Yes, it's correct.
So now that you know the real part is equal to a constant C
cos(y)(e^x - e^{-x}) = C
cos(y)(e^x - 1/e^x) = C
cos(y)(e^{2x} - 1) = Ce^x
cos(y) = Ce^x/(e^{2x} - 1)
y = arccos[Ce^x/(e^{2x} - 1)].
I'm sure you can sketch this.
The question:
For the mapping f(z) = sinh(z), find and sketch the image of Re(z) = c
My attempt:
Unless I'm mistaken, Re(z) = C is just a line.
I'm having problems with the f(z) = sinh(z) part. I set f(z) = w, and used the definition of sinh(z) as follows:
Now I'm stuck. :/ Have I attempted this correctly, and if so, how do I proceed? Any help would be greatly appreciated!