Cosine of Complex Number/Proof 1
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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 |
$\blacksquare$