find all x,y ∈R such that (x+yi)^(2019) =x−yi


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Question Number 67881 by mr W last updated on 01/Sep/19

$f i n d a l l x, y \in R s u c h t h a t$ ${(x + y i)}^{2019} = x - y i$
	Answered by mind is power last updated on 01/Sep/19
	$∣(x+iy)∣^(2019) =∣x−iy∣=∣x+iy∣ ⇒∣x+iy∣^(2019) =∣x+iy∣ ⇒∣x+iy∣=1 let e^(ia) =x+iy ⇒e^(i2019a) =e^(−ia+2kπ) ⇒a={((2kπ)/(2020))∣0≤k≤2019} S={e^(i((2kπ)/(2020))) ∣0≤k≤2019}$
	$∣ (x + i y) ∣^{2019} =∣ x - i y ∣=∣ x + i y ∣$ $\Rightarrow∣ x + i y ∣^{2019} =∣ x + i y ∣$ $\Rightarrow∣ x + i y ∣= 1$ $l e t e^{i a} = x + i y$ $\Rightarrow e^{i 2019 a} = e^{- i a + 2 k π}$ $\Rightarrow a = {\frac{2 k π}{2020} ∣ 0 ⩽ k ⩽ 2019}$ $S = {e^{i \frac{2 k π}{2020}} ∣ 0 ⩽ k ⩽ 2019}$
	Commented by mr W last updated on 01/Sep/19

	$t h a n k s s i r!$ $i . e .$ $(x, y) = (\cos \frac{k π}{1010}, \sin \frac{k π}{1010})$ $w i t h 0 ⩽ k ⩽ 2019$
	Commented by mind is power last updated on 01/Sep/19

	$y^{'} r e w e l c o m$
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