Euler's Formula/Examples/e^-i pi by 2
Example of Use of Euler's Formula
- $e^{-i \pi / 2} = -i$
Proof
| \(\ds e^{-i \pi / 2}\) | \(=\) | \(\ds \cos \frac {-\pi} 2 + i \sin \frac {-\pi} 2\) | Euler's Formula | |||||||||||
| \(\ds \) | \(=\) | \(\ds \cos \frac {3 \pi} 2 + i \sin \frac {3 \pi} 2\) | Cosine of Angle plus Full Angle, Sine of Angle plus Full Angle | |||||||||||
| \(\ds \) | \(=\) | \(\ds 0 + i \times \paren {-1}\) | Cosine of $\dfrac {3 \pi} 2$, Sine of $\dfrac {3 \pi} 2$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds -i\) |
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 2$. Geometrical Representations: $(2.19)$