Let's see how your function looks in a neighborhood around y = 0.
f = @(beta) 7*exp(-0.008*beta)+0.008*sin(7*beta)-7*cos(7*beta);
fplot(f);
yticks(0);
grid on
ylim([-0.5 0.5]);
beta = 0 does look like a solution, but it is not the only solution. Calling fzero with different initial guesses may allow you to calculate those other solutions. If you think about your function you should be able to calculate roughly how far apart the local minima of your function should be, and that can inform your choice of starting points for fzero.
