Graph of the function intersects the axis X at f = 0
so we need to solve the equation:
$$\cos{\left(\frac{x + \pi}{3} \right)} - 1 = 0$$
Solve this equationThe points of intersection with the axis X:
Analytical solution$$x_{1} = - \pi$$
$$x_{2} = 5 \pi$$
Numerical solution$$x_{1} = -474.380493870955$$
$$x_{2} = -59.6902589797336$$
$$x_{3} = -3.14159276062015$$
$$x_{4} = 91.1061865286018$$
$$x_{5} = 53.4070735966335$$
$$x_{6} = -59.690264170919$$
$$x_{7} = -3.14159134496669$$
$$x_{8} = 72.2566324813595$$
$$x_{9} = -97.3893733028467$$
$$x_{10} = -40.8407029507697$$
$$x_{11} = 53.4070756953521$$
$$x_{12} = 15.7079643832$$
$$x_{13} = 34.5575205743468$$
$$x_{14} = 53.4070766577001$$
$$x_{15} = -21.9911500132853$$
$$x_{16} = 53.4070738154238$$
$$x_{17} = -59.6902596099374$$
$$x_{18} = 72.2566318621985$$
$$x_{19} = 15.7079643063945$$
$$x_{20} = 15.7079625798234$$
$$x_{21} = 53.4070745520074$$
$$x_{22} = 72.2566298163471$$
$$x_{23} = 34.5575206523395$$
$$x_{24} = -78.5398163691121$$
$$x_{25} = 91.1061881917053$$
$$x_{26} = 91.1061869214634$$
$$x_{27} = -3.14159419270109$$
$$x_{28} = -59.6902613484779$$
$$x_{29} = -40.8407032751831$$
$$x_{30} = -3.14159113256562$$
$$x_{31} = -78.5398160333126$$
$$x_{32} = 72.2566278965339$$
$$x_{33} = -40.8407049460369$$
$$x_{34} = -97.3893708890293$$
$$x_{35} = -97.3893707853007$$
$$x_{36} = 34.5575181670294$$
$$x_{37} = 72.2566323921551$$
$$x_{38} = -21.9911485864388$$
$$x_{39} = -59.6902604578483$$
$$x_{40} = -78.5398156708263$$
$$x_{41} = 53.4070751199802$$
$$x_{42} = -78.5398150881882$$
$$x_{43} = -40.8407044242062$$
$$x_{44} = 91.1061857467651$$
$$x_{45} = 91.1061854151691$$
$$x_{46} = -21.9911459637955$$
$$x_{47} = 15.7079642483061$$
$$x_{48} = 72.2566301226729$$
$$x_{49} = -78.539815215081$$
$$x_{50} = -40.8407057203169$$
$$x_{51} = -97.3893715773867$$
$$x_{52} = -3.14159207598184$$
$$x_{53} = -78.5398147949047$$
$$x_{54} = -78.5398169218883$$
$$x_{55} = 53.4070762534334$$
$$x_{56} = 15.7079647959253$$
$$x_{57} = -59.6902615980313$$
$$x_{58} = -21.9911499262929$$
$$x_{59} = -21.9911470840848$$
$$x_{60} = 91.1061868853346$$
$$x_{61} = -59.6902590363808$$
$$x_{62} = 34.5575165845019$$
$$x_{63} = 53.4070761151575$$
$$x_{64} = -3.14159321181009$$
$$x_{65} = -40.8407043841756$$
$$x_{66} = 15.7079634550617$$
$$x_{67} = -59.6902619075171$$
$$x_{68} = 53.4070754421324$$
$$x_{69} = -40.840704052337$$
$$x_{70} = -78.5398176498381$$
$$x_{71} = -21.9911493872822$$
$$x_{72} = -78.5398160333896$$
$$x_{73} = 34.5575182254197$$
$$x_{74} = 34.557519895729$$
$$x_{75} = 34.5575190194442$$
$$x_{76} = -97.3893724570633$$
$$x_{77} = 15.707961894931$$
$$x_{78} = -40.8407060333612$$
$$x_{79} = -21.9911473480329$$
$$x_{80} = 91.1061884976653$$
$$x_{81} = 15.7079659774467$$
$$x_{82} = -3.14159358974287$$
$$x_{83} = -78.5398178465509$$
$$x_{84} = 72.2566295511795$$
$$x_{85} = -21.9911476483818$$
$$x_{86} = -3.14159296600646$$
$$x_{87} = 15.7079618120813$$
$$x_{88} = 72.2566310277172$$
$$x_{89} = -3.14159378098184$$
$$x_{90} = 34.5575176700454$$
$$x_{91} = 91.1061874222829$$
$$x_{92} = -97.3893745206956$$