A nearly single cycle intense terahertz (THz) pulse with peak electric and magnetic fields of 0.5 MV/cm and 0.16 T, respectively, excites both modes of spin resonances in the weak antiferromagnet FeBO3. The high frequency quasiantiferromagnetic mode is excited resonantly and its amplitude scales linearly with the strength of the THz magnetic field, whereas the low frequency quasiferromagnetic mode is excited via a nonlinear mechanism that scales quadratically with the strength of the THz electric field and can be regarded as a THz inverse Cotton-Mouton effect. THz optomagnetism is shown to be more energy efficient than similar effects reported previously for the near-infrared spectral range.