find-all-x-y-R-such-that-x-yi-2019-x-yi-

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 + 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