Mathematics

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De Moivre's Theorem:

De Moivre's Theorem is a formula that gives the value of the nth power of a complex number in polar form. It states that for any complex number z = r(cos θ + i sin θ), where r is the modulus (or magnitude) of the complex number and θ is its argument (or angle), and any positive integer n, we have:

(z)^n = r^n (cos nθ + i sin nθ)

In other words, to raise a complex number to a power, we simply raise its modulus to that power and multiply its argument by that power, expressed in polar form. This formula is useful for calculating the roots of complex numbers and for simplifying trigonometric identities. For example, we can use De Moivre's Theorem to show that:

(cos θ + i sin θ)^n = cos nθ + i sin nθ

which is known as Euler's Formula.