Can one rewrite $\sin(2t)$ to $\sin u$ and apply the formula $\sin u=\frac{e^{i\phi}-e^{-i\phi}}{2i}$?

Question

I have the Laplace transformation to be done for $f(t)=t\sin 4t$. Since the first translational theorem says

$\mathscr{L}\{f(t)\}=F(s)=\mathscr{L}\{e^{at}f(t)\}=F(s-a)$

I would like to put $f(t)=t$ and rewrite $\sin t$ in exponential form, by use of the formula

$\sin x=\frac{e^{i\phi}-e^{-i\phi}}{2i}$ to rewrite sin in the exponential form. But what if $x=4t$ in $\sin x$ here?

Thanks