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arccot

The ArcCotangent (Inverse Cotangent) Function

ArcCotangent returns the angle whose cotangent equals the given value. arccot(x) = arctan(1/x) with range (0, π). Domain: all reals. arccot(0) = π/2.

Concept Fundamentals
All reals
Domain
(0°, 180°)
Range
90°
arccot(0)
arccot(x) + arctan(x) = 90°
Complement

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arccot(0) = π/2 — when cot(θ) = 0, θ = π/2 (or 90°). arccot(x) + arctan(x) = π/2 for x > 0 — complementary angles. arccot(-x) = π - arccot(x) — reflection property.

Key quantities
All reals
Domain
Key relation
(0°, 180°)
Range
Key relation
90°
arccot(0)
Key relation
arccot(x) + arctan(x) = 90°
Complement
Key relation

Ready to run the numbers?

Why: ArcCotangent appears in calculus (d/dx[arccot(x)] = -1/(1+x²)) and when working with complementary angles.

How: arccot(x) = arctan(1/x) adjusted so the range is (0, π). For x > 0, arccot(x) = arctan(1/x). For x < 0, add π to get the principal value in (π/2, π).

arccot(0) = π/2 — when cot(θ) = 0, θ = π/2 (or 90°).arccot(x) + arctan(x) = π/2 for x > 0 — complementary angles.
Sources:Wolfram MathWorld — Inverse CotangentKhan Academy — Inverse Trig

Run the calculator when you are ready.

Start CalculatingEnter any value to find the angle whose cotangent equals it

Examples — Click to Load

arccotangent.sh
CALCULATED
$ arccot --value 1 --output degrees
arccot(x)
45°
arctan(x)
45°
Quadrant
Q1
Ref Angle
45°
cot(arccot)
1
Complement
45°
Range
(0°, 180°)
Domain
All reals ✓
Share:
ArcCotangent Calculator Result
arccot(1)
45°
Q1Ref 45°cot(arccot) = 1
numbervibe.com/calculators/mathematics/trigonometry/arccotangent-calculator

Inverse Trig Value Breakdown

arccot vs arctan

arccot vs arctan (Complementary)

Calculation Breakdown

INPUT VALIDATION
Input Value
1
Domain Check
All reals ✓
ext{arccot} ext{accepts} ext{any} ext{real} ext{number}
PRIMARY RESULT
Compute arccot
0.78539816 rad
arccot(x) = arctan(1/x) for x>0
ARCCOTANGENT RESULT
45°
arccot(1)
RELATED VALUES
Range
(0°, 180°)
ext{Open} ext{interval}, ext{excludes} 0 ext{and} 180
arctan(x)
45°
ext{arccot}(x) + ext{arctan}(x) = 90^{circ} ext{for} x > 0
Quadrant
Q1
Reference Angle
45°
VERIFICATION
cot(arccot(x))
1
ext{Verification}: ext{equals} ext{input}

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

🧮 Fascinating Math Facts

📐

arccot(x) = arctan(1/x) adjusted for range (0, π) — different from arctan convention.

— Wolfram MathWorld

↔️

arccot(0) = π/2 — the unique angle where cotangent equals zero.

— Paul's Notes

Key Takeaways

  • • arccot(x) = arctan(1/x) adjusted for range. Domain: all reals, Range: (0, π)
  • • arccot(0)=90°, arccot(1)=45°, arccot(√3)=30°, arccot(-1)=135°. arccot(-x) = π - arccot(x) for x > 0
  • • arccot(x) + arctan(x) = π/2 for x > 0 — complementary angles. cot(arccot(x)) = x
  • • Unlike arcsin and arccos, arccot accepts any real number — same as arctan
  • • The range (0, π) ensures a unique inverse. arccot(0) = π/2 by convention (cot approaches ±∞ at 0 and π)

Did You Know?

