Cosine of i
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Theorem
- $\cos i = \dfrac e 2 + \dfrac 1 {2 e}$
where $\cos$ denotes the complex cosine function and $i$ is the imaginary unit.
Proof 1
We have:
| \(\text {(1)}: \quad\) | \(\ds \cos i + i \sin i\) | \(=\) | \(\ds e^{i \times i}\) | Euler's Formula | ||||||||||
| \(\ds \) | \(=\) | \(\ds e^{-1}\) | Definition of Imaginary Unit | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 e\) |
Also:
| \(\text {(2)}: \quad\) | \(\ds \cos i - i \sin i\) | \(=\) | \(\ds \map \cos {-i} + i \map \sin {-i}\) | Cosine Function is Even and Sine Function is Odd | ||||||||||
| \(\ds \) | \(=\) | \(\ds e^{i \times \paren {-i} }\) | Euler's Formula | |||||||||||
| \(\ds \) | \(=\) | \(\ds e^1\) | Definition of Imaginary Unit | |||||||||||
| \(\ds \) | \(=\) | \(\ds e\) |
Then from $(1) + (2)$:
| \(\ds 2 \cos i\) | \(=\) | \(\ds \frac 1 e + e\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \cos i\) | \(=\) | \(\ds \frac 1 2 \paren {\frac 1 e + e}\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac e 2 + \frac 1 {2 e}\) |
$\blacksquare$
Proof 2
| \(\ds \cos i\) | \(=\) | \(\ds \cosh 1\) | Hyperbolic Cosine in terms of Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {e^1 + e^{-1} } 2\) | Definition of Hyperbolic Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac e 2 + \frac 1 {2 e}\) |
$\blacksquare$