Hi!
Good question! Like every other programmer, to replace sin/cos I simply used a Look-Up Table (LUT), that is a big table with all the sine / cosine values precalculated (See "LUT makes sprites go round!" section of the making of).
If you want to see an example on how to use this for a SNES project, you can check the source-code of my other game "The Last Super"
=> download game and source code from here: https://drludos.itch.io/the-last-super
=> Open the file "TheLastSuper.c" and go to line 222 for the LUT table itself and line 2549 for an example of how I set X/Y position of a sprite using those table.
Hope that helps!