= Solution
Write $z=x+iy$ with real $x,y$. The complex <hyperbolic sine> is $\sinh z=\sinh x\cos y+i\cosh x\sin y$. Its zero <real part> requires $\sinh x\cos y=0$. If $\cos y=0$, the magnitude of the <imaginary part> would be $\cosh x\ge1$, which cannot equal $1/2$. Therefore $x=0$ and $\sin y=1/2$. All solutions are
$$
\boxed{z=i\left(\frac\pi6+2\pi k\right)\quad\text{or}\quad z=i\left(\frac{5\pi}6+2\pi k\right),\qquad k\in\mathbb Z.}
$$
On the <Argand diagram> these are two interlaced arithmetic sequences on the imaginary axis; both repeat with imaginary period $2\pi i$.
\Image[/past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2014-complex-hyperbolic-sine-roots.png]
{title=All roots lie on the imaginary axis in two families repeating every $2\pi i$}
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