With the rapid growth of aircraft traffic and the new modes of transport such as Urban Air Mobility systems crowding the air space, aircraft noise is no longer a mere design constraint but an important factor to design and optimize for. Reducing aircraft noise however requires efficient coupling of simulation tools with design methods to be able to meet the stringent future aircraft noise requirements that allow for sustainable growth. Gradient-based design optimization based on adjoint method for sensitivity analysis offers a feasible design approach. Adjoint methods based on steady state physics have been widely in practice in industrial applications mainly for aerodynamic optimization so far. However, extending this approach to aeroacoustic optimization is not straightforward and not a common practice in industrial settings due the requirement of unsteady adjoint solutions that is prohibitively expensive. This dissertation presents temporal and spatial coarsening techniques for the computation of low-cost unsteady adjoints to obtain sensitivities for aeroacoustic shape optimization. The effects of the coarsening techniques on the accuracy of the gradients are analyzed by using different levels of temporal and spatial coarsening using multiple two dimensional and three dimensional test cases. Computational cost savings as well as reduction of memory storage requirements up to 10% of the base adjoint solutions are presented while maintaining reasonable accuracy in the gradients driving the optimization for these test cases. Finally, an extension to the temporal coarsening technique is proposed with non-uniform time stepping of adjoint solver based on the local flow truncation error estimates of the flow solver. The proposed extension is demonstrated to further improve the accuracy of the low-cost gradients providing motivations for the future directions of the work done in this thesis.