Coterminal Angles: Definition, Positive and Negative Examples, and Practice

What are coterminal angles?

Coterminal angles are angles that end on the same final ray.

They may spin different amounts, but their terminal side points in the same direction.

For example, 45°, 405°, and -315° are coterminal because each one ends on the same ray.

Coterminal angle rule

A full turn brings an angle back to the same direction.

In degrees, add or subtract 360°.

In radians, add or subtract 2π.

If the angle unit feels new, review degrees and radians first.

Formula in degrees

Use this formula to make more coterminal angles in degrees.

coterminal angle = θ + 360°k

The letter k is any integer, such as -2, -1, 0, 1, or 2.

Adding 360° makes one extra full turn. Subtracting 360° makes one full turn in the opposite direction.

Formula in radians

Radians use the same idea, but the full turn is written as 2π.

coterminal angle = θ + 2πk

For example, π/3, 7π/3, and -5π/3 are coterminal because they differ by full turns of 2π.

Degree examples

Start with 120°.

A positive coterminal angle is 120° + 360° = 480°.

A negative coterminal angle is 120° - 360° = -240°.

So 120°, 480°, and -240° all end in the same direction.

Radian examples

Start with π/4.

A positive coterminal angle is π/4 + 2π = 9π/4.

A negative coterminal angle is π/4 - 2π = -7π/4.

All three angles end on the same terminal side.

Coterminal angles and circle diagrams

Coterminal angles are easiest to see on a circle because each full turn comes back to the same starting line.

This is related to circle angle ideas such as a central angle, where the vertex sits at the center.

The difference is that coterminal angles focus on final direction after full turns.

Common mistakes

Do not add 180° when you need a coterminal angle. Adding 180° points the ray in the opposite direction.

Do not mix units. Use 360° with degrees and 2π with radians.

Do not stop at only positive answers. Negative coterminal angles are valid too.

Quick practice

1. One coterminal angle with 30° is 390°.

2. One negative coterminal angle with 30° is -330°.

3. One coterminal angle with π/2 is 5π/2.

4. 210° and -150° are coterminal.