Size of energy gap for an electron in an area where the potential is the dirac comb

[hulet]

1. The problem statement, all variables and given/known data
Hi, I am trying to solve a question involving band structure of solids... The electron is placed in an area filled with positive ions which is represented by a dirac delta potential....

2. Relevant equations

The problem asks me to find the equivalent of the Kronig-Penney equation which I found to be Cos[2 n Pi /N] = Cos[ka] - (Beta/ka) Sin[ka] and then asks me to:
a) find values for Beta for which half the states in the first band have negative energy
b) find values for Beta for which all states in the first band have negative energy
c) for Beta=5/2 what is the size of the energy gap between the 1st and 2nd energy band


3. The attempt at a solution

I have tried plotting the above equation for different values of Beta...

I understand that there are forbidden energies for any value of the RHS below -1 or above +1... I also understand that each value of n gives us a different energy level.

but I don't understand how the graph of the RHS of the above equation tells us whether the energy of the particle is positive or negative??

i have read kronig penney threads on this forum but none describe positive or negative energies... thanks for the help