Graph of the function intersects the axis X at f = 0
so we need to solve the equation:
$$3 \sin{\left(2 x \right)} - 16 \cos{\left(x \right)} = 0$$
Solve this equationThe points of intersection with the axis X:
Analytical solution$$x_{1} = - \frac{\pi}{2}$$
$$x_{2} = \frac{\pi}{2}$$
Numerical solution$$x_{1} = -139.800873084746$$
$$x_{2} = -1.5707963267949$$
$$x_{3} = -64.4026493985908$$
$$x_{4} = 76.9690200129499$$
$$x_{5} = -23.5619449019235$$
$$x_{6} = -58.1194640914112$$
$$x_{7} = 61.261056745001$$
$$x_{8} = 80.1106126665397$$
$$x_{9} = -48.6946861306418$$
$$x_{10} = -29.845130209103$$
$$x_{11} = -4.71238898038469$$
$$x_{12} = -86.3937979737193$$
$$x_{13} = -36.1283155162826$$
$$x_{14} = -98.9601685880785$$
$$x_{15} = 1.5707963267949$$
$$x_{16} = -39.2699081698724$$
$$x_{17} = 73.8274273593601$$
$$x_{18} = -92.6769832808989$$
$$x_{19} = 42.4115008234622$$
$$x_{20} = 67.5442420521806$$
$$x_{21} = -32.9867228626928$$
$$x_{22} = 14.1371669411541$$
$$x_{23} = 4.71238898038469$$
$$x_{24} = 32.9867228626928$$
$$x_{25} = -10.9955742875643$$
$$x_{26} = 70.6858347057703$$
$$x_{27} = 36.1283155162826$$
$$x_{28} = 20.4203522483337$$
$$x_{29} = -70.6858347057703$$
$$x_{30} = -26.7035375555132$$
$$x_{31} = 10.9955742875643$$
$$x_{32} = 23.5619449019235$$
$$x_{33} = 45.553093477052$$
$$x_{34} = 83.2522053201295$$
$$x_{35} = -67.5442420521806$$
$$x_{36} = -89.5353906273091$$
$$x_{37} = -54.9778714378214$$
$$x_{38} = 95.8185759344887$$
$$x_{39} = -17.2787595947439$$
$$x_{40} = 26.7035375555132$$
$$x_{41} = 17.2787595947439$$
$$x_{42} = 177.499984927823$$
$$x_{43} = -42.4115008234622$$
$$x_{44} = 54.9778714378214$$
$$x_{45} = -7.85398163397448$$
$$x_{46} = 48.6946861306418$$
$$x_{47} = -51.8362787842316$$
$$x_{48} = 89.5353906273091$$
$$x_{49} = 92.6769832808989$$
$$x_{50} = 58.1194640914112$$
$$x_{51} = -80.1106126665397$$
$$x_{52} = -73.8274273593601$$
$$x_{53} = 86.3937979737193$$
$$x_{54} = -76.9690200129499$$
$$x_{55} = 365.995544143211$$
$$x_{56} = 51.8362787842316$$
$$x_{57} = 350.287580875262$$
$$x_{58} = 39.2699081698724$$
$$x_{59} = -20.4203522483337$$
$$x_{60} = 64.4026493985908$$
$$x_{61} = -83.2522053201295$$
$$x_{62} = 98.9601685880785$$
$$x_{63} = 7.85398163397448$$
$$x_{64} = -95.8185759344887$$
$$x_{65} = -14.1371669411541$$
$$x_{66} = 29.845130209103$$
$$x_{67} = -45.553093477052$$
$$x_{68} = -61.261056745001$$