Coterminal Angle Calculator

Find coterminal angles quickly and accurately

Enter an angle to find its coterminal angles. Coterminal angles share the same terminal side but differ by full rotations (360°, 2π, or 400 gradians).

Examples

  • 30° → 390°, 750°, 1110° (positive)
  • 45° → -315°, -675°, -1035° (negative)
  • π/4 → 9π/4, 17π/4, 25π/4 (radians)
  • 100g → 500g, 900g, 1300g (gradians)

Important Note

Coterminal angles have the same trigonometric function values since they share the same terminal position on the unit circle.

Other Titles
Understanding Coterminal Angle Calculator: A Comprehensive Guide
Explore coterminal angles, their mathematical properties, and applications in trigonometry and periodic functions

Understanding Coterminal Angle Calculator: A Comprehensive Guide

  • Coterminal angles share the same terminal side position
  • They differ by complete rotations of 360°, 2π, or 400 gradians
  • Essential concept in trigonometry and periodic functions
Coterminal angles are angles that share the same terminal side when drawn in standard position, differing only by complete rotations.
Two angles are coterminal if their difference is a multiple of a full rotation: 360° in degrees, 2π in radians, or 400 in gradians.
Coterminal angles have identical trigonometric function values since they represent the same point on the unit circle.

Basic Examples

  • 30° and 390° are coterminal (differ by 360°)
  • π/6 and 13π/6 are coterminal (differ by 2π)
  • -45° and 315° are coterminal
  • 50g and 450g are coterminal (gradians)

Step-by-Step Guide to Using the Coterminal Angle Calculator

  • Input angles in your preferred unit
  • Generate multiple coterminal angles
  • Understand positive and negative coterminals
Our calculator generates both positive and negative coterminal angles by adding and subtracting full rotations.
Calculation Method:
  • Positive Coterminals: Add 360°, 720°, 1080° (or equivalent in other units)
  • Negative Coterminals: Subtract 360°, 720°, 1080° (or equivalent in other units)

Calculation Examples

  • For 45°: Positive coterminals are 405°, 765°, 1125°
  • For 45°: Negative coterminals are -315°, -675°, -1035°
  • For π/3: Add/subtract 2π, 4π, 6π for coterminals

Real-World Applications of Coterminal Angle Calculations

  • Engineering: Rotational machinery and cyclic processes
  • Physics: Periodic motion and wave analysis
  • Computer Graphics: Animation and rotation cycles
Coterminal angles are essential in analyzing periodic phenomena and rotational systems.
Engineering Applications:
  • Rotational Machinery: Understanding equivalent positions in rotating equipment
  • Control Systems: Managing cyclic processes and feedback loops

Application Examples

  • Turbine blade: 30° and 390° positions are functionally identical
  • Clock hands: 3:00 and 15:00 represent coterminal hour hand positions

Common Misconceptions and Correct Methods

  • Understanding the difference between coterminal and supplementary angles
  • Recognizing when angles are NOT coterminal
Students often confuse coterminal angles with other angle relationships.
Key Distinctions:
  • Coterminal: Differ by full rotations (360°, 2π, 400g)
  • Supplementary: Add up to 180° (completely different concept)

Clarification Examples

  • Correct: 50° and 410° are coterminal
  • Wrong: 50° and 130° are NOT coterminal (they're supplementary)

Mathematical Properties and Formulas

  • General formula for coterminal angles
  • Unit conversions and relationships
The general formula for coterminal angles: θ + n × (full rotation), where n is any integer.
Unit-Specific Formulas:
  • Degrees: θ + n × 360°
  • Radians: θ + n × 2π
  • Gradians: θ + n × 400g

Formula Examples

  • Formula application: 75° + 2×360° = 795°
  • Radian example: π/4 + 3×2π = 25π/4
  • Gradian example: 150g + 1×400g = 550g