Mathematical Methods for Physics and Engineering - Matematica.NET

3.4 DE MOIVRE’S THEOREMIm zr 1 e iθ 1r 2 e iθ 2r 1e i(θ 1−θ 2 )r 2Re zFigure 3.9 The division of two complex numbers. As in the previous figure,r 1 and r 2 are both greater than unity.immediately apparent. The division of two complex numbers in polar form isshown in figure 3.9.3.4 de Moivre’s theoremWe now derive an extremely important theorem. Since ( e iθ) n= e inθ , we have(cos θ + i sin θ) n =cosnθ + i sin nθ, (3.27)where the identity e inθ =cosnθ + i sin nθ follows from the series definition ofe inθ (see (3.21)). This result is called de Moivre’s theorem andisoftenusedinthemanipulation of complex numbers. The theorem is valid for all n whether real,imaginary or complex.There are numerous applications of de Moivre’s theorem but this sectionexamines just three: proofs of trigonometric identities; finding the nth roots ofunity; and solving polynomial equations with complex roots.3.4.1 Trigonometric identitiesThe use of de Moivre’s theorem in finding trigonometric identities is best illustratedby example. We consider the expression of a multiple-angle function interms of a polynomial in the single-angle function, and its converse.95