The question

For the mapping f(z) = sinh(z), find and sketch the image of Re(z) = c.

My attempt

Well I tried using the definition of sinh(z) in terms of exponentials:

=

I then used De Moivre's theorem:

The real part of this is:

I'm pretty sure I got this completely wrong.

Can someone explain to me how I'm supposed to attempt this? Any assistance would be greatly appreciated.