Cotangent Calculator

Enter an angle in degrees or radians to find its cotangent — cot(θ) = cos(θ) / sin(θ) = 1 / tan(θ) — along with the equivalent angle, tangent, and quadrant.

Quick Facts

Formula
cot(θ) = cos(θ) / sin(θ) = 1 / tan(θ)
Also equal to adjacent / opposite for an acute angle in a right triangle.
Undefined points
θ = 0°, 180°, 360°, ... (multiples of 180°)
Cotangent is undefined wherever sin(θ) = 0.
Period
180° (π radians)
cot(θ) repeats every 180°, half the period of sine or cosine.

Your Results

Calculated
Cotangent cot(θ)
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cos(θ) / sin(θ)
Equivalent angle
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Same angle, other unit
Tangent tan(θ)
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Reciprocal of cotangent
Quadrant
-
Position on the unit circle

Ready

Enter an angle and unit, then press Calculate.

Understanding the Cotangent Function

Cotangent is one of the six basic trigonometric ratios. For an angle θ, it is defined as the ratio of the cosine to the sine of that angle, or equivalently the adjacent side divided by the opposite side in a right triangle. This calculator evaluates cot(θ) directly from an angle you enter in degrees or radians.

The formula

Cotangent has three equivalent definitions that all give the same result:

  • Ratio form: cot(θ) = cos(θ) / sin(θ)
  • Reciprocal form: cot(θ) = 1 / tan(θ)
  • Right-triangle form: cot(θ) = adjacent / opposite, for an acute angle θ in a right triangle

Common reference values

  • cot(30°) = √3 ≈ 1.732
  • cot(45°) = 1
  • cot(60°) = 1/√3 ≈ 0.577
  • cot(90°) = 0
  • cot(0°) and cot(180°) are undefined

Why cotangent is sometimes undefined

Because cot(θ) = cos(θ) / sin(θ), the function is undefined wherever sin(θ) = 0 — that is, at θ = 0°, 180°, 360°, and every multiple of 180° (0, π, 2π, ... radians). Near those angles cot(θ) grows without bound (its graph has a vertical asymptote), and this calculator reports "Undefined" instead of a number when you enter one of those angles.

Reading the quadrant

The sign of cot(θ) depends on which quadrant the angle falls in: positive in Quadrant I and Quadrant III (where sine and cosine share the same sign), and negative in Quadrant II and Quadrant IV (where they have opposite signs). The calculator reports the quadrant alongside the value so you can quickly check whether the sign makes sense.

Frequently Asked Questions

What is cotangent?
Cotangent is a trigonometric ratio defined as cot(θ) = cos(θ) / sin(θ), which is the same as 1 / tan(θ). In a right triangle, it equals the length of the side adjacent to angle θ divided by the length of the side opposite it.
How is cotangent related to tangent?
Cotangent is the reciprocal of tangent: cot(θ) = 1 / tan(θ), provided tan(θ) is not zero. They always share the same sign, so cot(θ) is positive exactly where tan(θ) is positive (Quadrants I and III) and negative where tan(θ) is negative (Quadrants II and IV).
Why is cotangent undefined at 0° and 180°?
Cotangent equals cos(θ) / sin(θ), and division by zero is undefined. Since sin(θ) = 0 at θ = 0°, 180°, 360°, and every multiple of 180° (every multiple of π radians), cotangent has no value at those angles — its graph approaches a vertical asymptote instead.
What is the period of cotangent?
Cotangent repeats every 180° (π radians), which is half the period of sine or cosine (360°). That means cot(θ) = cot(θ + 180°) for any angle θ where both are defined.

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