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Complex Numbers question

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I'm going to assume you know de Moivre's theorem

 

so if z = cos x + i sin x then z = e^ix

 

1/z^n = z^-n = (z^n)^-1 = (cos (-x) + i sin (-x)) = cos x - i sin x (because cos x is an even function)

 

So, z + 1/z = cos x + i sin x + cos x - i sin x = 2cos x which is real. So Im(z + 1/z) = 0

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Ossih has a perfect answer for 5(a). However it seems that you also ask for help for question 5(b), so I’ll write down the whole worked solution here:

 

post-115475-0-01714000-1412977887_thumb.

 

Tell me if anything is unclear :)

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