The antiferromagnetic insulator La1.5Ca0.5CoO4 has been investigated by Co L-2,L-3-edge and O K-edge X-ray absorption spectroscopy (XAS) measurements and Co L-2,L-3-edge resonant soft X-ray magnetic scattering (RXMS) measurement to determine the Co electronic structures associated with magnetic ordering. Co L-2,L-3-edge linear-dichroic XAS shows that Co2+ takes a high-spin (HS) state and Co3+ takes a low-spin (LS) state. Using Co L-2,L-3-edge RXMS, we directly determined that an antiferromagnetic order is formed with a HS state of Co2+ ions. Moreover, the spin and orbital angular momenta of the Co2+ HS state are quantitatively estimated to be 1.1 +/- 0.1 and 1.0 +/- 0.1, respectively, and to align parallel in the ab plane by utilizing the cluster model calculation. The large orbital angular momentum of the Co2+ HS state originates from the small D-4h-symmetry crystal field splitting of t(2g) levels, which is comparable with the spin-orbit coupling constant of the Co 3d orbital.