Tangent of Complex Number/Formulation 1
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Theorem
Let $a$ and $b$ be real numbers.
Let $i$ be the imaginary unit.
Then:
- $\tan \paren {a + b i} = \dfrac {\sin a \cosh b + i \cos a \sinh b} {\cos a \cosh b - i \sin a \sinh b}$
where:
- $\tan$ denotes the complex tangent function
- $\sin$ denotes the real sine function
- $\cos$ denotes the real cosine function
- $\sinh$ denotes the hyperbolic sine function
- $\cosh$ denotes the hyperbolic cosine function.
Proof
\(\ds \tan \paren {a + b i}\) | \(=\) | \(\ds \frac {\sin \paren {a + b i} } {\cos \paren {a + b i} }\) | Definition of Complex Tangent Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\sin a \cosh b + i \cos a \sinh b} {\cos a \cosh b - i \sin a \sinh b}\) | Sine of Complex Number and Cosine of Complex Number |
$\blacksquare$