Calculator
Example Data Table
Sample values below show how Euler formula maps common angles onto the complex plane.
| Angle | Radians | cos(θ) | sin(θ) | e^(iθ) |
|---|---|---|---|---|
| 0° | 0 | 1 | 0 | 1 + 0i |
| 30° | π/6 | 0.8660 | 0.5000 | 0.8660 + 0.5000i |
| 45° | π/4 | 0.7071 | 0.7071 | 0.7071 + 0.7071i |
| 60° | π/3 | 0.5000 | 0.8660 | 0.5000 + 0.8660i |
| 90° | π/2 | 0 | 1 | 0 + 1i |
Formula Used
r·e^(iθ) = r cos(θ) + i r sin(θ)
Real Part = r cos(θ)
Imaginary Part = r sin(θ)
Magnitude = √(Real² + Imaginary²)
Phase = atan2(Imaginary, Real)
This calculator applies Euler’s formula to convert a complex number between exponential, trigonometric, rectangular, and polar forms. When r = 1, the point lies on the unit circle, so the magnitude of e^(iθ) remains 1 for every angle.
How to Use This Calculator
- Enter the angle θ in either degrees or radians.
- Enter the magnitude r to evaluate r·e^(iθ).
- Choose the number of decimal places to display.
- Press the calculate button to generate the result panel.
- Review the real part, imaginary part, magnitude, phase, and converted forms.
- Use the CSV or PDF buttons to save the computed output.
FAQs
1. What does Euler’s formula show?
It links exponential notation with trigonometric form. Specifically, e^(iθ) equals cos(θ) plus i·sin(θ), letting you move between complex-number representations easily.
2. Why is the magnitude of e^(iθ) always 1?
Because the real and imaginary parts are cos(θ) and sin(θ). Their squared sum is 1, so the point always stays on the unit circle.
3. What is the role of r in r·e^(iθ)?
The value r scales the unit-circle result. It changes the distance from the origin while preserving the angle, which makes polar-to-rectangular conversion straightforward.
4. Can I enter angles in radians?
Yes. The calculator accepts both degrees and radians. It converts the angle internally so every result stays consistent across forms.
5. What does the rectangular result mean?
It is the complex number written as a + bi. Here, a is the real component and b is the imaginary component.
6. Why is the recovered phase useful?
It verifies the angle implied by the calculated real and imaginary parts. This helps confirm conversions and catch input mistakes.
7. When is Euler’s formula used in practice?
It appears in signal processing, differential equations, Fourier analysis, control systems, and electrical engineering whenever oscillations or rotations are modeled.
8. What happens if r equals zero?
The rectangular result becomes 0 + 0i. Magnitude is zero, and the phase is numerically reported from atan2, though angle interpretation becomes degenerate.