0-360 Coterminal Calculator

Find coterminal angles between 0° and 360° with our precise calculator. Coterminal angles are angles that share the same terminal side when drawn in standard position. This calculator helps you determine all coterminal angles within the 0° to 360° range for any given angle.

What are coterminal angles?

Coterminal angles are angles that have the same terminal side when drawn in standard position. In other words, they are angles that differ by a full rotation (360°) of the unit circle. For example, 45° and 405° are coterminal because 405° - 360° = 45°.

Coterminal angles are useful in trigonometry, navigation, and various engineering applications where angles are measured in degrees or radians. By finding coterminal angles, you can simplify calculations and comparisons between angles.

Key Points

1. Coterminal angles share the same terminal side on the unit circle.

2. They differ by integer multiples of 360°.

3. The reference angle is the smallest positive angle that is coterminal with a given angle.

How to find coterminal angles

To find coterminal angles for a given angle θ, you can use the following formula:

Formula

Coterminal angle = θ + (360° × n), where n is any integer (positive, negative, or zero).

This formula allows you to find all coterminal angles by adding or subtracting full rotations (360°) to the original angle. For example:

Example

If θ = 45°, then:

For n = 1: 45° + (360° × 1) = 405°
For n = -1: 45° + (360° × -1) = -315°
For n = 2: 45° + (360° × 2) = 765°

When working with angles between 0° and 360°, you can find all coterminal angles by adding or subtracting 360° until you reach the desired range. For example, to find coterminal angles between 0° and 360° for θ = 405°:

Example

405° - 360° = 45° (which is between 0° and 360°)

This process ensures that you find all coterminal angles within the specified range.

Example calculations

Let's look at a few examples to illustrate how to find coterminal angles between 0° and 360°.

Example 1: θ = 90°

Find all coterminal angles between 0° and 360° for θ = 90°.

Solution:

90° is already between 0° and 360°.
Adding 360°: 90° + 360° = 450° (which is outside the range)
Subtracting 360°: 90° - 360° = -270° (which is outside the range)

Therefore, the only coterminal angle between 0° and 360° is 90° itself.

Example 2: θ = 405°

Find all coterminal angles between 0° and 360° for θ = 405°.

Solution:

Subtract 360°: 405° - 360° = 45° (which is between 0° and 360°)

Therefore, the coterminal angle between 0° and 360° is 45°.

Example 3: θ = -180°

Find all coterminal angles between 0° and 360° for θ = -180°.

Solution:

Add 360°: -180° + 360° = 180° (which is between 0° and 360°)

Therefore, the coterminal angle between 0° and 360° is 180°.

FAQ

What is the difference between coterminal and equivalent angles?

Coterminal angles are angles that share the same terminal side when drawn in standard position, differing by integer multiples of 360°. Equivalent angles, on the other hand, are angles that are equal in measure, such as 45° and 45° + 360° × n.

How do coterminal angles relate to the unit circle?

Coterminal angles all terminate at the same point on the unit circle. This means they have the same sine and cosine values, making them equivalent in trigonometric calculations.

Can coterminal angles be negative?

Yes, coterminal angles can be negative. For example, -90° and 270° are coterminal because 270° - 360° = -90°. However, when working with angles between 0° and 360°, you can convert negative angles to their positive coterminal equivalents by adding 360°.

What is the reference angle for coterminal angles?

The reference angle is the smallest positive angle that is coterminal with a given angle. It is used to simplify trigonometric calculations by reducing any angle to its equivalent between 0° and 90°.