Imaginary Part of Sine of Complex Number
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Theorem
Let $z = x + i y \in \C$ be a complex number, where $x, y \in \R$.
Let $\sin z$ denote the complex sine function.
Then:
- $\Im \paren {\sin z} = \cos x \sinh y$
where:
- $\Im z$ denotes the imaginary part of a complex number $z$
- $\sin$ denotes the complex sine function
- $\cos$ denotes the real cosine function
- $\sinh$ denotes the hyperbolic sine function.
Proof
From Sine of Complex Number:
- $\sin \paren {x + i y} = \sin x \cosh y + i \cos x \sinh y$
The result follows by definition of the imaginary part of a complex number.
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.5$. The Functions $e^z$, $\cos z$, $\sin z$