Solution
= Solution
Start with the <exponential function> definition of the <hyperbolic sine> and use <Euler's formula>:
$$
\begin{aligned}
\sinh(x+iy)&=\frac{e^xe^{iy}-e^{-x}e^{-iy}}2\\
&=\frac{e^x(\cos y+i\sin y)-e^{-x}(\cos y-i\sin y)}2\\
&=\frac{e^x-e^{-x}}2\cos y+i\frac{e^x+e^{-x}}2\sin y.
\end{aligned}
$$
Identifying the <hyperbolic sine> and <hyperbolic cosine> in the final line gives
$$
\boxed{\sinh(x+iy)=\sinh x\cos y+i\cosh x\sin y.}
$$