Euler's Formula/Examples/e^i pi by 4
Example of Use of Euler's Formula
- $e^{i \pi / 4} = \dfrac {1 + i} {\sqrt 2}$
Proof
| \(\ds e^{i \pi / 4}\) | \(=\) | \(\ds \cos \frac \pi 4 + i \sin \frac \pi 4\) | Euler's Formula | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {\sqrt 2} 2 + i \dfrac {\sqrt 2} 2\) | Cosine of $\dfrac \pi 4$, Sine of $\dfrac \pi 4$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {1 + i} {\sqrt 2}\) | simplification |
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 2$. Geometrical Representations: $(2.19)$