 0°]$$ coterminal with the given angle, calculated as $$\theta' - 360°$$ (or $$-360°$$ if $$\theta' = 0$$).

Generating Multiple Coterminal Angles

Once you have the standard coterminal angle $$\theta'$$, you can generate infinitely many coterminal angles by adding or subtracting multiples of 360°:

$$\text{Positive: } \theta' + 360°, \; \theta' + 720°, \; \theta' + 1080°, \ldots$$

$$\text{Negative: } (\theta' - 360°) - 360°, \; (\theta' - 360°) - 720°, \ldots$$

Why Coterminal Angles Are Important

Coterminal angles have identical trigonometric values because they point in the same direction. This property is fundamental to understanding the periodicity of trigonometric functions:

$$\sin(\theta + 360°n) = \sin\theta, \quad \cos(\theta + 360°n) = \cos\theta$$

In real-world applications, coterminal angles appear whenever rotation is involved. A wheel that turns 405° has ended in the same position as one that turned 45°. In navigation, a heading of −90° is the same as 270° (due west).

Radians and Coterminal Angles

In radians, coterminal angles differ by multiples of $$2\pi$$:

$$\theta_2 = \theta_1 + 2\pi n$$

For instance, $$\frac{\pi}{4}$$ and $$\frac{9\pi}{4}$$ are coterminal because $$\frac{9\pi}{4} = \frac{\pi}{4} + 2\pi$$.

How to Use This Calculator

Enter any angle in degrees. The calculator finds the standard positive coterminal angle (0°–360°), the standard negative coterminal angle (−360°–0°), and three additional coterminal angles in each direction. The quadrant and radian equivalent are also displayed.