Sine of i
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Theorem
- $\sin i = \paren {\dfrac e 2 - \dfrac 1 {2 e} } i$
where $\sin$ denotes the complex sine 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 $(1) - (2)$:
| \(\ds 2 i \sin i\) | \(=\) | \(\ds \frac 1 e - e\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \sin i\) | \(=\) | \(\ds \frac 1 {2 i} \paren {\frac 1 e - e}\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {\frac e 2 - \frac 1 {2 e} } i\) |
$\blacksquare$
Proof 2
| \(\ds \sin i\) | \(=\) | \(\ds i \sinh 1\) | Hyperbolic Sine in terms of Sine | |||||||||||
| \(\ds \) | \(=\) | \(\ds i \frac {e^1 - e^{-1} } 2\) | Definition of Hyperbolic Sine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {\frac e 2 - \frac 1 {2 e} } i\) |
$\blacksquare$