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: $(2.19)$