Question

28-30 комплексные числа

Answer 1

$28.\\z_1 = 0.6(cos120 + isin120) = 0.6e^{i\frac{2}{3}\pi}\\z_2 = 3(cos240+isin240) = 3e^{i\frac{4}{3}\pi}\\\frac{z_1}{z_2} = \frac{0.6e^{i\frac{2}{3}\pi}}{3e^{i\frac{4}{3}\pi}} = \frac{1}{5}e^{i(-\frac{2}{3}\pi)}=0.2(cos120-isin120) = 0.2(0.5 - 0.5\sqrt{3}i) = 0.1 - 0.1\sqrt{3}i\\Answer: 0.1 - 0.1\sqrt{3}i\\29.\\z = 2(cos60+isin60) => z^4 = 2^4(cos(4*60) + isin(4*60)) = 16(cos240 + isin240) = 16(-0.5-0.5\sqrt{3}i) = -8 - 8\sqrt{3}i\\Answer: -8 - 8\sqrt{3}i$

$30.\\z = 1 - i = \sqrt{2}(cos(-45)+isin(-45))\\\sqrt[3]{z} = z^{\frac{1}{3}} =\sqrt{2}^{\frac{1}{3}}(cos(-15+\frac{2}{3}\pi n)+ sin(-15+\frac{2}{3}\pi n)), n \in [0;1;2]\\Answer: 2^{\frac{1}{6}}(cos(-15+\frac{2}{3}\pi n)+ sin(-15+\frac{2}{3}\pi n)), n \in [0;1;2]$