Mathematics - Inverse Trigonometric Functions Question with Solution | TestHub

Concepts: ${\cot }^{-1}(x)$ is decreasing, so ${\cot }^{-1}(x)>\frac{\pi }{6}$ implies $x<\cot (\frac{\pi }{6})$.

Solution:

Since ${\cot }^{-1}(\frac{n}{\pi })>\frac{\pi }{6}$, then $\frac{n}{\pi }<\cot (\frac{\pi }{6})=\sqrt{3}$. Thus, $n<\pi \sqrt{3}\approx 5.44$. Since $n\in N$, the maximum value of $n$ is 5.

Final Answer: The final answer is $\boxed{5}$