Euler's Formula
The beautiful equation linking exponentials, trigonometry, and complex numbers.
Often called the most elegant formula in math.
The Formula
e^(iθ) = cos(θ) + i·sin(θ)
Euler's formula connects exponential functions with trigonometric functions through complex numbers. When θ = π, it gives the famous Euler's identity: e^(iπ) + 1 = 0.
Variables
| Symbol | Meaning |
|---|---|
| e | Euler's number (approximately 2.71828) |
| i | Imaginary unit (√(-1)) |
| θ | Angle in radians |
| cos(θ) | Real part of the complex number |
| sin(θ) | Imaginary part of the complex number |
Example 1
Evaluate e^(iπ/4)
e^(iπ/4) = cos(π/4) + i·sin(π/4)
= √2/2 + i·√2/2
= 0.7071 + 0.7071i
Example 2
Verify Euler's identity: e^(iπ) + 1 = 0
e^(iπ) = cos(π) + i·sin(π)
= -1 + i·0 = -1
-1 + 1 = 0 ✓ (connecting e, i, π, 1, and 0 in one equation)
When to Use It
Use Euler's formula when:
- Working with complex numbers in polar form
- Simplifying trigonometric calculations using exponentials
- Analyzing alternating current (AC) circuits in electrical engineering
- Solving differential equations with oscillatory solutions
Key Notes
- θ must be in radians — using degrees gives wrong results (e.g., e^(i×180°) ≠ e^(iπ))
- e^(iθ) always has magnitude 1, tracing a unit circle in the complex plane; for a circle of radius r, write r·e^(iθ)
- Euler's identity (e^(iπ) + 1 = 0) is a special case at θ = π, linking five fundamental constants: e, i, π, 1, and 0
- In electrical engineering, e^(iωt) represents a rotating phasor — the real part gives the cosine waveform and the imaginary part gives the sine waveform
Key Notes
- Formula: e^(iθ) = cos θ + i sin θ: Relates the complex exponential to sine and cosine. At θ = π: e^(iπ) = −1, giving Euler's identity e^(iπ) + 1 = 0 — often called the most beautiful equation in mathematics for connecting five fundamental constants.
- Geometric interpretation: e^(iθ) traces the unit circle in the complex plane as θ increases. The real part cos θ is the x-coordinate and the imaginary part sin θ is the y-coordinate. Every complex number can be written as re^(iθ) (polar form).
- De Moivre's theorem follows directly: (e^(iθ))^n = e^(inθ), which means (cosθ + i sinθ)^n = cos(nθ) + i sin(nθ). This simplifies computing powers and roots of complex numbers enormously.
- Cosine and sine from Euler: cos θ = (e^(iθ) + e^(−iθ)) / 2; sin θ = (e^(iθ) − e^(−iθ)) / (2i). This representation is used in signal processing and differential equations to replace trig functions with exponentials, which are easier to differentiate and integrate.
- Applications: Euler's formula underpins AC circuit analysis (phasors: V = Ve^(iωt)), Fourier transforms (signal decomposition into e^(iωt) components), quantum mechanics (wave functions ψ = Ae^(ikx)), and solving linear differential equations with constant coefficients.