CORDIC is a fairly simple integer-only algorithm to compute sine/cosine. It is rather cheap to implement, but it might still be too much for a 8.4MHz clock and generating 16.384 kS/s. (IOW, there are 512 cycles per sample.)

A nice description of how CORDIC works: https://github.com/francisrstokes/githublog/blob/main/2024/5/10/cordic.md