Now, I know there's two ways to go about this and it seems everywhere I look around on the web people are solving it in a way I think that seems longer, harder and more prone to mistakes in exams. It involves using the exponential identities and taking logs. I was shown another way, but unfortunately I haven't quite got the grasp of it.
sinh z = 0
sinh (x + iy) = sinh(x)cos(y) + i cosh(x)sin(y) = 0
1) sinh(x)cos(y) = 0
2) cosh(x)sin(y) = 0
(1) Either x = 0 or y = ± Pi/2
(2) cosh (x) is never 0, so therefore x is not 0. Hence y = 0
This is where I'm stuck, I do not know how to go from here.
The answer is z = i*k*Pi