Rock masses, especially those with different pre-existing cracks, are prone to instability and failure under tensile loading, resulting in different degrees of engineering disasters. Therefore, to better understand the effect of pre-existing cracks with different dip angles on the tensile instability failure behaviour of rocks, the mechanism of crack initiation, propagation and coalescence in precracked sandstone under radial compression loading is investigated through numerical simulations. The temporal and spatial evolution of acoustic emission (AE) events is investigated by the moment tensor (MT), and the fracture mode of micro-cracks is determined. The results show that the pre-existing cracks weaken the specimens. The strength, crack initiation points and macro-failure modes of the specimens differ significantly depending on the dip angle of the pre-existing crack. For different dip angles of the pre-existing cracks, all the micro-cracks at the crack initiation point are tensile cracks, which are dominant during the whole loading process, and mixed cracks are mainly generated near the upper and lower loading ends after the peak stress. Of the total number of events, more than 75% are tensile cracks; approximately 15% are shear mode cracks; and the remainder consist of mixed mode cracks. The study reveals the instability and failure mechanism of pre-cracked rock, which is of great significance to ensure the long-term stability of rock mass engineering.