Modulus 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:
- $\cmod {\sin z} = \sqrt {\sin^2 x + \sinh^2 y}$
where:
- $\cmod z$ denotes the modulus of a complex number $z$
- $\sin x$ denotes the real sine function
- $\sinh$ denotes the hyperbolic sine function.
Proof
\(\ds \sin \paren {x + i y}\) | \(=\) | \(\ds \sin x \cosh y + i \cos x \sinh y\) | Sine of Complex Number | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \cmod {\sin z}^2\) | \(=\) | \(\ds \paren {\sin x \cosh y}^2 + \paren {\cos x \sinh y}^2\) | Definition of Complex Modulus | ||||||||||
\(\ds \) | \(=\) | \(\ds \sin^2 x \paren {1 + \sinh^2 y} + \cos^2 x \sinh^2 y\) | Difference of Squares of Hyperbolic Cosine and Sine | |||||||||||
\(\ds \) | \(=\) | \(\ds \sin^2 x + \paren {\sin^2 x + \cos^2 x} \sinh^2 y\) | rearranging | |||||||||||
\(\ds \) | \(=\) | \(\ds \sin^2 x + \sinh^2 y\) | Sum of Squares of Sine and Cosine | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \cmod {\sin z}\) | \(=\) | \(\ds \sqrt {\sin^2 x + \sinh^2 y}\) |
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.5$. The Functions $e^z$, $\cos z$, $\sin z$