This paper presents a method for optimal trajectory planning with applications to drone cinematography. Aerial cinematography with drones is growing fast due to their maneuverability and ability to create unique visual effects. However, planning optimal drone trajectories and camera movements is still a major challenge to autonomous aerial filming. The trajectories must meet the objectives on aesthetic quality of the videos while satisfying several constraints imposed by drone dynamics, gimbal mechanical limits and surrounding obstacles. In this paper, we propose a novel formulation of the problem by decoupling the gimbal and the drone control systems. The problem is formulated as an optimization problem taking into account the gimbal rotation limits and collision avoidance constraints as well as camera angle driven objective functions to ensure feasible and smooth drone trajectories that generate visually pleasing videos. We evaluate the efficacy of our method through simulations and real-world experiments in an outdoor environment.