Semnan University International Journal of Nonlinear Analysis and Applications 2008-6822 15 11 2024 11 01 A trigonometric functional equation with an automorphism 403 415 8330 10.22075/ijnaa.2023.31988.4745 EN Youssef Aserrar Department of Mathematics, Faculty of Sciences, Ibn Zohr University, Agadir, Morocco Elhoucien Elqorachi Department of Mathematics, Faculty of Sciences, Ibn Zohr University, Agadir, Morocco Th.M. Rassias Department of Mathematics, National Technical University of Athens, Athens, Greece Journal Article 2023 09 05 Let $S$ be a semigroup. In the present paper, we determine the complex-valued solutions $(f,g)$ of the functional equation<br />$$g(x\sigma (y)) = g(x)g(y)-f(x)f(y)+\alpha f(x\sigma(y)),\  x,y\in S,$$<br />where $\sigma :S\rightarrow S$ is an automorphism that need not be involutive, and $\alpha \in \mathbb{C}$ is a fixed constant. Our results generalize and extend the ones by Stetk\ae r in The cosine addition law with an additional term. Aequat Math., no. 6, 90, 1147-1168 (2016), and also the ones by Aserrar and Elqorachi in A generalization of the cosine addition law on semigroups. Aequat Math. 97, 787–804 (2023). Some consequences of our results are presented. https://ijnaa.semnan.ac.ir/article_8330_390894b225ea101525451b357626a207.pdf