Floating structures have been proposed to support offshore wind turbines in deep water, where environmental forcing could subject the rotor to meaningful angular displacements in both precession and nutation, offering design challenges beyond conventional bottom-founded structures. This paper offers theoretical developments underlying an efficient methodology to compute the large-angle rigid body rotations of a floating wind turbine in the time domain. The tower and rotor-nacelle assembly (RNA) are considered as two rotational bodies in the space, for which two sets of Euler angles are defined and used to develop two systems of Euler dynamic equations of motion. Transformations between the various coordinate systems are derived in order to enable a solution for the motion of the tower, with gyroscopic, environmental, and restoring effects applied as external moments. An example is presented in which the methodology is implemented in matlab in order to simulate the time-histories of a floating tower with a RNA.