Due to the similarity between sine and cosine functions, transformations of y = cos (x) to y = a cos (n x − b) + c have the same effects as transformations of y = sin (x) to y = a sin (n x − b) + c.

Semigroups of transformations and endomorphisms have emerged as powerful algebraic frameworks to elucidate the underlying structures of graphs. By harnessing the principles of semigroup theory ...