Hyperbolic Sine in terms of Sine

Theorem
Let $z \in \C$ be a complex number.

Then:


 * $i \sinh z = \map \sin {i z}$

where:
 * $\sin$ denotes the complex sine
 * $\sinh$ denotes the hyperbolic sine
 * $i$ is the imaginary unit: $i^2 = -1$.

Also presented as
This identity is also seen in the form:


 * $\sinh z = -i \map \sin {i z}$

which can be seen to follow from the other form by multiplication by $-i$.

Also see

 * Hyperbolic Cosine in terms of Cosine
 * Hyperbolic Tangent in terms of Tangent
 * Hyperbolic Cotangent in terms of Cotangent
 * Hyperbolic Secant in terms of Secant
 * Hyperbolic Cosecant in terms of Cosecant