Largest Negative Coterminal Angle Calculator

Find the largest negative coterminal angle for any given angle using our calculator. Coterminal angles share the same terminal side in standard position and differ by full rotations (360° or 2π radians). This calculator helps you determine the largest negative angle that is coterminal with your input angle.

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° or 2π radians) to an initial angle.

For any given angle θ, its coterminal angles can be expressed as:

Coterminal Angle Formula

θ + 360° × n, where n is any integer

Or in radians: θ + 2π × n, where n is any integer

The largest negative coterminal angle occurs when n is the most negative integer that keeps the angle between -360° and 0° (or -2π and 0 radians).

How to Find the Largest Negative Coterminal Angle

To find the largest negative coterminal angle for a given angle θ:

Determine the reference angle by finding the remainder when θ is divided by 360° (or 2π radians).
If the reference angle is positive, subtract 360° (or 2π radians) to get the largest negative coterminal angle.
If the reference angle is negative, add 360° (or 2π radians) until you get the largest negative angle.

Example

For θ = 45°:

45° ÷ 360° = 0.125 with a remainder of 45°

Since 45° is positive, subtract 360° to get -315° as the largest negative coterminal angle.

Example Calculation

Let's find the largest negative coterminal angle for 210°:

210° ÷ 360° = 0.583 with a remainder of 210°
Since 210° is positive, subtract 360° to get -150°

The largest negative coterminal angle for 210° is -150°.

Note

In radians, the calculation would be similar, using 2π instead of 360°.

FAQ

What is the difference between coterminal and reference angles?

Coterminal angles share the same terminal side and differ by full rotations. Reference angles are the smallest positive angle that a terminal side makes with the x-axis, regardless of quadrant.

Can coterminal angles be negative?

Yes, coterminal angles can be negative. The largest negative coterminal angle is the one closest to -360° (or -2π radians) that maintains the same terminal side as the original angle.

How do I convert between degrees and radians for coterminal angles?

Use the conversion factor π radians = 180°. For example, to convert 360° to radians: 360° × (π/180°) = 2π radians.