Coterminal Angle Calculator
Find coterminal angles that share the same terminal side in standard position
Calculate Coterminal Angles
The angle to find coterminal angles for
Results
Normalized Angle (0° to 360° or 0 to 2π)
Positive Coterminal Angles
Negative Coterminal Angles
General formula: θ ± k × 360° (where k is any integer)
Your angle: 0° ± k × 360°
Example Calculations
Finding Coterminal Angles
Given angle: 420°
Normalized: 420° - 360° = 60°
Positive coterminal: 60° + 360° = 420°, 60° + 720° = 780°
Negative coterminal: 60° - 360° = -300°, 60° - 720° = -660°
Checking if Coterminal
Angle 1: 45°
Angle 2: 405°
Difference: 405° - 45° = 360°
Result: 360° ÷ 360° = 1 (integer) → Coterminal ✓
Negative Angle Example
Given angle: -858°
Calculation: -858° + 3×360° = -858° + 1080° = 222°
Normalized: 222° (between 0° and 360°)
Quick Reference
Definition
Coterminal angles share the same terminal side when in standard position
Formula
θ ± k × 360° (degrees)
θ ± k × 2π (radians)
where k is any integer
Standard Position
Initial side along positive x-axis, vertex at origin
Common Examples
Tips & Applications
Coterminal angles have identical trigonometric values
Use the smallest positive coterminal angle for calculations
Important in physics for rotational motion
Used in navigation and GPS calculations
Understanding Coterminal Angles
What are Coterminal Angles?
Coterminal angles are angles that share the same terminal side when placed in standard position. The standard position means one side (initial side) lies along the positive x-axis, and the vertex is at the origin. Despite having different measures, coterminal angles end up pointing in the same direction.
Key Properties
- •Differ by complete rotations (360° or 2π rad)
- •Have identical trigonometric function values
- •Can be positive or negative
- •Infinitely many coterminal angles exist for any angle
How to Find Coterminal Angles
For degrees: θ ± k × 360°
For radians: θ ± k × 2π
where k is any positive integer
Applications
- Trigonometry: Simplifying calculations using equivalent angles
- Physics: Rotational motion and periodic phenomena
- Engineering: Gear ratios and mechanical systems
- Navigation: Compass bearings and directional calculations
Important Notes
Difference from Reference Angles: Reference angles are always between 0° and 90°, while coterminal angles can be any size but share the same terminal side.
Trigonometric Equivalence: sin(θ) = sin(θ ± 360°k), and the same applies to all trigonometric functions.