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.
Why This Mathematical Concept Matters
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.
- ●arccot(-x) = π - arccot(x) — reflection property.
Examples — Click to Load
Inverse Trig Value Breakdown
arccot vs arctan
arccot vs arctan (Complementary)
Calculation Breakdown
⚠️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?
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
| Feature | ArcCotangent | ArcTangent | ArcCosine |
|---|---|---|---|
| Domain | All reals | All reals | [-1, 1] |
| Range | (0, π) | (-π/2, π/2) | [0, π] |
| Complement | arctan(x) | arccot(x) | arcsin(x) |
| At x=0 | 90° | 0° | 90° |
| At x=1 | 45° | 45° | 0° |
| Reflection | arccot(-x)=π-arccot(x) | arctan(-x)=-arctan(x) | arccos(-x)=π-arccos(x) |
| Quadrants | Q1, Q2 | Q1, Q4 | Q1, Q2 |
| Common use | Phase, control | Slope, atan2 | Dot 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
Official & Educational Sources
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.