Cotangent (cot)

Cotangent $\cot\theta = \frac{1}{\tan\theta} = \frac{\cos\theta}{\sin\theta}$.

Domain: $\theta \neq k\pi$. Range: all real numbers.

Right triangle: $\cot\theta = \frac{\text{adjacent}}{\text{opposite}}$.

Period: $\pi$ (same as tangent).

Pythagorean identity: $1 + \cot^2\theta = \csc^2\theta$.

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

Integral: $\int \cot x \, dx = \ln|\sin x| + C$.

Cotangent has vertical asymptotes at $\theta = k\pi$ and zeros at $\theta = \pi/2 + k\pi$. It's a "decreasing" version of tangent: from just past $0$ to just before $\pi$, $\cot$ decreases from $+\infty$ to $-\infty$.

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