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Question Number 67881 by mr W last updated on 01/Sep/19
find all x,y ∈R such that  (x+yi)^(2019) =x−yi
$${find}\:{all}\:{x},{y}\:\in{R}\:{such}\:{that} \\ $$$$\left({x}+{yi}\right)^{\mathrm{2019}} ={x}−{yi} \\ $$
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}
$$\mid\left({x}+{iy}\right)\mid^{\mathrm{2019}} =\mid{x}−{iy}\mid=\mid{x}+{iy}\mid \\ $$$$\Rightarrow\mid{x}+{iy}\mid^{\mathrm{2019}} =\mid{x}+{iy}\mid \\ $$$$\Rightarrow\mid{x}+{iy}\mid=\mathrm{1} \\ $$$${let}\:{e}^{{ia}} \:={x}+{iy} \\ $$$$\Rightarrow{e}^{{i}\mathrm{2019}{a}} ={e}^{−{ia}+\mathrm{2}{k}\pi} \\ $$$$\Rightarrow{a}=\left\{\frac{\mathrm{2}{k}\pi}{\mathrm{2020}}\mid\mathrm{0}\leqslant{k}\leqslant\mathrm{2019}\right\} \\ $$$${S}=\left\{{e}^{{i}\frac{\mathrm{2}{k}\pi}{\mathrm{2020}}} \:\:\:\mid\mathrm{0}\leqslant{k}\leqslant\mathrm{2019}\right\} \\ $$
Commented by mr W last updated on 01/Sep/19
thanks sir!  i.e.  (x, y)=(cos ((kπ)/(1010)), sin ((kπ)/(1010)))  with 0≤k≤2019
$${thanks}\:{sir}! \\ $$$${i}.{e}. \\ $$$$\left({x},\:{y}\right)=\left(\mathrm{cos}\:\frac{{k}\pi}{\mathrm{1010}},\:\mathrm{sin}\:\frac{{k}\pi}{\mathrm{1010}}\right) \\ $$$${with}\:\mathrm{0}\leqslant{k}\leqslant\mathrm{2019} \\ $$
Commented by mind is power last updated on 01/Sep/19
y′re welcom
$${y}'{re}\:{welcom} \\ $$

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