2022TrialShort answer

Q16 (a) (i)

1 mark

Question

Using Euler’s theorem, show that if $z=e^{i\theta}=\cos\theta+i\sin\theta$ , then
$z^n+z^{-n}=2\cos n\theta.$

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Solution

Q.16(a)(i) Euler's theorem

$z^n=e^{in\theta}=\cos n\theta+i\sin n\theta$
$z^{-n}=e^{-in\theta}=\cos(-n\theta)+i\sin(-n\theta)$
$=\cos n\theta-i\sin n\theta\quad [\cos\text{ even, }\sin\text{ odd}]$

$\therefore\ z^n+z^{-n}=2\cos n\theta.$

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Complex Numbers

Other representations of complex numbersUse Euler's formula and convert between Cartesian, polar and exponential forms to solve problems.

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