Euler's Formula/Real Domain/Corollary

Corollary to Euler's Formula: Real Domain

Let $\theta \in \R$ be a real number.

Then:

$e^{-i \theta} = \cos \theta - i \sin \theta$

where:

$e^{-i \theta}$ denotes the complex exponential function

$\cos \theta$ denotes the real cosine function

$\sin \theta$ denotes the real sine function

$i$ denotes the imaginary unit.

Proof

\(\ds e^{-i \theta}\)

\(=\)

\(\ds \cos \paren {-\theta} + i \sin \paren {-\theta}\)

Euler's Formula

\(\ds \)

\(=\)

\(\ds \cos \theta + i \sin \paren {-\theta}\)

Cosine Function is Even

\(\ds \)

\(=\)

\(\ds \cos \theta - i \sin \theta\)

Sine Function is Odd

$\blacksquare$

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 2$. Geometrical Representations
• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 7$: Relationship between Exponential and Trigonometric Functions: $7.16$