trigonometry - Sum of two trig function's identity

We all know that

$\sin(x) + \sin(y) = 2\sin((x+y)/2)\cos((x-y)/2)$

But is there an identity for

$\sin(x) + z\sin(y) = ?$

Or do I need to figure it out using Euler's formula
$\sin(x) = (e^{ix} - e^{-ix})/2$ and put it back into trigonometric form?