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