Currently, the number of smartphones with an embedded gyroscope sensor has been increasing due games whose performance relies on 3D augmented reality. In general, teaching papers on the gyroscope sensor address very simple spatial configuration, where the fixed rotation axis coincides to the z-axis of the smartphone. This work presents five experimental setups with different spatial orientations of a smartphone on a turntable. The total angular velocity vector of the turntable is constant, but its projections on the three fixed Cartesian axes of the smartphone varies for the five spatial orientations investigated. For the sake of consistency, the magnitude of the vector sum of the three angular velocities components yielded by the gyroscope sensor was checked by video analysis using the free software, Tracker. The internal consistency between the results provided by this video analysis technique and data from the gyroscope assures us this sensor is reliable for teaching proposals. The main goals of this paper are (1) encourage teachers to effectively incorporate the gyroscope sensor of smartphones in to their classrooms, especially in high schools and undergraduate physics and engineering courses; (2) spread in the academic syllabus the Tait-Bryan convention, a very intuitive way to perform 3D rotations and; (3) explain how to interpret data of the three components of angular velocity obtained by the gyroscope sensor in different spatial orientations.