The only variable on which E may depend is R, the distance from the line of charge. There is no angular dependence because the line is cylindrically symmetric, i.e. it does not matter at which angle about the line you view it. There is no axial dependence because the line of charge is infinitely long, i.e. there is no preferred position along the line.
Similar arguments determine that the direction of E is directly away from the line, perpendicular to it at all locations. There may be no azimuthal or axial components because the line is infinite and has no extent in the transverse directions.
Please note that these results were also obtained from the direct evaluation of the charge distribution!