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:

sinh(z) = \frac{e^z - e^{-z}}{2}

= \frac{e^xe^{iy} - e^{-x}e^{-iy}}{2}

I then used De Moivre's theorem:

\frac{e^xcos(y) - e^{-x}cos(-y) + e^xisin(y) + e^{-x}sin(-y)}{2}

The real part of this is:

\frac{e^xcos(y) - e^{-x}cos(-y)}{2}

cos(y)(e^x - e^{-x}) = 2c

c = cos(y)sinh(x)

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.