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: $(2.19)$