This study numerically investigates the effects of fluid-structure interaction (FSI) on the trailing-edge noise, particularly for the cases of wake instability and Karman vortex shedding. The trailing edge is modeled as a flat plate with an elastic cantilever end and its flow-induced vibration is solved by an eigenmode analysis with the Galerkin method. The flow and sound coupled in the FSI analysis are computed on the moving grid by a direct numerical simulation (DNS) procedure. The computed result of wake instability shows that when the first-eigenmode natural frequency ${omega}_c$, of the cantilever is close to be resonant with the wake characteristic frequency We, the sound pressure level (SPL) is significantly reduced by 20 dB at ${omega}_n/{omega}_c=0.95$, or increased by 15 dB at ${omega}_n/{omega}_c=1.05$, for all angles. For the Karman vortex shedding, a similar frequency modulation occurs via FSI, if ${omega}_n$ is close to ${omega}_c$. The flow and acoustic details are somewhat different for this case but a considerable noise reduction was also possible for angles from $-120^{circ}$ to $+120^{circ}$.