Arccotangent Function

Approach: We need an angle $\theta \in (0, \pi)$ with $\cot \theta = \sqrt{3}$.

Steps:

1. Among the special angles, $\cot \frac{\pi}{6} = \sqrt{3}$;
2. $\frac{\pi}{6} \approx 0.524$, which lies in $(0, \pi)$;
3. By uniqueness on the principal interval, $\operatorname{arccot} \sqrt{3} = \frac{\pi}{6}$.

Answer: $\operatorname{arccot} \sqrt{3} = \frac{\pi}{6}$.

Approach:

The arccotangent is the inverse of the cotangent function restricted to its principal interval $(0, \pi)$: the domain comes from the range of $\cot x$, and the range is the principal interval itself.

Steps:

1. The cotangent function $y = \cot x$ takes every real value on $(0, \pi)$, so its range is $\mathbb{R}$ and the domain of $y = \operatorname{arccot} x$ is $\mathbb{R}$.
2. On that interval the cotangent function decreases from $+\infty$ to $-\infty$, so choosing $(0, \pi)$ as the principal interval makes the inverse exist and stay single-valued.
3. The inverse maps the angles of the principal interval back onto $\mathbb{R}$, so the range of $y = \operatorname{arccot} x$ is $(0, \pi)$.

Answer:

Domain: $\mathbb{R}$; Range: $(0, \pi)$.