The sine addition formula on a semigroup S is the functional equation \(f(xy) = f(x)g(y) + g(x)f(y)\) for all \(x,y \in S\). For some time the solutions have been known on groups, regular semigroups, and semigroups which are generated by their squares. The obstacle to finding the solution on all semigroups arose in the special case that g is a multiplicative function. We overcome this obstacle and find the general solution on all semigroups using a transfinite induction argument. A new type of solution appears which is not seen on regular semigroups or semigroups generated by their squares.

We also give the general solution of the sine subtraction formula \(f(x\sigma(y)) = f(x)g(y) - g(x)f(y)\) on monoids, where \(\sigma\) is an automorphic involution. The solutions of both equations can be described in terms of additive and multiplicative functions, with a slight new twist. The general continuous solutions on topological semigroups are also found. A variety of examples are given to illustrate the results.

半群S上的正弦加法公式是函数方程\(f(xy) = f(x)g(y) + g(x)f(y)\)对于所有\(x,y \in S\) . 一段时间以来，人们已经知道群、正则半群和由它们的平方生成的半群的解。在 g 是乘法函数的特殊情况下，在所有半群上找到解的障碍出现了。我们克服了这个障碍，并使用超限归纳论证找到了所有半群的一般解。出现了一种新的解，它在规则半群或由它们的平方生成的半群上是看不到的。

我们还给出了幺半群上正弦减法公式\(f(x\sigma(y)) = f(x)g(y) - g(x)f(y)\) 的一般解，其中\(\sigma \)是一个自守对合。两个方程的解都可以用加法函数和乘法函数来描述，并有一些新的变化。还找到了拓扑半群上的一般连续解。给出了各种例子来说明结果。