One of the sources of the energy waste in wireless sensor networks is idle listening, the time in which a node monitors the free channel. In applications where the events occur sporadically, energy consumption due to idle listening can be further reduced by dual-radio cooperation. In dual-radio cooperation, nodes in the network have two stacks. One stack makes use of a low-power wake-up radio for event-driven communication over the main radio. The other stack may employ any sensor networking medium access control protocol only over the main radio. One of the two stacks can be dynamically operational depending on the rate of the events or the packet arrival rate. When the event rate becomes small, the event-driven stack takes over the operation. If the event rate increases, it could be more efficient to operate the legacy single radio stack. In this paper, we investigate the performance of the dual-radio cooperation in wireless sensor networks. A medium access control protocol is proposed for the dual-radio cooperation to maximize the energy efficiency of the wireless network. We define the critical event rate as the event rate threshold above which the single-radio stack performs better than the dual-radio stack. We analyzed and validated the critical event rate by simulations. We show that 70-97% energy conservation is possible by employing the dual-radio cooperation.