DeMoivre's Theorem (Leaving Cert Mathematics): Revision Notes

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Using Demoivre to Prove Trigonometric Identities

De Moivre's can be used to prove trigonometric identities.

Example

infoNote

Use de Moivre's theorem to prove the identity (cos⁡θ+isin⁡θ)3=cos⁡(3θ)+isin⁡(3θ)(\cos \theta + i \sin \theta)^3 = \cos(3\theta) + i \sin(3\theta).

Expand the left hand side :

(cos⁡θ+isin⁡θ)3=cos⁡3θ−3cos⁡θsin⁡2θ+i(3cos⁡2θsin⁡θ−sin⁡3θ)(\cos \theta + i \sin \theta)^3 = \cos^3 \theta - 3 \cos \theta \sin^2 \theta + i (3 \cos^2 \theta \sin \theta - \sin^3 \theta)

Compare real with real, imaginary with imaginary :

cos⁡(3θ)=cos⁡3θ−3cos⁡θsin⁡2θ\cos(3 \theta) = \cos^3 \theta - 3 \cos \theta \sin^2 \theta sin⁡(3θ)=3cos⁡2θsin⁡θ−sin⁡3θ\sin(3 \theta) = 3 \cos^2 \theta \sin \theta - \sin^3 \theta

Explore Leaving Cert Mathematics Model Answers by Topics

Fundamentals of Complex Numbers

The Basics

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Solving Equations

DeMoivre's Theorem

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Explore Leaving Cert Mathematics Quizzes by Topics

Fundamentals of Complex Numbers

The Basics

Argand Diagrams & Modulus

Solving Equations

DeMoivre's Theorem

Transformations

Explore Leaving Cert Mathematics Flashcards by Topics

Fundamentals of Complex Numbers

The Basics

Argand Diagrams & Modulus

Solving Equations

DeMoivre's Theorem

Transformations

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