Physiological and toxicological effects of metallic ions depend on their speciation and on the structure of their associated bioligand complexes. In the field of chemical and/or nuclear toxicological studies, we are investigating cobalt complexes with biorelevant ligands such as amino acids or peptides. The aqueous reaction of cobalt dichloride with an excess of cysteine (Cys, C$_3$H$_5$NSO$_2$$^{2-}$) in a basic medium under an anaerobic atmosphere and subsequent oxidation by O$_2$ , afforded the mononuclear complexes Co(II):3Cys and Co(III):3Cys, respectively. A combination of X-ray absorption spectroscopy (XAS) measurements and Car-Parrinello molecular dynamics (CPMD) simulations allowed us to assess structural features of the already explored Co(III):3Cys complex. Inclusion of the temperature effects in the CPMD calculations gives an implicit access to disorder effects in the extended X-ray absorption fine structure (EXAFS) equation. The very good agreement between the measured and the simulated data showed the accuracy of these models provided by CPMD. The present investigation is completed by new UV-visible, X-ray absorption near edge structure (XANES) and electron paramagnetic resonance (EPR) data of Co(II):3Cys. These data are consistent with a Co(II) high-spin d$^7$ complex in a distorted octahedral geometry. This work contributes to the knowledge of topics such as metal-bioligand interaction which is of major interest in the field of bioinorganic chemistry.