  Given a semigroup $S$ equipped with an involutive automorphism $\sigma$, we
determine the complex-valued solutions $f,g,h$ of the functional equation
\begin{equation*}f(x\sigma(y))=f(x)g(y)+g(x)f(y)+h(x)h(y),\,\,x,y\in
S,\end{equation*} in terms of multiplicative functions and solutions of the
special cases of sine and cosine-sine functional equations.
