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

Question

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?

Answer 1

The answer lies in the sum-to-product formula for sine,

$$\sin(a) + \sin(b) = 2 \sin \left( \frac{a+b}{2} \right) \cos \left( \frac{a-b}{2} \right)$$

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.

(Reference sheet for trig identities.)