Proving surjectivity of $\cos(z)$ and $\sin(z)$ and find all $z : \cos(z) \in \mathbb R$ and all $z: \sin(z) \in \mathbb R$

1) Note that one of $\frac{2w+w_0}{2}, \frac{2w-w_0}{2}$ must be nonzero, and that $e^{iz}$ achieves all values except $0$, since for any $re^{i\theta}\in \Bbb C$ we have $re^{i\theta}=e^{i(\theta-i\ln r)}$ thus we have some $z$ such that $e^{iz}=\frac{2w+w_0}{2}$ or $e^{iz}=\frac{2w-w_0}{2}$.

2) If we write $z=x+iy$ then $$\cos(z)=\frac{e^{ix-y}+e^{-ix+y}}{2}=\frac{e^{-y}e^{ix}+e^y\overline{e^{ix}}}{2}$$ which has conjugate $$\frac{e^{-y}\overline{e^{ix}}+e^ye^{ix}}{2}$$ so $\cos(z)$ is real iff $$e^{-y}e^{ix}+e^y\overline{e^{ix}}=e^{-y}\overline{e^{ix}}+e^ye^{ix}.$$ If $e^{ix}$ is real then $x=n\pi$ for some $n\in\Bbb N$. Otherwise $e^{ix}$ and $\overline{e^{ix}}$ are linearly independent over $\mathbb R$, so $e^{-y}=e^y$ thus $y=0$.

The case of $\sin$ is similar for both problems.