Hyperbolic Sine in terms of Sine

Theorem

 * $\sin \left({ix}\right) = i \sinh x $

where $\sin$ is the sine, $\sinh$ is the hyperbolic sine, and $i^2=-1$.

Proof
Recall the Sine Exponential Formulation:


 * $ \displaystyle \sin x = \frac 1 2 i \left({ e^{-ix} - e^{ix} }\right) $

Then: