filter design - 3dB-Cut off frequency of moving average

I've read the thread: Cut off frequency of moving average

And I use the second answer in my algorithm to calculate the 3dB cut off frequency of my filter, which works great, as my filter length is usually above 300. I verified it with the step response.

But I would like to have a source or derivation for this formula.

I tried by hand with taylor series stopping after the second and third term. I come close but not exactly to the formula and mapple gives me a valid but extremly complicated result.

Hope you guys can help.

kind regards

Slev1n