Application of De Moivre’s Theorem (Complex Numbers)

Shemar, Alisa, Mohamed, Cooper

What is De Moivre’s Theorem?

The equation is:

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• It can be used to both find complex numbers raised to a real power or all the roots of an equation
• Based on how the resultant products of increasing complex numbers rotate counterclockwise evenly

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Guided Problem

Using De Moivre’s Theorem, simplify the following expression:

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First, convert to trigonometric form.

Next, apply De Moivre’s Theorem by raising the modulus to the sixth power and multiplying the argument by six.

Lesson 7

• Plug in every value for K from 0 till n-1
• Each complex number that results from this equation is one of the roots

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Finding the nth root of a complex number

With Z being the complex number and n being the power it is raised to

The fourth roots of -1

First find the modulus, r = 1 and then the argument, arg = 𝜋. This gives us -1 in polar form, 1 cis(𝜋)

Next plug the values into the formula, where r = 1, 𝛳 = 𝜋, n = 4, and k = n-1

Find the fifth roots of -9-9i√2

First convert to polar form by finding the modulus, r=√((-9)²+(-9)²)=√(243) = 9√3

Then find the argument, arg = tan^-1 (y/x) = tan^-1((-9√2)/(-9)) = tan^-1(√2) ≈ 0.9553

However, since the angle is in the third quadrant you add 𝜋 to that so the argument would be ≈ 4.0969

Polar Form: 9√3 cis(4.0969) or 9√3 (cos(4.0969)+ i sin(4.0969))

Then use this equation to find the roots by plugging in values for k given that k = n-1

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