Let $f:\mathbb{C}\rightarrow \mathbb{C}$ be the map given by $f(z)=\sin z-z$. Then image of $f$ is $\mathbb{C}$. (T/F)

Question

Let $f:\mathbb{C}\rightarrow \mathbb{C}$ be the map given by $f(z)=\sin z-z$. Then image of $f$ is $\mathbb{C}$. (T/F)

I think the statement is true, both $\sin z$ and $z$ is entire, so is $f$ and also by Picard theorem entire function can omit at most one point. Is my reasoning correct?