Complex Numbers Past Paper Questions CAPE Pure Maths

Exponential Notation

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Arguments

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Loci

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Sum and Product of the Roots

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De Moivre

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Square Roots of Complex Numbers

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Loci

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Use de Moivre's theorem to prove that 

 $\sin5x=16\sin^5x-20\sin^3x+5\sin x$  

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(i)  Write the complex number $z=\left(-1+\sqrt{3}\ i\right)^7$ in the form  $z=r\ e^{i\theta}$  where  $r=\left|z\right|$  and  $\theta=\arg\left(z\right)$  . 

(ii) Hence, prove that  $\left(-1+\sqrt{3}\ i\right)^7=64\left(-1+\sqrt{3}\ i\right)$  

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