Measurement of the viscoelastic properties of a cell using microscopic tracer particles has been complicated given that the medium viscosity is dependent upon the size of the measurement probe leading to reliability issues. Further, a technique for direct calibration of optically trapped particles in vivo has been elusive due to the frequency dependence and spatial inhomogeneity of the cytoplasmic viscosity, and the requirement of accurate knowledge of the medium refractive index. Here, we employ a recent extension of Jeffery's model of viscoelasticity in the microscopic domain to fit the passive motional power spectra of micrometer-sized optically trapped particles embedded in a viscoelastic medium. We find excellent agreement between the 0 Hz viscosity in MCF7 cells and the typical values of viscosity in literature, between 2 to 16 mPa sec expected for the typical concentration of proteins inside the cytoplasmic solvent. This bypasses the dependence on probe size by relying upon small thermal displacements. Our measurements of the relaxation time also match values reported with magnetic tweezers, at about 0.1 s. Finally, we calibrate the optical tweezers and demonstrate the efficacy of the technique to the study of in vivo translational motion. © 2020 IOP Publishing Ltd.