Graph of the function intersects the axis X at f = 0
so we need to solve the equation:
$$e^{x} \cos{\left(x \right)} = 0$$
Solve this equationThe points of intersection with the axis X:
Analytical solution$$x_{1} = - \frac{\pi}{2}$$
$$x_{2} = \frac{\pi}{2}$$
Numerical solution$$x_{1} = 14.1371669411541$$
$$x_{2} = -92.6769832808989$$
$$x_{3} = -48.6946861306418$$
$$x_{4} = -23.5619449019235$$
$$x_{5} = -86.3937979737193$$
$$x_{6} = -17.2787595947439$$
$$x_{7} = 20.4203522483337$$
$$x_{8} = -26.7035375555132$$
$$x_{9} = -64.4026493985908$$
$$x_{10} = -83.2522053201295$$
$$x_{11} = -4.71238898038469$$
$$x_{12} = -42.4115008234622$$
$$x_{13} = -29.845130209103$$
$$x_{14} = 17.2787595947439$$
$$x_{15} = 1.5707963267949$$
$$x_{16} = -67.5442420521806$$
$$x_{17} = -36.1283155162826$$
$$x_{18} = -80.1106126665397$$
$$x_{19} = -73.8274273593601$$
$$x_{20} = -10.9955742875643$$
$$x_{21} = -1.5707963267949$$
$$x_{22} = -98.9601685880785$$
$$x_{23} = -20.4203522483337$$
$$x_{24} = -105.243353895258$$
$$x_{25} = -32.9867228626928$$
$$x_{26} = -39.2699081698724$$
$$x_{27} = -58.1194640914112$$
$$x_{28} = -61.261056745001$$
$$x_{29} = 4.71238898038469$$
$$x_{30} = -76.9690200129499$$
$$x_{31} = -95.8185759344887$$
$$x_{32} = -51.8362787842316$$
$$x_{33} = 23.5619449019235$$
$$x_{34} = -14.1371669411541$$
$$x_{35} = -7.85398163397448$$
$$x_{36} = 7.85398163397448$$
$$x_{37} = -45.553093477052$$
$$x_{38} = 26.7035375555132$$
$$x_{39} = -89.5353906273091$$
$$x_{40} = 10.9955742875643$$
$$x_{41} = -70.6858347057703$$
$$x_{42} = -54.9778714378214$$
$$x_{43} = 29.845130209103$$