Tangent of Complex Number

Theorem
Let $a$ and $b$ be real numbers.

Let $i$ be the imaginary unit.

Then:
 * $\tan \left({a + b i}\right) = \dfrac {\sin a \cosh b + i \cos a \sinh b} {\cos a \cosh b - i \sin a \sinh b}$
 * $\tan \left({a + b i}\right) = \dfrac {\tan a + i \tanh b} {1 - i \tan a \tanh b}$
 * $\tan \left({a + b i}\right) = \dfrac {\tan a - \tan a \tanh ^2 b} {1 + \tan ^2 a \tanh ^2 b} + \dfrac {\tanh b + \tan ^2 a \tanh b} {1 + \tan ^2 a \tanh ^2 b} i$

where $\sin$ denotes the complex sine function, $\cos$ denotes the complex cosine function and $\tan$ denotes the complex tangent function.

Also See

 * Sine of Complex Number
 * Cosine of Complex Number
 * Cosecant of Complex Number
 * Secant of Complex Number
 * Cotangent of Complex Number