📐arccot gives the angle of a line with "run/rise" instead of "rise/run". Useful when you have horizontal/vertical ratiosSource: Khan Academy
🔄arccot(x) and arctan(x) are complementary: arccot(x) + arctan(x) = 90° for x > 0. For x < 0, arccot gives Q2 angleSource: Paul's Notes
📊Two conventions exist: (0, π) used here vs (-π/2, π/2) excluding 0. The (0, π) convention is common in US textbooksSource: Wolfram MathWorld
🔬d/dx[arccot(x)] = -1/(1+x²). Same magnitude as arctan derivative but opposite signSource: MIT OCW
📡In control theory and signal processing, arccot appears in phase margin and Bode plot calculationsSource: IEEE
🎓arccot(0) = 90° by convention because cot(θ)→∞ as θ→0⁺ and cot(θ)→-∞ as θ→π⁻. We choose 90° as the "principal" valueSource: NIST

How ArcCotangent Works

Since cot(θ) = 1/tan(θ), we have arccot(x) = arctan(1/x) with quadrant adjustment. For x < 0, we add π to keep the result in (0, π). arccot(0) = π/2 by convention.

Range (0, π)

The range (0, π) ensures a unique inverse. Cotangent has period π, so we restrict to one period. This gives angles in Q1 and Q2 only.

Relation to arctan

arccot(x) + arctan(x) = π/2 for x > 0. For x < 0, arccot(x) = π + arctan(1/x) to stay in (0, π). They are complementary.

arccot(0) Convention

cot(θ) approaches ±∞ as θ approaches 0 or π. By convention, arccot(0) = π/2 (90°) — the midpoint of the range.

Expert Tips

Memorize Key Values

arccot(0)=90°, arccot(1)=45°, arccot(√3)=30°, arccot(1/√3)=60°. Use the Unit Circle Calculator.

No Domain Restrictions

arccot accepts any real number, like arctan. Try the ArcTangent Calculator for the related function.

Reflection for Negative

arccot(-x) = π - arccot(x) for x > 0. So arccot(-1) = 180° - 45° = 135°.

Verify with cot(arccot(x))

cot(arccot(x)) = x for all real x. Use the Cotangent Calculator to confirm.

Inverse Trig Calculator Comparison

FeatureArcCotangentArcTangentArcCosine
DomainAll realsAll reals[-1, 1]
Range(0, π)(-π/2, π/2)[0, π]
Complementarctan(x)arccot(x)arcsin(x)
At x=090°0°90°
At x=145°45°0°
Reflectionarccot(-x)=π-arccot(x)arctan(-x)=-arctan(x)arccos(-x)=π-arccos(x)
QuadrantsQ1, Q2Q1, Q4Q1, Q2
Common usePhase, controlSlope, atan2Dot product

Frequently Asked Questions

Why is arccot(0) = 90°?

Cotangent approaches ±∞ as the angle approaches 0° or 180°. By convention, arccot(0) is defined as π/2 (90°) — the midpoint of the principal range (0, π).

How is arccot related to arctan?

arccot(x) = arctan(1/x) for x ≠ 0, with quadrant adjustment so the result lies in (0, π). For x > 0: arccot(x) + arctan(x) = π/2.

Why does arccot use range (0, π)?

Cotangent has period π. The range (0, π) ensures a unique inverse — one output per input. This convention covers Q1 and Q2.

What is arccot(-1)?

135° (or 3π/4 rad). arccot(-1) = π - arccot(1) = 180° - 45° = 135°. The reflection property gives Q2 angles for negative inputs.

Can arccot output negative angles?

No. The principal range is (0, π), so arccot always returns 0° to 180°. For negative angles, use arctan.

What is the derivative of arccot?

d/dx[arccot(x)] = -1/(1+x²). Same form as arctan but with a negative sign. The integral ∫ -dx/(1+x²) = arccot(x) + C.

Are there different arccot conventions?

Yes. Some use range (-π/2, π/2) excluding 0. The (0, π) convention used here is common in US calculus textbooks and matches the principal value for continuity.

When would I use arccot?

In control theory (phase margin), signal processing, and when working with cotangent-based ratios. Less common than arctan in everyday use but completes the six inverse trig functions.

ArcCotangent by the Numbers

ℝ
Domain
(0°, 180°)
Range
π/2
arccot+arctan
Q1, Q2
Quadrants

Disclaimer: This calculator provides results based on standard IEEE 754 floating-point arithmetic. Results are accurate to approximately 15 significant digits. We use the (0, π) principal value convention for arccot.

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