Graph of the function intersects the axis X at f = 0
so we need to solve the equation:
$$- \tan{\left(\frac{x}{3} + \frac{\pi}{6} \right)} = 0$$
Solve this equationThe points of intersection with the axis X:
Analytical solution$$x_{1} = - \frac{\pi}{2}$$
Numerical solution$$x_{1} = 64.4026493985908$$
$$x_{2} = 54.9778714378214$$
$$x_{3} = 73.8274273593601$$
$$x_{4} = -20.4203522483337$$
$$x_{5} = 102.101761241668$$
$$x_{6} = -10.9955742875643$$
$$x_{7} = 7.85398163397448$$
$$x_{8} = -67.5442420521806$$
$$x_{9} = -48.6946861306418$$
$$x_{10} = -76.9690200129499$$
$$x_{11} = -29.845130209103$$
$$x_{12} = 17.2787595947439$$
$$x_{13} = 45.553093477052$$
$$x_{14} = 83.2522053201295$$
$$x_{15} = -58.1194640914112$$
$$x_{16} = -39.2699081698724$$
$$x_{17} = -86.3937979737193$$
$$x_{18} = 92.6769832808989$$
$$x_{19} = -95.8185759344887$$
$$x_{20} = 36.1283155162826$$
$$x_{21} = 26.7035375555132$$
$$x_{22} = -1.5707963267949$$