Coterminal Angles — Definition, Formula & Examples

Q: How do you find coterminal angles?

Add or subtract 360° (or 2π radians) as many times as needed. For any angle θ, the expression θ + 360°n gives every coterminal angle, where n is any integer. Positive n rotates you forward; negative n rotates you backward.

Q: Can two coterminal angles have different signs?

Yes. A positive angle and a negative angle can be coterminal. For example, 120° and −240° are coterminal because 120° − 360° = −240°. They start from the same initial side and end on the same terminal side, just measured in opposite rotational directions.

Q: How many coterminal angles does an angle have?

Every angle has infinitely many coterminal angles. Since you can add or subtract 360° any number of times, there is no limit to the set of coterminal angles. However, exactly one coterminal angle falls in any given 360° interval, such as [0°, 360°).

Coterminal Angles

Angles which, drawn in standard position, share a terminal side. For example, 60°, -300°, and 780° are all coterminal.

Circle showing coterminal angles 60°, 780°, and -300° sharing the same terminal side, with initial side along positive x-axis.

See also

Measure of an angle, side of a polygon

Key Formula

\begin{gathered}\theta_{\text{coterminal}} = \theta + 360°\cdot n \qquad \text{(degrees)}\\\theta_{\text{coterminal}} = \theta + 2\pi n \qquad \text{(radians)}\end{gathered}

Where:

$\theta$ = The original angle
$n$ = Any integer (positive, negative, or zero)
$\theta_{\text{coterminal}}$ = An angle coterminal with θ

Worked Example

Problem: Find one positive and one negative angle coterminal with 75°.

Step 1: Use the formula with n = 1 to find a positive coterminal angle.

75° + 360°(1) = 75° + 360° = 435°

Step 2: Use the formula with n = −1 to find a negative coterminal angle.

75° + 360°(-1) = 75° - 360° = -285°

Step 3: Verify: All three angles (75°, 435°, and −285°) land on the same terminal side when drawn in standard position, so they are coterminal.

Answer: A positive coterminal angle is 435° and a negative coterminal angle is −285°.

Another Example

This example works in radians rather than degrees and shows how to reduce a large angle to its standard equivalent in the interval [0, 2π).

Problem: Determine the coterminal angle of 17π/4 that lies between 0 and 2π.

Step 1: Figure out how many full rotations (2π) fit into 17π/4. Divide 17π/4 by 2π.

\frac{17\pi/4}{2\pi} = \frac{17}{8} = 2.125

Step 2: The whole number part is 2, so subtract 2 full rotations from the original angle.

\frac{17\pi}{4} - 2\pi(2) = \frac{17\pi}{4} - \frac{16\pi}{4} = \frac{\pi}{4}

Step 3: Check that the result is in the desired range: 0 ≤ π/4 < 2π. It is, so we are done.

Answer: The coterminal angle between 0 and 2π is π/4.

Frequently Asked Questions

How do you find coterminal angles?

Add or subtract 360° (or 2π radians) as many times as needed. For any angle θ, the expression θ + 360°n gives every coterminal angle, where n is any integer. Positive n rotates you forward; negative n rotates you backward.

Can two coterminal angles have different signs?

Yes. A positive angle and a negative angle can be coterminal. For example, 120° and −240° are coterminal because 120° − 360° = −240°. They start from the same initial side and end on the same terminal side, just measured in opposite rotational directions.

How many coterminal angles does an angle have?

Every angle has infinitely many coterminal angles. Since you can add or subtract 360° any number of times, there is no limit to the set of coterminal angles. However, exactly one coterminal angle falls in any given 360° interval, such as [0°, 360°).

Coterminal Angles vs. Reference Angles

	Coterminal Angles	Reference Angles
Definition	Angles that share the same terminal side when drawn in standard position	The acute angle formed between the terminal side and the x-axis
Range of values	Can be any real number (positive, negative, greater than 360°)	Always between 0° and 90° (0 and π/2)
How many exist	Infinitely many for any given angle	Exactly one for any given angle
Formula	θ + 360°n	Depends on the quadrant of the terminal side
Purpose	Identify angles that represent the same position	Simplify trig evaluation using known acute-angle values

Why It Matters

Coterminal angles appear constantly in trigonometry because trig functions repeat every full rotation: sin(θ) = sin(θ + 360°). When solving trig equations, you express all solutions as a set of coterminal angles (e.g., θ = 30° + 360°n). Understanding coterminality is also essential in physics and engineering when working with rotating objects, phase angles in circuits, or periodic motion.

Common Mistakes

Mistake: Adding or subtracting 180° instead of 360° to find coterminal angles.

Correction: A half rotation (180°) places you on the opposite side of the circle, not the same terminal side. Always add or subtract a full rotation: 360° in degrees or 2π in radians.

Mistake: Forgetting that n can be any integer, not just 1 or −1.

Correction: Students sometimes find only one coterminal angle. Remember, you can use n = 2, 3, −2, etc. to generate infinitely many coterminal angles. When a problem asks for the angle in a specific interval (like 0° to 360°), keep adding or subtracting 360° until the result falls in that range.

Related Terms

Angle — The fundamental concept coterminal angles build on
Standard Position — The required positioning for defining coterminality
Terminal Side of an Angle — The ray that coterminal angles share
Measure of an Angle — Coterminal angles differ in measure by multiples of 360°
Reference Angle — Often found after reducing to a coterminal angle in [0°, 360°)
Radian — Alternate unit where coterminality uses multiples of 2π
Unit Circle — Visual tool showing why coterminal angles share trig values