Sine of i/Proof 1
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Theorem
- $\sin i = \paren {\dfrac e 2 - \dfrac 1 {2 e} } i$
Proof
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 $(1) - (2)$:
\(\ds 2 i \sin i\) | \(=\) | \(\ds \frac 1 e - e\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \sin i\) | \(=\) | \(\ds \frac 1 {2 i} \paren {\frac 1 e - e}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\frac e 2 - \frac 1 {2 e} } i\) |
$\blacksquare$