For which values of the parameter n>0 does the curve

For which values of the parameter $n>0$ does the curve $y=\arctan nx$ intersect the $Ox$ axis at an angle greater than $89°$?

Step 1: Intersection and slope

$y=0 \Leftrightarrow x=0$. With $y'=\dfrac{n}{1+(nx)^2}$, the slope at the origin is $k=y'(0)=n$.

Step 2: Angle condition

The angle with $Ox$ satisfies $\tan\theta=|k|=n$; requiring $\theta>89°$: \[n > \tan89° = \cot1° \approx 57.3. \tag{1}\]