Cosine of Complex Number
Jump to navigation
Jump to search
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 1
| \(\ds \cos \paren {a + b i}\) | \(=\) | \(\ds \cos a \cos \paren {b i} - \sin a \sin \paren {b i}\) | 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 |
$\blacksquare$
Proof 2
| \(\ds \cos a \cosh b - i \sin a \sinh b\) | \(=\) | \(\ds \frac {e^{i a} + e^{-i a} } 2 \frac {e^b + e^{-b} } 2 - i \frac {e^{i a} - e^{-i a} } {2 i} \frac {e^b - e^{-b} } 2\) | Euler's Cosine Identity, Definition of Hyperbolic Cosine, Euler's Sine Identity, Definition of Hyperbolic Sine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {e^{b + i a} + e^{-b + i a} + e^{b - i a} + e^{-b - i a} - e^{b + i a} + e^{-b + i a} + e^{b - i a} - e^{-b - i a} } 4\) | simplifying | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {e^{-b + i a} + e^{b - i a} } 2\) | simplifying | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {e^{i \paren {a + b i} } + e^{-i \paren {a + b i} } } 2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \map \cos {a + b i}\) | Euler's Cosine Identity |
$\blacksquare$