We observed the A1Σ+∼b3Π→a3Σ+/X1Σ+laser-induced fluorescence (LIF) of the RbCs molecule excited from the ground X1Σ+state by the Ti:Sapphire laser. The LIF spectra from the common perturbed levels of the singlet-triplet A ∼ b complex was recorded by the Fourier-transform (FT) spectrometer with the instrumental resolution of 0.03 cm−1. The relative intensity distribution in the rotationally resolved A∼b→a3Σ+(va)/X1Σ+(vX)progressions was measured, and their branching ratio was found to be about of 1÷5×10−4in the bound region of the a3Σ+and X1Σ+states. The experiment was complemented with the scalar- and full-relativistic calculations of the A/b−X/atransition dipole moments (TDMs) as functions of internuclear distance. The relative systematic error in the resulting ab initio TDM functions evaluated for the strong A−Xtransition was estimated as few percent in the energy region, where the experimental LIF intensities are relevant. The relative spectral sensitivity of the FT registration system, operated with the InGaAs diode detector and CaF beam-splitter, was calibrated in the range [6 500, 12 000] cm−1by a comparison of experimental intensities in the long A ∼ b → X(vX) LIF progressions of the K2 and KCs molecules with their theoretical counterparts evaluated using the ab initioA−XTDMs. Both experimental and theoretical transition probabilities can be employed to improve the stimulated Raman adiabatic passage process, a → A ∼ b → X, which is exploited for a laser assembling of ultracold RbCs molecules.