The full elastic constants of a carbon fiber in a unidirectional carbon fiber reinforced plastic composite were investigated by using a resonant ultrasound spectroscopy technique. The equivalent elastic properties of the composite were determined by comparing resonance frequency obtained from experiment and analysis. The subspace method which is the kind of the iterative method for calculating eigenvalues and eigenvectors of the real symmetric matrix such as the stiffness and mass matrix, was used to estimate the resonance frequency of the composite. The error function was calculated using the natural frequency obtained by the experiment and finite element analysis. It was found that the estimated elastic constants of the carbon fiber in the composite based on both Eshelby-Mori-Tanaka theory were reasonably consistent with the previously-reported values. The effectiveness of the method used here was confirmed by comparing the experimental results and those obtained by numerical analysis for the carbon fiber in the composite. The technique could be applicable to any other continuous fibers including those with much smaller diameter, as well as randomly oriented discontinuous fibers.