De Moivre’s Theorem Calculator

The De Moivre’s Theorem Calculator is a mathematical 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))

Where:

• 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)	Argument (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 time.

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.

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