Euler's Formula

📊 intermediate

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

Relates complex exponentials to trigonometric functions.

Variables & Units

$e$

Euler's number

Unit: $\text{unitless}$

$i$

Imaginary unit ($i^2=-1$)

Unit: $\text{unitless}$

$\theta$

Angle (radians)

Unit: $\text{rad}$

Key Information

📐 Derived From

Taylor series expansions of $e^{i\theta}$, $\cos\theta$, $\sin\theta$.

💡 Example

For $\theta = \pi$: $$e^{i\pi} = \cos\pi + i\sin\pi = -1$$ so $e^{i\pi} + 1 = 0$.

🔧 Applications

Signal processing, quantum mechanics, AC circuit analysis.

⚠️ Constraints

$\theta$ real.

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