  We observed the $A^1\Sigma^+\sim b^3\Pi\to a^3\Sigma^+/X^1\Sigma^+$
laser-induced fluorescence (LIF) spectra of the RbCs molecule excited from the
ground $X^1\Sigma^+$ state by the Ti:Sapphire laser. The LIF radiation from the
common perturbed levels of the singlet-triplet $A\sim b$ 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 $A\sim b\to a^3\Sigma^+(v_a)/X^1\Sigma^+(v_X)$ progressions was
measured, and their branching ratio was found to be about of 1$\div$5$
\times$10$^{-4}$ in the bound region of the $a^3\Sigma^+$ and $X^1\Sigma^+$
states. The experiment was complemented with the scalar and full relativistic
calculations of the $A/b - X/a$ transition dipole moments (TDMs) as functions
of internuclear distance. The relative systematic error in the resulting
\emph{ab initio} TDM functions evaluated for the strong $A - X$ transition 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$^{-1}$ by a
comparison of experimental intensities in the long $A\sim b\to X(v_X)$ LIF
progressions of the K$_2$ and KCs molecules with their theoretical counterparts
evaluated using the \emph{ab initio} $A - X$ TDMs. Both experimental and
theoretical transition probabilities can be employed to improve the stimulated
Raman adiabatic passage process, $a\to A\sim b \to X$, which is exploited for a
laser assembling of ultracold RbCs molecules.
