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:

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

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

I then used De Moivre's theorem:

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

The real part of this is:

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

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

$\displaystyle c = cos(y)sinh(x)$

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.