Sine of Complex Number/Proof 1

Theorem

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

Let $i$ be the imaginary unit.

Then:

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

where:

$\sin$ denotes the sine function (real and complex)

$\cos$ denotes the real cosine function

$\sinh$ denotes the hyperbolic sine function

$\cosh$ denotes the hyperbolic cosine function.

Proof

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

\(=\)

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

Sine of Sum

\(\ds \)

\(=\)

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

Hyperbolic Cosine in terms of Cosine

\(\ds \)

\(=\)

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

Hyperbolic Sine in terms of Sine

$\blacksquare$

Also see

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

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.5$. The Functions $e^z$, $\cos z$, $\sin z$