What values of m as a function of q satisfy this trigonometric equation?

[LmdL]

I have a trigonometric equation
2\sin \left ( \frac{q\pi }{m} \right )-\sin \left ( \frac{q\pi }{2} \right )=0
and want to know what values m as a function of q could take to satisfy the equation. Both terms zero is the obvious solution: q=2n; m=2; n is an integer. But there are more solutions. I tried to use different kinds of trigonometric identities, with no luck.
The best I could get is
m=\frac{q\pi}{\arcsin \left ( \frac{1}{2}\sin \left ( \frac{q\pi }{2} \right ) \right )}
which for q=2n gives q/m is an integer.
Is there a more elegant general solution?

Thanks!

 

[gmax137]

Are q and m integers?

 

[LmdL]

In general, for my purpose, both are continuous.

 

[mfb]

You found an expression of m as function of q. What else do you need?
You can add +2 pi * k at a suitable place (where you take the arcsine) to get the other solutions for m.

 

[LmdL]

I'm in doubt if there is a simpler expression involving q,m,π and probably n (integer). Without trigonometric functions.

 

[mfb]

That would surprise me.