Least Positive Coterminal Angle Calculator

This calculator helps you find the least positive coterminal angle for any given angle in degrees. Coterminal angles share the same terminal side in standard position and differ by full rotations (360°). Understanding coterminal angles is essential in trigonometry, navigation, and engineering applications.

What is a Coterminal Angle?

Coterminal angles are angles that share the same terminal side when drawn in standard position. In other words, they can be obtained by adding or subtracting full rotations (360°) to an initial angle.

For example, 45° and 405° are coterminal because 405° - 360° = 45°. Similarly, -225° and 135° are coterminal because -225° + 360° = 135°.

The concept of coterminal angles is fundamental in trigonometry and periodic functions. It helps simplify calculations involving angles that are not within the standard range of 0° to 360°.

How to Find the Least Positive Coterminal Angle

To find the least positive coterminal angle for a given angle θ:

If θ is already positive and less than 360°, it is its own least positive coterminal angle.
If θ is positive and greater than or equal to 360°, subtract 360° repeatedly until the result is between 0° and 360°.
If θ is negative, add 360° repeatedly until the result is between 0° and 360°.

This process ensures you find the smallest positive angle that is coterminal with the original angle.

Formula

The formula to find the least positive coterminal angle θ' for a given angle θ is:

θ' = θ mod 360° If θ' is negative, add 360° to θ'

Where:

θ is the original angle in degrees
θ' is the least positive coterminal angle
mod represents the modulo operation

Worked Example

Let's find the least positive coterminal angle for -120°.

1. Calculate -120° mod 360°:

-120° + 360° = 240°

2. Since 240° is positive and less than 360°, it is the least positive coterminal angle.

Result: 240°

Another example: 420°

1. Calculate 420° mod 360°:

420° - 360° = 60°

2. 60° is positive and less than 360°, so it is the least positive coterminal angle.

Result: 60°

FAQ

What is the difference between coterminal and equivalent angles?: Coterminal angles share the same terminal side and differ by full rotations (360°). Equivalent angles are coterminal angles that are equal in measure, such as 0° and 360°.
How do coterminal angles relate to trigonometric functions?: Since trigonometric functions are periodic with a period of 360°, coterminal angles produce the same function values. For example, sin(45°) = sin(405°).
Can coterminal angles be negative?: Yes, coterminal angles can be negative. However, the least positive coterminal angle is always non-negative and less than 360°.
What are some practical applications of coterminal angles?: Coterminal angles are used in navigation, engineering, and physics to simplify angle calculations and ensure consistency in measurements.
How do I find all coterminal angles for a given angle?: To find all coterminal angles, add or subtract any integer multiple of 360° to the original angle. For example, all coterminal angles for 45° are 45° + 360°n, where n is any integer.