Sine in terms of Hyperbolic Sine

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Theorem

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

Then:

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

where:

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


Proof

\(\ds \map \sinh {i z}\) \(=\) \(\ds \frac {e^{i z} - e^{-i z} } 2\) Definition of Hyperbolic Sine
\(\ds \) \(=\) \(\ds i \frac {e^{i z} - e^{-i z} } {2 i}\) multiplying top and bottom by $i$
\(\ds \) \(=\) \(\ds i \sin z\) Euler's Sine Identity

$\blacksquare$


Also see


Sources

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