Graph of the function intersects the axis X at f = 0
so we need to solve the equation:
$$2 \sin{\left(\frac{x}{2} + \frac{\pi}{6} \right)} + 1 = 0$$
Solve this equationThe points of intersection with the axis X:
Analytical solution$$x_{1} = - \frac{2 \pi}{3}$$
$$x_{2} = 2 \pi$$
Numerical solution$$x_{1} = -1024.15920507027$$
$$x_{2} = -18.8495559215388$$
$$x_{3} = -56.5486677646163$$
$$x_{4} = 43.9822971502571$$
$$x_{5} = 69.1150383789755$$
$$x_{6} = 10.471975511966$$
$$x_{7} = -94.2477796076938$$
$$x_{8} = 182.212373908208$$
$$x_{9} = 85.870199198121$$
$$x_{10} = -43.9822971502571$$
$$x_{11} = 60.7374579694027$$
$$x_{12} = -90.0589894029074$$
$$x_{13} = -31.4159265358979$$
$$x_{14} = -14.6607657167524$$
$$x_{15} = -6.28318530717959$$
$$x_{16} = 31.4159265358979$$
$$x_{17} = 45312.2380402768$$
$$x_{18} = 94.2477796076938$$
$$x_{19} = 81.6814089933346$$
$$x_{20} = 56.5486677646163$$
$$x_{21} = -2.0943951023932$$
$$x_{22} = 35.6047167406843$$
$$x_{23} = 98.4365698124802$$
$$x_{24} = 23.0383461263252$$
$$x_{25} = -69.1150383789755$$
$$x_{26} = -52.3598775598299$$
$$x_{27} = -102.625360017267$$
$$x_{28} = -27.2271363311115$$
$$x_{29} = -81.6814089933346$$
$$x_{30} = -64.9262481741891$$
$$x_{31} = 48.1710873550435$$
$$x_{32} = -39.7935069454707$$
$$x_{33} = 18.8495559215388$$
$$x_{34} = -77.4926187885482$$
$$x_{35} = 73.3038285837618$$
$$x_{36} = 6.28318530717959$$