Cotangent (cot)

What is cotangent

Cotangent is the reciprocal of tangent, i.e. cos α / sin α. It expresses the inverse ratio – adjacent to opposite leg.

Domain and properties

The domain is all real numbers except k·180° (where sin = 0). The range is all real numbers. The period is 180° and the function is odd.

Notable values

cot 30° = √3 ≈ 1.732, cot 45° = 1, cot 60° = √3/3 ≈ 0.577, cot 90° = 0

Where cotangent is used in practice

• A companion to tangent when computing angles and slopes, whenever the reciprocal is more convenient.
• Geodesy and surveying when computing angles.
• Electrical engineering – phase shifts in AC circuits.

The unit circle and graphs

The best way to understand trigonometric functions is the unit circle – a circle of radius 1 centred at the origin. Moving along it by an angle α, the coordinates of the point are exactly (cos α, sin α). This gives the fundamental identity sin²α + cos²α = 1 (the Pythagorean theorem for that point). The graph of the function is obtained by plotting the function value against the angle – producing the characteristic wave.

Relationships between the functions

tan α = sin α / cos α  •  cot α = 1 / tan α  •  sin²α + cos²α = 1