Coterminal Angle Calculator

Find any coterminal angle, visualize standard position on the unit circle, and understand the fundamental concept of infinite equivalent angles.

30° 135° -60° 420° -300° π/3 rad 3π/2 rad -π/4 rad -2π/3 rad -5π/6 rad
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Local & private: All calculations and drawings happen in your browser. Zero data transfer.You can type expressions like π/3, -2π/3, 3π/2 (use "pi" or "π"). Then click π button or just press Enter.
Given angle: 45.00° | Standard position (0°–360°): 45.00°
Principal coterminal angles

Minimum positive coterminal: 45.00°

Maximum negative coterminal: -315.00°

General formula: θ + 360°·k, k ∈ ℤ

Custom coterminal: 405.00°
Terminal side Standard initial side Angle arc

Understanding Coterminal Angles: The Infinite Winding Path

In trigonometry, coterminal angles are angles that share the same initial side (positive x‑axis) and the same terminal side. They differ only by an integer multiple of a full rotation — 360° (or 2π radians). This concept is essential for simplifying angle measures, evaluating trigonometric functions of any angle, and solving periodic equations.

Coterminal Angle Theorem

Two angles α and β are coterminal iff α − β = 360°·k (degrees) or α − β = 2π·k (radians) for some integer k.

How to find coterminal angles – step by step

  1. Identify the given angle (in degrees or radians).
  2. To find a positive coterminal angle, add 360° (or 2π) repeatedly until the result is between 0° and 360° (or 0 and 2π).
  3. To find a negative coterminal angle, subtract 360° (or 2π) until the result lies between -360° and 0° (or -2π and 0).
  4. General expression: θ ± 360°·k (degrees) or θ ± 2π·k (radians), where k is any integer.

Why coterminal angles matter

  • Trigonometric functions: sin(θ) = sin(θ + 360°k) because sine and cosine have a period of 360°. Understanding coterminal angles allows you to evaluate functions for any angle.
  • Navigation & rotation: Angles greater than 360° appear in physics (angular velocity) and engineering. Representing them as coterminal angles simplifies analysis.
  • Precalculus & calculus: Using reference angles and coterminal forms makes evaluating integrals of periodic functions easier.

Coterminal angles: typical examples & patterns

Given angle Min positive coterminal Max negative coterminal General form (degrees)
390° 30° -330° 30° + 360°k
-120° 240° -120° 240° + 360°k
750° 30° -330° 30° + 360°k
5π/3 rad 5π/3 rad -π/3 rad 5π/3 + 2πk
-π/2 rad 3π/2 rad -π/2 rad 3π/2 + 2πk
Real‑world application: robotics & angular motion

In robotics, a joint rotating 780° from its home position is indistinguishable from a rotation of 60° (since 780° − 720° = 60°) when only final orientation matters. Engineers use coterminal angles to normalize sensor readings and simplify inverse kinematics. Our visual unit circle confirms that 780° and 60° share the exact terminal side — a powerful conceptual bridge.

Degrees vs. Radians: converting & normalizing

Coterminal angles work identically in both systems. To convert an angle from degrees to radians, multiply by π/180. For radian measures, add/subtract 2π multiples until the angle lies in the desired range [0, 2π) or (-2π, 0]. The calculator automatically normalizes and shows both representations where relevant.

A reference angle is always acute (between 0° and 90°) and measures the smallest angle to the x‑axis. A coterminal angle is another angle that shares the same terminal side but can have any magnitude. Both help simplify trig evaluation, but reference angle focuses on the acute distance, while coterminal angles represent equivalence under full rotations.

Infinitely many. By adding or subtracting any integer multiple of 360° (or 2π rad) you generate a new coterminal angle. For example, 30°, 390°, 750°, -330°, -690°, etc., are all coterminal.

Yes, as long as they represent the same geometric rotation. For instance, 90° and π/2 rad are coterminal because they describe the same terminal side. Our calculator converts automatically between units so you can compare.

The unit circle shows the terminal side, angle arc, and helps you verify that coterminal angles map to identical points (cosθ, sinθ). Visual learners grasp the periodic nature instantly.

Fourier series, signal processing, complex exponentials (e^{iθ}), and solving trigonometric equations all rely on angle periodicity. Coterminal recognition reduces arguments to principal values.
References: Wolfram MathWorld — Coterminal Angle; Sullivan, M. "Trigonometry: A Unit Circle Approach" (Pearson); OpenStax Precalculus. Verified by GetZenQuery team Team — May 2026.