Euler's Sine Identity/Proof 3
Jump to navigation
Jump to search
Theorem
- $\sin z = \dfrac {e^{i z} - e^{-i z} } {2 i}$
Proof
\(\text {(1)}: \quad\) | \(\ds e^{i z}\) | \(=\) | \(\ds \cos z + i \sin z\) | Euler's Formula | ||||||||||
\(\text {(2)}: \quad\) | \(\ds e^{-i z}\) | \(=\) | \(\ds \cos z - i \sin z\) | Euler's Formula: Corollary | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds e^{i z} - e^{-i z}\) | \(=\) | \(\ds \paren {\cos z + i \sin z} - \paren {\cos z - i \sin z}\) | $(1) - (2)$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 2 i \sin z\) | simplifying | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {e^{i z} - e^{-i z} } {2 i}\) | \(=\) | \(\ds \sin z\) |
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.5$. The Functions $e^z$, $\cos z$, $\sin z$: $\text{(ii)}$: $(4.17)$