Solve $\sinh z = i$

Question

I am trying to solve $\sinh z = i$; this has to result in $2\pi n + (1/2) \pi i$. When applying the exponential form of $$\sinh z = \frac{e ^ z - e ^ {-z}} 2$$ and later the quadratic equation, this does not give me the result as described above. I would appreciate your collaboration guiding me to find the error.

Answer 1

You have\begin{align}\sinh z=i&\iff e^z-e^{-z}=2i\\&\iff(e^z)^2-2ie^z-1=0\\&\iff(e^z-i)^2=0\\&\iff e^z=i\\&\iff z=\frac\pi2i+2k\pi i\end{align}for some $k\in\mathbb Z$.