Coterminal Angle Calculator

How to Use This Calculator

1. Enter any angle in degrees (positive, negative, or > 360°).
2. The smallest positive and negative coterminals are shown instantly.
3. Use Multiple tab to list several coterminals in a range.
4. Use Verify to check if two angles are coterminal.

Formula

Example

Frequently Asked Questions

• What are coterminal angles? ▼
  Two angles are coterminal if they share the same terminal side when placed in standard position on the coordinate plane — that is, their initial side is along the positive x-axis and they point in the same final direction. Coterminal angles differ by integer multiples of 360° (or 2π radians). For example, 30°, 390°, 750°, and −330° are all coterminal because each one points to the same location in the plane. Visually, rotating 30° once, twice, or backward by 330° all end at the same terminal ray. Any angle has infinitely many coterminal angles. This concept is important in trigonometry because trig functions are periodic — they repeat every full rotation — so coterminal angles always produce identical sine, cosine, and tangent values.
• How do I find the smallest positive coterminal angle? ▼
  To find the smallest positive coterminal angle, use the formula: positive coterminal = ((θ mod 360) + 360) mod 360, where θ is the input in degrees. The double-mod step handles both positive and negative inputs correctly. Example 1: θ = 405°. 405 mod 360 = 45, then (45 + 360) mod 360 = 45°. Example 2: θ = −100°. −100 mod 360 = −100 (in many languages), so (−100 + 360) mod 360 = 260°. Example 3: θ = −400°. −400 mod 360 = −40, so (−40 + 360) mod 360 = 320°. A common pitfall is simply subtracting 360 once from a large negative angle — for values more negative than −360° that single subtraction still leaves a negative result.
• What is the general formula for all coterminal angles? ▼
  The general formula for all angles coterminal with a given angle θ is: θ + 360° × n, where n is any integer (positive, negative, or zero). In radians the formula is θ + 2π × n. Example: θ = 50°. The coterminal angles are …, −670°, −310°, 50°, 410°, 770°, … corresponding to n = …, −2, −1, 0, 1, 2, …. In radians: θ = π/6. Coterminals are π/6 + 2πn for all integers n, giving π/6, π/6 + 2π = 13π/6, π/6 − 2π = −11π/6, etc. A common mistake is using 180° instead of 360° as the period — that would give supplementary or reference-angle relationships, not coterminal ones.
• Do coterminal angles have the same trig values? ▼
  Yes — all coterminal angles produce identical values for every trigonometric function because sine, cosine, and tangent are periodic with period 2π (360°). Formally: sin(θ + 360°·n) = sin(θ), cos(θ + 360°·n) = cos(θ), and tan(θ + 360°·n) = tan(θ) for all integers n. Example: sin(30°) = 0.5. Coterminal angles 390° and −330° both give sin = 0.5. This periodicity is why calculators always reduce an input to a standard interval before computing trig values. Note that tangent and cotangent are also periodic with period 180° (π), so they repeat twice per full rotation. This means angles 180° apart can share the same tan value even though they are not coterminal.
• What is the application of coterminal angles? ▼
  Coterminal angles arise in any situation involving repeated rotation or periodic motion. In physics, the angular position of a spinning wheel is described modulo 360° — after 7 full rotations plus 45°, the wheel is at the same physical orientation as 45°. In electrical engineering, AC voltage is modeled as V(t) = Vₘ sin(ωt + φ), and the phase angle φ is always reduced to [0°, 360°) because coterminal phases are indistinguishable. In robotics and animation, joint angles are normalized to avoid unnecessary winding — a robot arm at 370° is repositioned to 10° without changing its pose. A common pitfall in programming is accumulating angle values without normalizing, which can lead to floating-point precision errors over many iterations.

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