# Trigonometry equation restructuring!

I would prefer help tonight, if possible, as huge test tomorrow!!!

How do I rewrite this in terms of n?

$$\frac{4A}{s^2n} = \cot \frac{\pi }{n}$$

Thank you!!!!

EDIT***: n is always a whole number greater than or equal to 3, as it represents the number of sides of a shape. Also, A and s must both be positive as they represent area and side length.

EDIT 2: Now, I am just looking for the inverse of function $y = x\cot \frac{\pi }{x}$ after using substitution. I would also accept something for $y = x\tan \frac{\pi }{x}$ if that makes any difference

-
With difficulty. I strongly doubt there's a nice closed form here. – user61527 Feb 26 '14 at 5:19
Where did this problem come from? – Mhenni Benghorbal Feb 26 '14 at 5:28
Remember the famous formula for area of any regular polygon, Area =(1/2)(apothem)(side length)(number of sides)? My precalc honors teacher offered extra credit to anyone who could find eliminate the variable of the apothem and create three equations: one to solve for each variable. S and A were easy; for reference, the equation for area given the side length and number is – louie mcconnell Feb 26 '14 at 5:46
whoops. I posted a little too soon. here is the real comment: Remember the famous formula for area of any regular polygon, Area =(1/2)(apothem)(side length)(number of sides)? My precalc honors teacher offered extra credit to anyone who could find eliminate the variable of the apothem and create three equations: one to solve for each variable. S and A were easy; for reference, the equation for area given the side length and number is $$A = \left ( \frac{s^2n}{4} \right ) \tan \frac{\left ( \pi \left ( n-2 \right ) \right )}{2n}$$ maybe I went wrong somewhere in the derivation? – louie mcconnell Feb 26 '14 at 5:58
i mean the derivation of the equation in the question, not the comment. I had that one checked. – louie mcconnell Feb 26 '14 at 5:59

There is no explicit solution for this equation in $n$. In order tomake this equation "nicer", you could always rewrite is as $$\tan (x)=\frac{s^2}{4\pi A} x=k x$$ with $x=\frac{\pi}{n}$ and look for solution of $x$ in the range $0<x<\frac{\pi}{2}$. For sure, there is no solution if $k<1$ except $x=0$.
Ik $k>1$, there is one solution in the considered interval but it needs to be searched by numerical methods such as Newton.