Hyperbolic Sine of Complex Number/Proof 1

Theorem

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

Let $i$ be the imaginary unit.

Then:

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

where:

$\sin$ denotes the real sine function

$\cos$ denotes the real cosine function

$\sinh$ denotes the hyperbolic sine function

$\cosh$ denotes the hyperbolic cosine function.

Proof

\(\ds \sinh \paren {a + b i}\)

\(=\)

\(\ds \sinh a \cosh \paren {b i} + \cosh a \sinh \paren {b i}\)

Hyperbolic Sine of Sum

\(\ds \)

\(=\)

\(\ds \sinh a \cos b + \cosh a \sin \paren {b i}\)

Cosine in terms of Hyperbolic Cosine

\(\ds \)

\(=\)

\(\ds \sinh a \cos b + i \cosh a \sin b\)

Sine in terms of Hyperbolic Sine

$\blacksquare$

Also see

• Hyperbolic Cosine of Complex Number
• Hyperbolic Tangent of Complex Number
• Hyperbolic Cosecant of Complex Number
• Hyperbolic Secant of Complex Number
• Hyperbolic Cotangent of Complex Number