Question

Тема: Комплексные числа в тригонометрической форме.
Запишите комплексное число в тригонометрической и показательной формах:
z=2-2i

Answer 1

$z=Re+i*Im\\\\ r=|z|=\sqrt{(Re)^2+(Im)^2}\\\\ cos(\phi)=\frac{Re}{r}=\frac{Re}{\sqrt{(Re)^2+(Im)^2}}\\\\ sin(\phi)=\frac{Im}{r}=\frac{Im}{\sqrt{(Re)^2+(Im)^2}}\\\\ z=r*[cos(\phi)+i*sin(\phi)]$

$z=2-2*i\\\\ Re=2\ \ \ Im=-2\\\\ r=|z|=\sqrt{(Re)^2+(Im)^2}=\sqrt{2^2+(-2)^2}=2\sqrt{2}\\\\ cos(\phi)=\frac{Re}{r}=\frac{2}{2\sqrt{2}}=\frac{\sqrt{2}}{2}\\\\ sin(\phi)=\frac{Im}{r}=\frac{-2}{2\sqrt{2}}=-\frac{\sqrt{2}}{2}\\\\ \phi=-\frac{\pi}{4}\\\\ z=r*[cos(\phi)+i*sin(\phi)]\\\\ z=2\sqrt{2}*[cos(-\frac{\pi}{4})+i*sin(-\frac{\pi}{4})]\\\\ z=|z|*e^{i\phi}=2\sqrt{2}*e^{i*(-\frac{\pi}{4})}$

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Ответ: $2\sqrt{2}*[cos(-\frac{\pi}{4})+i*sin(-\frac{\pi}{4})]=2\sqrt{2}*e^{i*(-\frac{\pi}{4})}$