Is there a single-valued function linear in $\sin(x)$ and $\cos(x)$ that is invertible on $[-\pi, \pi)$ or $[0,2\pi)$?

Question

What about $\sin(x/2)$ and $\cos(x/2)$? My intuition is no, for both, because $f = \sin(x)+\cos(x)$, $f = \sin(x)-\cos(x)$, $f = \sin(x/2)+\cos(x/2)$, and $f = \sin(x/2)-\cos(x/2)$ all don't do the trick, and scaling the terms shouldn't change that. Can anyone address this more formally?

Thanks in advance.