# Why does $\sin(x)$ + $\sin(x+a)$ always come out as a sine function? [duplicate]

How is it possible mathematically? In short, this says, $$\sin(x)$$ + $$\sin(x+a)$$ must be equal to some function like $$b$$ $$\sin(x+c)$$ somehow, but how?

$$\sin(a) + \sin(b) = 2 \sin \left( \frac{a+b}{2} \right) \cos \left( \frac{a-b}{2} \right)$$
Replace $$a$$ with $$x$$ and $$b$$ with $$x+a$$ in the above formula to get
\begin{align*} \sin(x) + \sin(x+a) &= 2 \sin \left( \frac{x+x+a}{2} \right) \cos \left( \frac{x-(x+a)}{2} \right) \\ &= 2 \sin \left( x + \frac a 2 \right) \cos \left( \frac a 2 \right) \end{align*}
Note that $$\cos(a/2)$$ is a constant, so you really have a sine wave, with amplitude $$2 \cos(a/2)$$, and shifted left by $$a/2$$ units.