The De Moivre’s Theorem Calculator is a 数学的 tool used to compute powers and roots of complex numbers. It simplifies the process of raising complex numbers to a power by using trigonometric functions, making calculations easier for students, engineers, and mathematicians.
This calculator is useful in fields like electrical engineering, quantum mechanics, and signal processing, where complex numbers are essential.
Formula of De Moivre’s Theorem Calculator
De Moivre’s Theorem states that for a complex number expressed in polar form:
z = r (cos θ + i sin θ)
its n-th power is given by:
z^n = r^n (cos(nθ) + i sin(nθ))
どこ：
- z is the complex number
- r is the modulus (absolute value) of the complex number
- θ (theta) is the argument (angle) of the complex number in radians
- n is the power to which the complex number is raised
This formula allows complex number exponentiation without expanding binomial expressions manually.
De Moivre’s Theorem Table
The table below provides commonly used values for quick reference.
| Modulus (r) | 引数(θ) | Power (n) | Result (Polar Form) |
|---|---|---|---|
| 2 | π/ 4 | 2 | 4 (cos π/2 + i sin π/2) |
| 3 | π/ 3 | 3 | 27 (cos π + i sin π) |
| 1 | π/ 6 | 4 | 1 (cos 2π/3 + i sin 2π/3) |
| 5 | π/ 2 | 5 | 3125 (cos 5π/2 + i sin 5π/2) |
This table helps quickly verify results without needing to calculate every 時間.
Example of De Moivre’s Theorem Calculator
Let’s calculate (1 + i)³ using De Moivre’s Theorem.
Step 1: Convert to Polar Form
The modulus is:
r = √(1² + 1²) = √2
The argument is:
θ = tan⁻¹ (1/1) = π/4
Step 2: Apply De Moivre’s Theorem
(1 + i)³ = (√2)³ × (cos (3 × π/4) + i sin (3 × π/4))
= 2√2 (cos (3π/4) + i sin (3π/4))
= 2√2 (-1/√2 + i 1/√2) = -2 + 2i
Thus, (1 + i)³ = -2 + 2i.
最も一般的な FAQ
De Moivre’s Theorem simplifies complex number exponentiation and is essential in solving trigonometric equations, signal processing, and physics problems.
Yes. By using fractional exponents (n = 1/2, 1/3, etc.), you can compute complex roots of numbers.