De Moivre's theorem
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De Moivre's Theorem
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De Moivre's theorem is a result that enables us to find powers and roots of complex numbers. Recall that complex numbers can be expressed in Cartesian or in polar form:[Equation goes here - download the original to see it.] De Moivre's Theorem tells us how to evaluate powers of a complex number - that is zn. It can be expressed in Cartesian and polar form: Cartesian form [Equation goes here - download the original to see it.] Polar form [Equation goes here - download the original to see it.] The proof follows by mathematical induction and exploits the property of multiplication of complex numbers. In polar form [Equation goes here - download the original to see it.] The proof in polar form is particularly straight-forward and elegant Proof of De Moivre's Theorem To prove [Equation goes here - download the original to see it.] Proof by mathematical induction Particular step [Equation goes here - download the original to see it.] Induction step The induction hypothesis is: For n = k [Equation goes here - download the original to see it.] To prove for n = k + 1 [Equation goes here - download the original to see it.] [Diagram goes here - download the original to see it. Conclusion Hence the induction step holds and the result is true for all [Equation goes here - download the original to see it.] Converting into Cartesian form gives: [Equation goes here - download the original to see it.]
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Contents of
De Moivre's theorem
1 De Moivre's Theorem
2 Interpretation of De Moivre's Theorem and the n roots of unity
3 Applications of De Moivre's Theorem to Trigonometric Identities
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