Question

Комплексные числа. Срочно нужно. 35 баллов

Answer 1

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