Smallest Positive and Largest Negative Coterminal Angles Calculator

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°) or an integer multiple of 360°.

For example, 45° and 405° are coterminal because 405° - 360° = 45°. Similarly, -315° and 45° are coterminal because -315° + 360° = 45°.

How to Find Coterminal Angles

To find coterminal angles for a given angle θ:

1. Add or subtract 360° (or any multiple of 360°) to θ to find other coterminal angles.
2. The smallest positive coterminal angle is found by adding 360° until the result is between 0° and 360°.
3. The largest negative coterminal angle is found by subtracting 360° until the result is between -360° and 0°.

Example Calculation

For θ = 45°:

• Smallest positive coterminal angle: 45° mod 360° = 45°
• Largest negative coterminal angle: 45° - 360° = -315°

For θ = 405°:

• Smallest positive coterminal angle: 405° mod 360° = 45°
• Largest negative coterminal angle: 405° - 360° = 45° (but this is the same as the smallest positive, so we subtract another 360° to get -315°)

Using the Calculator

Our calculator makes it easy to find coterminal angles:

1. Enter the angle in degrees in the input field.
2. Click "Calculate" to see the results.
3. The calculator will display the smallest positive and largest negative coterminal angles.
4. Use the "Reset" button to clear the form.

Applications of Coterminal Angles

Coterminal angles are used in various fields:

• Trigonometry: Simplifying trigonometric calculations by working with angles between 0° and 360°.
• Navigation: Determining directions and positions using periodic functions.
• Engineering: Analyzing periodic systems and signals.
• Computer Graphics: Creating smooth animations and rotations.