Cosine of Complex Number/Proof 1

Theorem

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

Let $i$ be the imaginary unit.

Then:

$\cos \left({a + b i}\right) = \cos a \cosh b - i \sin a \sinh b$

where:

$\cos$ denotes the cosine function (real and complex)

$\sin$ denotes the real sine function

$\sinh$ denotes the hyperbolic sine function

$\cosh$ denotes the hyperbolic cosine function

Proof

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

\(=\)

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

Cosine of Sum

\(\ds \)

\(=\)

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

Hyperbolic Cosine in terms of Cosine

\(\ds \)

\(=\)

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

Hyperbolic Sine in terms of Sine

$\blacksquare$

Also see

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