A rigorous analysis of sound radiation by a pulsating sphere forming a resonator together with a semi-spherical cavity is presented. Both hard/soft boundaries are considered, as well as mixed. The problem is solved by dividing the entire region into two subregions, one surrounding the sphere and containing the cavity and the other for the remaining half-space. Continuity conditions are applied to obtain the acoustic pressure. Then the acoustic radiation resistance is calculated both in the near- and far-field. The acoustic radiation reactance is calculated in the impedance approach. The resonance frequencies are determined, for which a significant growth of the sound pressure level is observed as well as the sound field directivity. These rigorous results are presented in the form of highly convergent, accurate and numerically efficient series.