🔄 Coterminal Angle Calculator
Calculate coterminal angles (angles with the same terminal side)
Can be positive or negative, any value
Number of coterminal angles to show (1-20)
How to Use This Calculator
Enter Angle
Input any angle in degrees. It can be positive, negative, or greater than 360°.
Set Number of Results
Choose how many positive and negative coterminal angles you want to see (1-20).
Calculate
Click "Calculate Coterminal Angles" to see the normalized angle and all coterminal angles.
Formula
Coterminal Angle = Angle ± (360° × n)
Where n is any integer (0, ±1, ±2, ±3, ...)
Definition:
Coterminal angles are angles that share the same terminal side when drawn in standard position. They differ by multiples of 360° (or 2π radians).
Normalization:
To find the angle in the 0° to 360° range: Normalized = Angle % 360 (adjusting for negative angles)
Examples:
- 45° is coterminal with: 405° (45° + 360°), 765° (45° + 720°), -315° (45° - 360°), etc.
- 150° is coterminal with: 510° (150° + 360°), 870° (150° + 720°), -210° (150° - 360°), etc.
- -30° is coterminal with: 330° (-30° + 360°), 690° (-30° + 720°), -390° (-30° - 360°), etc.
Key Properties:
- All coterminal angles have the same trigonometric function values
- There are infinitely many coterminal angles (add or subtract any multiple of 360°)
- Each angle has exactly one equivalent angle between 0° and 360°
- Coterminal angles differ by integer multiples of 360°
About Coterminal Angle Calculator
The Coterminal Angle Calculator finds all angles that share the same terminal side as a given angle. Coterminal angles are important in trigonometry because they have identical trigonometric function values.
When to Use This Calculator
- Trigonometry: Find equivalent angles for trigonometric calculations
- Precalculus: Understand angle relationships and standard position
- Physics: Calculate equivalent rotation angles
- Engineering: Normalize angles in design calculations
- Navigation: Convert bearings to standard 0-360° range
- Computer Graphics: Normalize rotation angles for rendering
Why Use Our Calculator?
- ✅ Any Angle: Works with positive, negative, or large angles
- ✅ Multiple Results: Shows many coterminal angles
- ✅ Normalization: Finds the 0-360° equivalent
- ✅ Degrees & Radians: Results in both units
- ✅ 100% Accurate: Precise calculations
- ✅ Completely Free: No registration required
Understanding Coterminal Angles
Coterminal angles are angles that have the same initial and terminal sides when drawn in standard position:
- Standard Position: An angle is in standard position when its vertex is at the origin and its initial side lies along the positive x-axis.
- Terminal Side: The ray that rotates from the initial side to form the angle.
- Same Terminal Side: Coterminal angles end at the same ray, just after different numbers of full rotations.
- Infinite Many: Every angle has infinitely many coterminal angles (add or subtract any multiple of 360°).
- Trigonometric Equality: sin(θ) = sin(θ + 360°n), cos(θ) = cos(θ + 360°n) for any integer n.
Real-World Applications
Navigation: Convert compass bearings to standard 0-360° format. For example, -30° is the same as 330°.
Rotations: In mechanics and robotics, normalize rotation angles to standard ranges.
Trigonometry: Simplify calculations by using the normalized angle (0-360°) instead of larger or negative angles.
Frequently Asked Questions
What are coterminal angles?
Coterminal angles are angles that share the same terminal side when drawn in standard position. They differ by multiples of 360° (or 2π radians).
How do you find coterminal angles?
Add or subtract any multiple of 360° from the given angle. Formula: Coterminal Angle = Given Angle ± (360° × n), where n is any integer.
Are coterminal angles the same?
Coterminal angles have the same trigonometric function values (sin, cos, tan, etc.) and point in the same direction, but they may have different degree measures.
How many coterminal angles does an angle have?
Every angle has infinitely many coterminal angles. You can add or subtract any multiple of 360° to get another coterminal angle.
What is the difference between coterminal and reference angles?
Coterminal angles share the same terminal side. A reference angle is the acute angle (0-90°) between the terminal side and the nearest x-axis. Coterminal angles have the same reference angle.
How do you normalize an angle to 0-360°?
Use modulo: Normalized = Angle % 360. If the result is negative, add 360. For example, 450° % 360 = 90°, and -30° % 360 = 330° (after adding 360).
Do coterminal angles have the same sine and cosine?
Yes! All coterminal angles have identical trigonometric function values because they point in the same direction. sin(45°) = sin(405°) = sin(-315°).
Can angles in radians be coterminal?
Yes! Coterminal angles in radians differ by multiples of 2π. For example, π/4, 9π/4, -7π/4 are all coterminal (they differ by multiples of 2π).