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Euler's Formula (e^{iθ})

Complex exponential e^{iθ} calculator. Enter an angle θ to get its trigonometric form via Euler's formula, with real part, imaginary part, modulus and argument—for complex numbers.

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e^{iθ} = cos θ + i·sin θ
enter degrees, auto-converted

📖 Tutorial | Euler's Formula (e^{iθ})

1. Definition
Euler's formula e^{iθ}=cosθ+i·sinθ links complex exponentials to trig: on the unit circle, angle θ maps to (cosθ, sinθ) with modulus always 1.
2. Symbols
SymbolMeaning
θangle (radians)
cosθreal part
sinθimaginary part
|e^{iθ}|=1point on the unit circle
3. How it works
  • Degrees are converted to radians;
  • Real=cosθ, imaginary=sinθ;
  • Modulus=√(cos²+sin²)=1;
  • The famous special case θ=π gives e^{iπ}+1=0.
4. Steps
  1. Enter the angle in degrees, e.g. 60;
  2. Click Calculate;
  3. Read real, imaginary and modulus;
  4. Understand rotation on the unit circle.
5. Example
Example: θ=60°. e^{i60°}=cos60°+i·sin60°=0.5+0.866i, modulus=1.
6. Pitfalls
Input is in degrees; the formula uses radians;
The modulus is always 1;
e^{iπ}+1=0 is the θ=π special case.

❓ FAQ | Euler's Formula (e^{iθ})

What does Euler’s formula say?
e^{iθ}=cosθ+i·sinθ, linking complex exponentials to trig.
Why is the modulus always 1?
√(cos²θ+sin²θ)=1, so it lies on the unit circle.
Where does e^{iπ}+1=0 come from?
Set θ=π: cosπ=−1, sinπ=0, so e^{iπ}=−1.
Degrees or radians?
Input degrees here; converted to radians internally.
Where is it used?
Signal processing, circuits and Fourier analysis for rotation/phase.