Research Article
Five trigonometric addition laws on semigroups
- By Youssef Aserrar, Elhoucien Elqorachi - 23 Dec 2025
- Applied Mathematics on Science and Engineering, Volume: 2(2025), Issue: 2, Pages: 55 - 66
- https://doi.org/10.65036/amse229
- Received: 08.10.2025; Accepted: 15.12.2025; Published: 23.12.2025
Abstract:
In this paper, we determine the complex-valued solutions of the following functional equations:
g(xσ(y)) = g(x)g(y)+ f (x) f (y), x,y ∈ S,
f (xσ(y)) = f (x)g(y)+ f (y)g(x), x,y ∈ S,
f (xσ(y)) = f (x)g(y)+ f (y)g(x)−g(x)g(y), x,y ∈ S,
f (xσ(y)) = f (x)g(y)+ f (y)g(x)+α g(xσ(y)), x, y ∈ S,
f (xσ(y)) = f (x)g(y)− f (y)g(x)+α g(xσ(y)), x, y ∈ S,
where S is a semigroup, α ∈ C\ {0} is a fixed constant, and σ : S → S is an involutive automorphism. We also provide some applications.