Cotangent Calculator - Calculate cot(x) and arccot(x)

What is the Cotangent Function?

The cotangent function, denoted as cot(x), is one of the six fundamental trigonometric functions. It is defined as the ratio of the adjacent side to the opposite side in a right triangle, or equivalently, as the reciprocal of the tangent function.

The cotangent function is widely used in mathematics, physics, engineering, and various scientific applications. It's particularly important in calculus, where it appears in derivatives and integrals of trigonometric functions.

The mathematical definition of cotangent is:

Key properties of the cotangent function include:

• Domain: cot(x) is defined for all real numbers except x = nπ, where n is any integer.
• Range: The cotangent function has a range of all real numbers (-∞, ∞).
• Periodicity: cot(x) is periodic with period π.
• Symmetry: cot(-x) = -cot(x), making it an odd function.
• Asymptotes: Vertical asymptotes occur at x = nπ.
• Derivative: The derivative of cot(x) is -csc²(x).

The cotangent function is essential in solving trigonometric equations, analyzing periodic phenomena, and in applications involving right triangles and circular motion.

What is Inverse Cotangent (Arccotangent)?

The inverse cotangent function, denoted as arccot(x) or cot⁻¹(x), is the inverse function of the cotangent. It answers the question: 'What angle has a cotangent of x?' The inverse cotangent function returns the angle whose cotangent is the given value.

The mathematical definition of inverse cotangent is:

Properties of the inverse cotangent function:

• Domain: The inverse cotangent is defined for all real numbers.
• Range: The principal value range is (0, π).
• Monotonicity: arccot(x) is strictly decreasing on its domain.
• Special values: arccot(0) = π/2, arccot(1) = π/4, arccot(√3) = π/6.
• Derivative: The derivative of arccot(x) is -1/(1 + x²).

The inverse cotangent function is particularly useful in solving trigonometric equations and in applications where you need to find angles from cotangent values.

Common Cotangent Values

Here are some important cotangent values for common angles:

• cot(0°) = undefined
• cot(30°) = √3 ≈ 1.732
• cot(45°) = 1
• cot(60°) = 1/√3 ≈ 0.577
• cot(90°) = 0
• cot(120°) = -1/√3 ≈ -0.577
• cot(135°) = -1
• cot(150°) = -√3 ≈ -1.732