Arccot Calculator

Calculate arccot values for any number. Find angles whose cotangent equals a given value. Visualize arccot function and understand inverse trigonometric properties.

Inverse Relationship: arccot(x) = θ where cot(θ) = x

Alternatively: arccot(x) = π/2 - arctan(x) for x ≥ 0, and arccot(x) = 3π/2 - arctan(x) for x < 0 (depending on range convention)

x
Enter a real number to find arccot(x)
Common values:
0
1/√3
1
√3
∞
-1
-√3
Unit for the calculated angle
Different mathematical texts use different principal value ranges for arccot
When checked, shows multiple angles that satisfy cot(θ) = x
Calculating...

Understanding the Arccot Function

The arccotangent function (arccot or cot⁻¹) is the inverse function of the cotangent. It returns the angle whose cotangent is a given number. Since the cotangent function is not one-to-one over its entire domain, we restrict its domain to (0, π) radians to define the principal value of arccot.

Mathematical Definition:

For a real number x, arccot(x) is defined as:

arccot(x) = θ, where θ ∈ (0, π) and cot(θ) = x

Alternative definitions based on arctangent:

arccot(x) = π/2 - arctan(x) for all real x (most common convention)

The arccot function is continuous and decreasing on its entire domain (-∞, ∞), approaching π as x → -∞ and 0 as x → ∞.

Arccot Function Properties

Property Description Mathematical Expression
Domain All real numbers (-∞, ∞) x ∈ ℝ
Range (Principal Value) (0, π) radians (0° to 180°) 0 < θ < π
Monotonicity Strictly decreasing If x₁ > x₂, then arccot(x₁) < arccot(x₂)
Symmetry arccot(-x) = π - arccot(x) For x > 0
Relationship with arctan arccot(x) = π/2 - arctan(x) For all real x
Derivative d/dx arccot(x) = -1/(1 + x²) For all real x
Integral ∫arccot(x) dx = x·arccot(x) + ½ ln(1 + x²) + C Indefinite integral
Limits limx→±∞ arccot(x) = 0 or π Depends on direction

Common Arccot Values

x (cot θ) arccot(x) in degrees arccot(x) in radians Exact Value Angle in Triangle
√3 ≈ 1.732 30° π/6 π/6 30°-60°-90° triangle
1 45° π/4 π/4 Isosceles right triangle
1/√3 ≈ 0.577 60° π/3 π/3 30°-60°-90° triangle
0 90° π/2 π/2 Right angle
-1/√3 ≈ -0.577 120° 2π/3 2π/3 Obtuse angle
-1 135° 3π/4 3π/4 Obtuse angle
-√3 ≈ -1.732 150° 5π/6 5π/6 Obtuse angle
→∞ 0° 0 0 Acute angle limit
→-∞ 180° π π Straight angle limit

Arccot Function Applications

1

Geometry and Trigonometry: Arccot is used to find angles when the ratio of adjacent to opposite sides is known in right triangles. Essential in solving triangles and trigonometric equations.

2

Physics and Engineering: Used in calculations involving slopes, angles of elevation/depression, force components, and in electrical engineering for phase angle calculations.

3

Calculus and Analysis: Arccot appears in integration problems, differential equations, and in series expansions. Its derivative -1/(1+x²) is important in mathematical analysis.

4

Computer Science and Graphics: Used in computer graphics for calculating angles, in game development for character movement and camera angles, and in robotics for joint angle calculations.

Frequently Asked Questions

arccot and cot⁻¹ represent the same inverse cotangent function. Both notations are used interchangeably in mathematics. arccot emphasizes the "arc" concept (the angle whose cotangent is x), while cot⁻¹ uses the inverse function notation.

Different mathematical texts and computer systems use different principal value ranges for arccot. The most common convention is (0, π) which makes arccot continuous and decreasing on (-∞, ∞). An alternative is (-π/2, π/2] excluding 0, which maintains symmetry with arctan. The choice affects the function's values for negative inputs.

For all real numbers x, arccot(x) = π/2 - arctan(x) in the (0, π) range convention. This relationship comes from the cofunction identity: cot(θ) = tan(π/2 - θ). For the alternative range convention, arccot(x) = arctan(1/x) for x > 0, and arccot(x) = π + arctan(1/x) for x < 0.

In the (0, π) range convention, arccot(0) = π/2 (90°), since cot(π/2) = 0. In the alternative convention (-π/2, π/2] excluding 0, arccot(0) is undefined because 0 is excluded from the range. This calculator uses the (0, π) convention by default.

The principal value of arccot is always within (0, π) radians (or the selected range convention). However, considering the periodic nature of cotangent, the general solution to cot(θ) = x is θ = arccot(x) + kπ where k is any integer. So while the principal value is restricted, infinite other solutions exist outside the principal range.