Master Complex Number Calculations With Demoivre’S Theorem | Explained Simply

DeMoivre’s Theorem

[r(cosx + isinx)]^n = r^n(cos(nx)+isin(nx))

DeMoivre’s theorem is a mathematical formula that is used to calculate the powers of complex numbers. It is named after Abraham De Moivre, an 18th-century mathematician who lived in England.

The theorem states that if a complex number is represented in polar form as r(cos θ + i sin θ), then its nth power can be expressed as
r^n (cos nθ + i sin nθ).

In simpler terms, the formula tells us that when we raise a complex number to a power, we can multiply its magnitude by that power and add the power of the angle at which it lies. The result gives the polar form of the complex number raised to that power.

For example, suppose we want to calculate the cube of the complex number 3(cos(π/3) + i sin(π/3)). Using DeMoivre’s Theorem, we can write:

(3(cos(π/3) + i sin(π/3)))^3 = 3^3(cos(3π/3) + i sin(3π/3))
= 27(cos(π) + i sin(π))
= -27

Thus, we see that the cube of the given complex number is -27.

DeMoivre’s theorem is a powerful tool that simplifies calculations involving complex numbers significantly. It is used extensively in many branches of mathematics, including trigonometry, calculus, and engineering.

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