Coterminal Angle — Definition, Formula & Examples
A coterminal angle is an angle that shares the same initial side and terminal side as another angle but differs by one or more full rotations. For example, 30° and 390° are coterminal because 390° is just 30° plus one complete 360° turn.
Two angles in standard position are coterminal if and only if they differ by an integer multiple of (or radians). That is, angles and are coterminal when for some integer .
Key Formula
Where:
- = The original angle in degrees
- = Any integer (positive, negative, or zero)
How It Works
To find a coterminal angle, add or subtract (or radians) as many times as needed. Adding gives a positive coterminal angle; subtracting gives a negative one. If you need the coterminal angle between and , keep adding or subtracting until the result falls in that range. Since coterminal angles point in the same direction, they produce identical values for all six trigonometric functions.
Worked Example
Problem: Find a positive coterminal angle for −150° that lies between 0° and 360°.
Step 1: Add 360° to the given angle.
Step 2: Check that the result is between 0° and 360°. Since 210° falls in this range, we are done.
Answer: The positive coterminal angle is 210°.
Why It Matters
Coterminal angles appear constantly in precalculus and physics whenever you need to simplify an angle to a standard range. Evaluating trig functions, solving trig equations, and analyzing periodic motion all rely on recognizing that angles separated by full rotations are functionally the same.
Common Mistakes
Mistake: Adding or subtracting 180° instead of 360° to find a coterminal angle.
Correction: A half-turn (180°) gives a supplementary direction, not the same terminal side. Always use full rotations of 360° (or radians).