Question

Комплексные числа. Срочно нужно. 35 баллов

Answer 1

$z=x+yi \ |z|= sqrt{x^2+y^2} \ z_1=1+1i\ |z_1|= sqrt{1^2+1^2} = sqrt{2} \ |z_2|= sqrt{(-2)^2}= 2 \ |z_3|= sqrt{ (sqrt{3})^2+(-1)^2 }=2 \ |z_4|= sqrt{0^2+2^2}=2$
$Arg(z_1)= frac{ pi }{4} \ Arg(z_2)=- pi \ Arg(z_3)=arctg( frac{-1}{ sqrt{3} })=- frac{ pi }{6} \ Arg(z_4)= frac{ pi }{2}$
$z=|z|(cos alpha +i sin alpha ), alpha =Arg(z) \ z_1= sqrt{2} (cos frac{ pi }{4} +isin frac{ pi }{4}) \ z_2=2(cos( - pi)+isin(- pi ) \ z_3=2e^{-i frac{ pi }{6} } \ z_4=2e^{i frac{ pi }{2}}$
$frac{z_2}{z_1}= frac{2(cos( - pi)+isin(- pi )}{sqrt{2} (cos frac{ pi }{4} +isin frac{ pi }{4})} =i-1 \ i-1=|i-1|(cos frac{3 pi }{4} +isinfrac{3 pi }{4})= sqrt{2} (cos frac{3 pi }{4} +isinfrac{3 pi }{4})$
$z_3times z_4=2e^{-i frac{ pi }{6} }times2e^{i frac{ pi }{2}}=4e^{i frac{ pi }{3}} \ e^{ifrac{ pi }{3}}=cosfrac{ pi }{3}+isinfrac{ pi }{3} \ x=|z|cosfrac{ pi }{3} \ y=|z|sinfrac{ pi }{3} \ x=4times 0,5=2 \ y=4times frac{ sqrt{3} }{2}=2 sqrt{3} \ z=2+2 sqrt{3} i$
$sqrt[6]{-2}= sqrt[6]{2} (cos frac{- pi }{6} +isin frac{- pi }{6})=sqrt[6]{2}(± frac{ sqrt{3} }{2} ±0,5i) \ sqrt[6]{-2}=± sqrt[6]{2} i$
$Z_3^4=(2e^{-i frac{ pi }{6} })^4=16e^{-i frac{2 pi }{3}$
$frac{(i-1)^3}{i^{12}+i^{31}}= frac{2+2i}{1-i}=2i=2(cos frac{ pi }{2} +isin frac{ pi }{2})$