First Positive Coterminal Angle Calculator

Coterminal angles are angles that share the same terminal side when drawn in standard position. The first positive coterminal angle is the smallest positive angle that is coterminal with a given angle. This calculator helps you find it quickly and understand the concept.

What is a Coterminal Angle?

Coterminal angles are angles that have the same terminal side when drawn in standard position. In other words, they differ by a full rotation (360° or 2π radians) of each other.

For example, 45° and 405° are coterminal because 405° - 360° = 45°. Similarly, -90° and 270° are coterminal because -90° + 360° = 270°.

Key Points

Coterminal angles share the same terminal side
They differ by integer multiples of 360° (degrees) or 2π (radians)
All angles coterminal with a given angle have the same sine and cosine values

How to Find the First Positive Coterminal Angle

The first positive coterminal angle is the smallest positive angle that is coterminal with the given angle. To find it:

If the angle is already positive and less than 360°, it is its own first positive coterminal angle
If the angle is negative, add 360° repeatedly until you get a positive angle
If the angle is positive but greater than or equal to 360°, subtract 360° repeatedly until you get an angle between 0° and 360°

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

Formula

First Positive Coterminal Angle Formula

For an angle θ in degrees:

If θ ≥ 0 and θ < 360°: First positive coterminal angle = θ

If θ < 0: First positive coterminal angle = θ + 360° × ceil(-θ/360°)

If θ ≥ 360°: First positive coterminal angle = θ - 360° × floor(θ/360°)

The same logic applies to radians, using 2π instead of 360°.

Worked Example

Let's find the first positive coterminal angle for -120°:

Since -120° is negative, we use the formula: -120° + 360° × ceil(120/360°)
ceil(120/360°) = ceil(0.333) = 1
First positive coterminal angle = -120° + 360° × 1 = 240°

Therefore, the first positive coterminal angle for -120° is 240°.

FAQ

What is the difference between coterminal and equivalent angles?

Coterminal angles share the same terminal side, while equivalent angles are the same in measure (e.g., 90° and 90°). All coterminal angles are equivalent in their trigonometric values (sine, cosine, tangent).

Can the first positive coterminal angle be greater than 360°?

No, by definition, the first positive coterminal angle must be between 0° and 360° (or 0 and 2π radians).

How do I find all coterminal angles for a given angle?

To find all coterminal angles, add or subtract 360° (or 2π radians) repeatedly to the original angle. The first positive coterminal angle is the smallest positive angle in this set.