Tangent of Complex Number

Theorem

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

Let $i$ be the imaginary unit.

Then:

Formulation 1

$\tan \paren {a + b i} = \dfrac {\sin a \cosh b + i \cos a \sinh b} {\cos a \cosh b - i \sin a \sinh b}$

Formulation 2

$\tan \paren {a + b i} = \dfrac {\tan a + i \tanh b} {1 - i \tan a \tanh b}$

Formulation 3

$\tan \paren {a + b i} = \dfrac {\tan a - \tan a \tanh ^2 b} {1 + \tan ^2 a \tanh ^2 b} + \dfrac {\tanh b + \tan ^2 a \tanh b} {1 + \tan ^2 a \tanh ^2 b} i$

Formulation 4

$\tan \paren {a + b i} = \dfrac {\sin 2 a + i \sinh 2 b} {\cos 2 a + \cosh 2 b}$

where:

$\tan$ denotes the tangent function (real and complex)

$\sin$ denotes the real sine function

$\cos$ denotes the real cosine function

$\sinh$ denotes the hyperbolic sine function

$\cosh$ denotes the hyperbolic cosine function

$\tanh$ denotes the hyperbolic tangent function.

Also see

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