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Coterminal Angles — Same Terminal Side

Find angles that share the same terminal side in standard position. Coterminal angles differ by multiples of 360° (or 2π radians) and have identical trig values.

Concept Fundamentals
θ₂ = θ₁ + n×360°
Coterminal formula
θ₂ = θ₁ + n×2π
In radians
360° or 2π
Period
sin, cos, tan identical
Same trig values

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Coterminal angles differ by 360° or 2π radians They have identical trig function values Used to simplify angles to standard range 0°–360° Negative angles: subtract 360° to get positive coterminal Custom range: θ_cot = θ − ⌊(θ−a)/(b−a)⌋×(b−a)

Key quantities
θ₂ = θ₁ + n×360°
Coterminal formula
Key relation
θ₂ = θ₁ + n×2π
In radians
Key relation
360° or 2π
Period
Key relation
sin, cos, tan identical
Same trig values
Key relation

Ready to run the numbers?

Why: Coterminal angles are fundamental in trigonometry — they share the same point on the unit circle and thus identical sine, cosine, and tangent. Used for simplifying angles and understanding periodicity.

How: Add or subtract multiples of 360° (or 2π) to get coterminal angles. The calculator can find angles in a custom range [a, b) or list positive/negative coterminals.

Coterminal angles differ by 360° or 2π radiansThey have identical trig function values
Sources:Wolfram MathWorld - Coterminal AngleKhan Academy - Trigonometry

Run the calculator when you are ready.

Coterminal Angle CalculatorEnter any angle to find coterminal angles
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θ₂ = θ₁ + n×360°

Coterminal Angle Calculator

Find angles that share the same terminal side. Enter any angle and get coterminal angles in your chosen range.

📐 Sample Examples — Click to Load

Input

°
1–10
°
°
coterminal.sh
CALCULATED
$ coterminal --angle=45° --range=[0,360)
Original
45°
In Range
45°
Negative
-315°
Others
-315°, 405°
Share:
Coterminal Angles
θ = 45°
45°in range
Negative: -315°Formula: θ + n×360°
numbervibe.com/calculators/mathematics/angle/coterminal-angle-calculator

Coterminal Angles Comparison

Coterminals by Quadrant

Angle Magnitudes (Scaled)

Step-by-Step Breakdown

INPUT
Given angle
θ = 45°
INPUT
Target range
[0°, 360°)
CONVERT
Range size
360° - 0° = 360°
RESULT
Angle already in range
45°
DERIVED
Negative coterminal
45° - 360° = -315°
FORMULA
General formula
θ_coterminal = 45° + n × 360°
ext{where} n ext{is} ext{any} ext{integer}

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

🔄

Coterminal angles land on the same point of the unit circle

— Trigonometry

📐

sin(390°) = sin(30°) — coterminal angles share trig values

— Khan Academy

⏱️

Angles can be simplified to 0°–360° or -180°–180° for analysis

— Wolfram MathWorld

🎯

Negative angles: -30° is coterminal with 330°

— OpenStax

Key Takeaways

  • Coterminal angles share the same terminal side when placed in standard position on the coordinate plane
  • Formula: θ₂ = θ₁ + n × 360° (or n × 2π in radians), where n is any integer
  • Coterminal angles have identical trigonometric function values: sin(45°) = sin(405°) = sin(-315°)
  • Every angle has infinitely many coterminal angles — add or subtract multiples of 360°
  • Principal angle is the coterminal angle in the range [0°, 360°) — the most common representation

Did You Know?

🔄45° and 405° are coterminal — one full rotation (360°) separates them. Both point to the same spot on the unit circle.Source: Khan Academy
📐The tangent function has period π (180°), so coterminal angles differing by 180° also share the same tan value.Source: Wolfram MathWorld
✈️Aircraft headings use coterminal angles: 370° is the same as 10° — both indicate slightly east of north.Source: Aviation applications
⚡AC circuit phase angles use coterminal equivalence: 45° and 405° represent the same phase relationship between voltage and current.Source: Electrical engineering
🎯Polar coordinates (r, θ) and (r, θ + 2nπ) represent the same point — coterminal angles in polar form.Source: Coordinate geometry
⏰Clock hands repeat positions every 12 hours — coterminal in the time-angle analogy (360° per 12 hours).Source: Time measurement

How Coterminal Angles Work

Coterminal angles differ by multiples of 360° (or 2π radians). To find a coterminal angle in a specific range [a, b):

Step 1: Compute range size

s = b - a (e.g., 360° - 0° = 360°)

Step 2: Normalize the angle

Add or subtract multiples of s until the angle falls in [a, b). Use: θ_coterminal = θ - ⌊(θ - a) / s⌋ × s

Step 3: Negative coterminal

Subtract the range size: θ_negative = θ_coterminal - s

General formula

All coterminal angles: θ + n × 360° (or n × 2π), where n ∈ ℤ

Expert Tips

Use Principal Angle for Trig

When evaluating sin/cos/tan, convert to [0°, 360°) first — it simplifies lookup on the unit circle.

Physics Uses ±180°

Many physics problems use [-180°, 180°) for angles — e.g., phase angles and directional bearings.

Radians: Use 2π

In radians, the period is 2π. Coterminal angles differ by n × 2π. Convert degrees to radians: θ_rad = θ_deg × π/180.

Reference vs Coterminal

Reference angle is the acute angle to the x-axis (0°–90°). Coterminal angles share the same reference angle.

Coterminal vs Reference Angles

ConceptDefinitionExample (θ = 135°)
CoterminalSame terminal side; differ by n×360°135°, 495°, -225°
ReferenceAcute angle to x-axis (0°–90°)45°

Frequently Asked Questions

What is the principal coterminal angle?

The coterminal angle in the range [0°, 360°). For 420°, it is 60°; for -45°, it is 315°.

How many coterminal angles does an angle have?

Infinitely many. Every angle θ has coterminal angles θ + n×360° for all integers n.

Can negative angles be coterminal with positive angles?

Yes. -45° is coterminal with 315° because -45° + 360° = 315°.

Why do coterminal angles have the same trig values?

They share the same terminal side on the unit circle, so (cos θ, sin θ) is identical. This is the basis of trig periodicity.

What range do physicists use?

Often [-180°, 180°) for phase angles and directional measurements.

How do I find coterminal in a custom range [a, b)?

Use θ_cot = θ - ⌊(θ - a) / (b - a)⌋ × (b - a). Add or subtract the range size until the angle falls in [a, b).

What is the difference between coterminal and reference angles?

Coterminal angles have the same terminal side. Reference angle is the acute angle (0°–90°) to the x-axis — used to find trig values in any quadrant.

How accurate is this calculator?

Uses standard floating-point. Results are displayed with 2 decimal places. For critical applications, verify with domain tools.

Coterminal Angles by the Numbers

360°
One full rotation
2π
Radians equivalent
∞
Coterminals per angle
Same
sin, cos values

Disclaimer: This calculator provides coterminal angles using the standard formula θ₂ = θ₁ + n×360°. Results use floating-point precision. For critical trigonometry or engineering applications, verify with domain-specific tools. Not a substitute for professional analysis.

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