1. What is De Moivre's Theorem?

De Moivre's Theorem is a formula that connects complex numbers and trigonometry. It states that for any real number θ and integer n, the following holds:

2. How Does the Calculator Work?

The calculator applies De Moivre's Theorem to compute:

Where:

• \( r \) — Magnitude of the complex number (always positive)
• \( \theta \) — Angle in radians
• \( n \) — Exponent (can be any real number)
• \( i \) — Imaginary unit (√-1)

Explanation: The theorem allows easy computation of powers and roots of complex numbers in polar form.

3. Importance of De Moivre's Theorem

Details: This theorem is fundamental in complex number theory, electrical engineering, and physics. It simplifies complex number exponentiation and helps find roots of complex numbers.

4. Using the Calculator

Tips: Enter the magnitude (r > 0), angle in radians, and the exponent n. The calculator provides results in both polar and rectangular (a + bi) forms.

5. Frequently Asked Questions (FAQ)

Q1: Can n be a fractional number?
A: Yes, De Moivre's Theorem works for any real number n, including fractions (which correspond to roots).

Q2: What if my angle is in degrees?
A: Convert degrees to radians first (multiply by π/180) before using the calculator.

Q3: Can r be negative?
A: No, r represents magnitude and must be positive. Negative magnitudes are not meaningful in polar form.

Q4: How is this used in real applications?
A: It's used in signal processing, electrical engineering (AC circuit analysis), and quantum mechanics.

Q5: What about Euler's formula?
A: De Moivre's Theorem is essentially the integer version of Euler's formula \( e^{iθ} = \cos θ + i \sin θ \).