In this video, we dive into De Moivre's Theorem, a powerful tool for raising complex numbers to positive integer powers.
We begin by illustrating how squaring a complex number (z = a + bi) in polar form (z = r(cos θ + i sin θ)) simplifies to z² = r²(cos 2θ + i sin 2θ) using trigonometric identities.
Next, we introduce De Moivre's Theorem, which generalizes this pattern for any positive integer n: zⁿ = rⁿ(cos nθ + i sin nθ). We provide a rigorous proof of the theorem using mathematical induction.
Finally, we demonstrate the practical application of De Moivre's Theorem by working through two clear examples, showing you how to raise complex numbers in rectangular form to various powers.
Perfect for students learning complex numbers and mathematical induction!