An adaptive state observer is presented for a class of LTI systems that have a sinusoidal disturbance in the measurement output. Adaptation rules are developed for identifying the unknown sinusoidal disturbance signal from the system output. For the application of the identification result to the state estimation problem, the sinusoidal signal with arbitrary initial phase has been considered in this paper. In order to test the performance of proposed algorithm, comparative computer simulations have been carried out with an existing robust observer. The simulation results show the effectiveness of the proposed method.