Cotangent Calculator

What is Cotangent?

The reciprocal of tangent, cotangent is written cot(θ) = cos(θ) / sin(θ) = 1 / tan(θ). Inside a right triangle, it is the ratio of the adjacent to the opposite side.

Some notable values: cot(30°) ≈ 1.7321, cot(45°) = 1, cot(60°) ≈ 0.5774. Note that cotangent goes undefined at 0°, 180° and integer multiples of 180°.

Unit conversions: 1° = π/180 rad ≈ 0.01745 rad. The other way around, 1 rad = 180°/π ≈ 57.296°. And 400 gradians correspond to 360°.

The cotangent function

By definition, cot(θ) = cos(θ)/sin(θ) = 1/tan(θ). It shares the tangent's period of π, but the vertical asymptotes land in different places: wherever the sine drops to zero, that is, at θ = kπ. The function is odd. A few values worth memorizing: cot(30°) = √3, cot(45°) = 1, cot(60°) = √3/3 and cot(90°) = 0. To see where one of them comes from, take θ = 45°: cot(45°) = cos(45°)/sin(45°) = (√2/2)/(√2/2) = 1.

Applications: integrals, signals and optics

You won't meet the cotangent as early as sine, cosine or tangent in Brazilian secondary school, but it turns up often once you reach integrals, integral transforms and series expansions. Snell's law for refraction can be rewritten in terms of cotangents when the geometry is pinned to the surface normal. Signal processing relies on it for filter design and phase relations, and in numerical methods it gives you alternative closed forms for derivatives that sidestep awkward divisions by zero.

FAQ

Is cotangent just 1/tan? Wherever both are defined, yes. Where they part ways is the domain: cot has a value at θ = 90° (it's 0) but blows up at θ = 0°, and the tangent does exactly the reverse.

Why isn't it taught more in school? Most right-triangle geometry can be handled with sine, cosine and tangent alone. Cotangent, secant and cosecant earn their keep in calculus and physics, which is why they show up later in the curriculum.

What's the derivative? (cot x)' = −csc²x.

Common identity? 1 + cot²θ = csc²θ, which is just the Pythagorean identity after you divide everything by sin²θ.