The De Moivre's Theorem Calculator helps you easily raise complex numbers to a power. This is very useful in subjects like engineering, physics, and mathematics, where complex numbers are often used. The calculator simplifies the process using De Moivre's Theorem, which works with complex numbers in polar form.
Purpose and Functionality
De Moivre's Theorem: This theorem allows you to raise complex numbers to a power efficiently using polar coordinates. It's given by the formula:
zn=rn(cos(nθ)+isin(nθ))z^n = r^n (\cos(n\theta) + i\sin(n\theta))zn=rn(cos(nθ)+isin(nθ))
Where:
- zzz is the complex number.
- rrr is the magnitude (or modulus) of the complex number.
- θ\thetaθ is the angle the complex number makes with the positive real axis.
- nnn is the power to which the complex number is raised.
Inputs for the Calculator
- Magnitude (r): This is the distance from the origin in the complex plane.
- Angle (θ): This is the angle in degrees or radians that the complex number makes with the positive real axis.
- Power (n): This is the exponent to which the complex number is raised.
Calculation Steps
- Input the Magnitude rrr: Enter the magnitude of the complex number.
- Input the Angle θ\thetaθ: Enter the angle in degrees or radians. If the angle is in degrees, convert it to radians using the formula: Radians=Degrees×π180\text{Radians} = \text{Degrees} \times \frac{\pi}{180}Radians=Degrees×180π
- Input the Power nnn: Enter the power to which the complex number is to be raised.
- Calculate the New Magnitude: Compute the new magnitude as rnr^nrn.
- Calculate the New Angle: Compute the new angle as nθn\thetanθ. Convert it back to degrees if needed.
- Compute the Result using De Moivre's Theorem: Use the formula to get zn=rn(cos(nθ)+isin(nθ))z^n = r^n (\cos(n\theta) + i\sin(n\theta))zn=rn(cos(nθ)+isin(nθ)).
Step-by-Step Example
Suppose you have a complex number with a magnitude of 2 and an angle of 30°, and you want to calculate its 3rd power.
- Magnitude rrr: 2
- Angle θ\thetaθ: 30° (which is π6\frac{\pi}{6}6π radians)
- Power nnn: 3
Calculation:
- New Magnitude: 23=82^3 = 823=8
- New Angle: 3×30°=90°3 \times 30° = 90°3×30°=90° (which is π2\frac{\pi}{2}2π radians)
Result: z3=8(cos(π2)+isin(π2))=8(0+i⋅1)=8iz^3 = 8 \left( \cos\left(\frac{\pi}{2}\right) + i\sin\left(\frac{\pi}{2}\right) \right) = 8(0 + i \cdot 1) = 8iz3=8(cos(2π)+isin(2π))=8(0+i⋅1)=8i
This shows how a complex number can be efficiently raised to a power using De Moivre's Theorem, simplifying calculations significantly.
Relevant Information Table
| Input | Description |
|---|---|
| Magnitude (r) | Distance from the origin in the complex plane (scalar) |
| Angle (θ) | Angle in degrees or radians |
| Power (n) | Exponent (integer) |
Conclusion
The De Moivre's Theorem Calculator is a powerful tool for efficiently raising complex numbers to a power. It simplifies complex calculations and is particularly useful in fields where complex numbers are frequently used. This calculator can handle inputs in both degrees and radians, making it versatile and user-friendly. By converting the problem into polar coordinates and using De Moivre's Theorem, you can achieve accurate results quickly and easily.