Unmanned aerial vehicles (UAVs)-assisted communications have been an essential complement of conventional wireless networks. In this paper, we consider a UAV-enabled multicasting system, where a UAV with a directional antenna of adjustable beamwidth is employed to disseminate a common file to a group of ground users. By minimizing the mission completion time, we investigate how beamwidth control would affect the UAV's three-dimensional (3D) location/trajectory. First, we consider the quasi-stationary UAV scenario, where the UAV is deployed at a static location. In this case, we jointly optimize the 3D UAV location and antenna beamwidth under the practical constraints on the UAV's altitude and beamwidth, while ensuring that all users are covered by the main lobe of the UAV antenna. Although this problem is nonconvex, its global optimality can be obtained by using a two-step algorithm, where semi-closed solutions of the 3D location and beamwidth are derived. Next, in the mobile UAV scenario, a joint 3D UAV trajectory and beamwidth design is proposed, additionally constrained by the horizontal and vertical speed. To tackle this nonconvex problem, we develop an iterative optimization algorithm based on successive convex approximation techniques, which updates the 3D trajectory and beamwidth simultaneously in each iteration. Numerical results demonstrate the effectiveness of the proposed design as compared to benchmark schemes.