trigonometry

Cotangent (cot)

Cotangent is the reciprocal of tangent: cot(θ) = cos(θ)/sin(θ). Domain excludes angles where sin = 0.

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Cotangent cot⁡θ=1tan⁡θ=cos⁡θsin⁡θ\cot\theta = \frac{1}{\tan\theta} = \frac{\cos\theta}{\sin\theta}.

Domain: θ≠kπ\theta \neq k\pi. Range: all real numbers.

Right triangle: cot⁡θ=adjacentopposite\cot\theta = \frac{\text{adjacent}}{\text{opposite}}.

Period: π\pi (same as tangent).

Pythagorean identity: 1+cot⁡2θ=csc⁡2θ1 + \cot^2\theta = \csc^2\theta.

Derivative: ddxcot⁡x=−csc⁡2x\frac{d}{dx}\cot x = -\csc^2 x.

Integral: ∫cot⁡x dx=ln⁡∣sin⁡x∣+C\int \cot x \, dx = \ln|\sin x| + C.

Cotangent has vertical asymptotes at θ=kπ\theta = k\pi and zeros at θ=π/2+kπ\theta = \pi/2 + k\pi. It's a "decreasing" version of tangent: from just past 00 to just before π\pi, cot⁡\cot decreases from +∞+\infty to −∞-\infty.

Like csc and sec, cotangent appears mostly in calculus and trig identity manipulation. For arithmetic, convert to cos⁡/sin⁡\cos/\sin.