Cotangent Calculator

Calculate cotangent values for any angle in degrees or radians. Visualize cotangent function and understand trigonometric properties with interactive graphs.

Cotangent Function Definition: In a right triangle, cot(θ) = adjacent / opposite

Using sine and cosine functions: cot(θ) = cos(θ) / sin(θ)

Cotangent is the reciprocal of tangent: cot(θ) = 1 / tan(θ)

Relationship to Tangent: cot(θ) × tan(θ) = 1 for all angles where both functions are defined.

When tan(θ) is large, cot(θ) is small, and vice versa.

Enter an angle value in your chosen unit
Common angles:
0°
30°
45°
60°
90°
120°
135°
180°
Radians:
0
π
π/2
π/3
π/4
2π
Cotangent Function Parameters
Vertical scaling factor (default: 1)
Number of cycles per unit (default: 1)
Horizontal shift (default: 0)
Calculating...

Understanding the Cotangent Function

The cotangent function is one of the reciprocal trigonometric functions, relating an angle of a right triangle to the ratio of the length of the adjacent side to the opposite side. It is the reciprocal of the tangent function.

Mathematical Definition:

In a right triangle with angle θ:

cot(θ) = adjacent / opposite

Using sine and cosine functions:

cot(θ) = cos(θ) / sin(θ)

As reciprocal of tangent:

cot(θ) = 1 / tan(θ)

The cotangent function is periodic with period π radians (180°), and has vertical asymptotes where sin(θ) = 0.

Cotangent Function Properties

Property Description Mathematical Expression
Domain All real numbers except where sin(θ) = 0 θ ≠ nπ, n ∈ ℤ
Range All real numbers (-∞, ∞)
Period Repeats every π radians cot(θ + π) = cot(θ)
Amplitude No amplitude (unbounded) N/A
Symmetry Odd function cot(-θ) = -cot(θ)
Zeros Where cot(θ) = 0 θ = π/2 + nπ, n ∈ ℤ
Asymptotes Vertical asymptotes where undefined θ = nπ, n ∈ ℤ
Relationship to tan Reciprocal function cot(θ) × tan(θ) = 1

Common Cotangent Values

Angle (degrees) Angle (radians) cot(θ) Exact Value
0° 0 ∞ Undefined
30° π/6 1.7321 √3
45° π/4 1 1
60° π/3 0.5774 √3/3
90° π/2 0 0
120° 2π/3 -0.5774 -√3/3
135° 3π/4 -1 -1
180° π ∞ Undefined

Cotangent Function Applications

1

Engineering & Physics: Cotangent appears in various physics equations, particularly in wave mechanics, alternating current circuits, and structural engineering calculations.

2

Coordinate Geometry: The cotangent of an angle represents the slope of a line perpendicular to one with slope tan(θ). In coordinate geometry, lines with slopes m₁ and m₂ are perpendicular if m₁ × m₂ = -1, which relates to the cotangent-tangent relationship.

3

Calculus & Analysis: The derivative of tangent is sec²θ, while the derivative of cotangent is -csc²θ. These relationships are fundamental in calculus when differentiating trigonometric functions.

4

Trigonometric Identities: Cotangent is essential in trigonometric identities and simplifying complex trigonometric expressions. It often appears in integration problems and series expansions.

Frequently Asked Questions

Cotangent is defined as cot(θ) = cos(θ)/sin(θ). At 0° (0 radians) and 180° (π radians), sin(θ) = 0, which would result in division by zero. Therefore, the cotangent function is undefined at these angles, creating vertical asymptotes in its graph.

Cotangent and tangent are reciprocal functions. This means cot(θ) = 1/tan(θ) and tan(θ) = 1/cot(θ), provided neither function is zero. They are also complementary: tan(θ) = cot(90° - θ) or tan(θ) = cot(π/2 - θ).

The cotangent function has a period of π radians (180°), which is the same as the tangent function. This means cot(θ + π) = cot(θ) for all values of θ where the function is defined. This period is half the period of sine and cosine functions.

The cotangent function has vertical asymptotes at θ = nπ, where n is any integer. These occur where sin(θ) = 0, making the function undefined. The graph approaches positive or negative infinity as it nears these asymptotes from either side.

For common angles (0°, 30°, 45°, 60°, 90°), memorize the exact values. For other angles, you can calculate cot(θ) = cos(θ)/sin(θ) using known sine and cosine values. You can also use the reciprocal relationship cot(θ) = 1/tan(θ) if you know the tangent value. Another method is to use trigonometric identities or reference angles. For example, cot(120°) = cot(180° - 60°) = -cot(60°) = -√3/3.