Coterminal Angle Calculator
Find coterminal angles, reference angles, and analyze quadrant positions with step-by-step solutions. Our trigonometry calculator supports both degrees and radians for comprehensive angle analysis.
Last updated: December 15, 2024
Need a custom trigonometry calculator for your educational platform? Get a Quote
Enter angle in degrees
Generate 1-10 coterminal angles
Angle Analysis
Coterminal Angles:
405.0000°
765.0000°
1125.0000°
Reference Angle:
45.0000°
Quadrant:
I
Analysis:
Positive coterminal angles are found by adding multiples of 360° to the original angle. These angles have the same terminal side and trigonometric function values.
Calculation Steps:
- Original angle: 45°
- Formula: θ + n × 360° where n > 0
- 45 + 1 × 360 = 405.0000°
- 45 + 2 × 360 = 765.0000°
- 45 + 3 × 360 = 1125.0000°
Coterminal Angles:
- • Definition: Angles that share the same terminal side
- • Formula: θ ± n × 360° (n = integer)
- • Same trig values: sin(θ) = sin(θ + 360°)
- • Reference angle: Acute angle to x-axis
Quick Example Result
For angle 45°, positive coterminal angles:
405°, 765°, 1125°
Reference angle: 45°, Quadrant: I
How This Calculator Works
Our coterminal angle calculator applies fundamental trigonometric principles to find angles that share the same terminal side. The calculator uses angle relationshipsto generate coterminal angles, determine reference angles, and analyze quadrant positions.
Coterminal Angle Formulas
θ ± n × 360°θ ± n × 2πAcute angle to x-axis (0° ≤ θᵣ ≤ 90°)These formulas generate infinitely many coterminal angles by adding or subtracting full rotations. The reference angle is always the acute angle between the terminal side and the x-axis, regardless of which quadrant the angle is in.
Shows coterminal angles and their shared terminal sides on the unit circle
Trigonometric Foundation
Coterminal angles are fundamental to understanding trigonometric functions and their periodic nature. Since trigonometric functions repeat every 360° (or 2π radians), coterminal angles have identical sine, cosine, and tangent values. This concept is essential for solving trigonometric equations and understanding function behavior.
- Coterminal angles have identical trigonometric function values
- Reference angles help determine the magnitude of trigonometric functions
- Quadrant analysis determines the signs of trigonometric functions
- Standard position provides a consistent framework for angle measurement
Sources & References
- Trigonometry - Ron Larson, Robert P. Hostetler (10th Edition)Comprehensive treatment of angle relationships and coterminal angles
- National Council of Teachers of Mathematics - Trigonometry Teaching StandardsEducational guidelines for teaching angle concepts and relationships
- Khan Academy - Trigonometry and Unit CircleEducational resources on coterminal angles and trigonometric concepts
Need help with other trigonometry calculations? Check out our inverse tangent calculator and angle between vectors calculator.
Get Custom Calculator for Your PlatformExample Analysis
Given Angle:
- Initial bearing: 135° (Southeast)
- Need: Equivalent bearings
- Application: Navigation systems
Coterminal Analysis:
- Positive: 135° + 360° = 495°
- Negative: 135° - 360° = -225°
- Reference angle: 180° - 135° = 45°
- Quadrant: III (both sin and cos negative)
Result: All angles (135°, 495°, -225°) represent the same direction
In navigation, coterminal angles represent the same compass direction. Whether you use 135°, 495°, or -225°, you're pointing southeast. The reference angle of 45° helps calculate distances and trigonometric values for navigation computations.
Frequently Asked Questions
Found This Calculator Helpful?
Share it with others who need help with trigonometry and angle calculations
Suggested hashtags: #Trigonometry #Mathematics #CoterminalAngles #Angles #Calculator