Spin-1/2 Heisenberg antiferromagnets Cs$_2$CuCl$_4$ and Cs$_2$CuBr$_4$ with distorted triangular-lattice structures are studied by means of electron spin resonance spectroscopy in magnetic fields up to the saturation field and above. In the magnetically saturated phase, quantum fluctuations are fully suppressed, and the spin dynamics is defined by ordinary magnons. This allows us to accurately describe the magnetic excitation spectra in both materials and, using the harmonic spin-wave theory, to determine their exchange parameters. The viability of the proposed method was proven by applying it to Cs$_2$CuCl$_4$, yielding $J/k_B=4.7(2)$ K, $J/k_B=1.42(7)$ K [$J/Jsimeq 0.30$] and revealing good agreement with inelastic neutron-scattering results. For the isostructural Cs$_2$CuBr$_4$, we obtain $J/k_B=14.9(7)$ K, $J/k_B=6.1(3)$ K, [$J/Jsimeq 0.41$], providing exact and conclusive information on the exchange couplings in this frustrated spin system.
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