This would be classed as homework (although I am not a student, I'm going through an old textbook of mine).
The question is in two parts:
Express $\sinh(x+iy)$ in the form $u+iv$, where $x,y,u,v$ are real, and show that $|\sinh(x+iy)|^{2} = \frac{1}{2}(\cosh(2x) - \cos(2y))$
if $\sinh(x+iy) = e^{\frac{i\pi}{3}}$ prove either
$x = \ln(\frac{\sqrt 6 + \sqrt 2}{2}), y= 2n\pi + \pi/4$
or
$x = \ln(\frac{\sqrt 6 - \sqrt 2}{2}), y= 2n\pi + 3\pi/4$
where $n$ is an integer.
I can solve part 1. $\sinh(x+iy) = \cos(y)\sinh(x)+ i\sin(y)\cosh(x)$, and by each of the summands, expanding the hyperbolic parts and rearranging I can prove the equality.
However, I cannot prove part 2. It's becoming quite annoying. I would assume part 1 should be used to prove part 2, but cannot find a way to use it.
If $ e^{\frac{i\pi}{3}}$ is expressed in cartesian coords, it is $1/2 + i\sqrt 3/2$.
So
$\cos(y)\sinh(x) = 1/2$, and $\sin(y)\cosh(x) = \sqrt 3 /2$.
I tried expanding the hyperbolics, and substituting $u=e^{x}$, and end up with two very ugly quadratics, that seem to get me noewhere.