Mathematics: HELP ME PLEASE!! PLEASE

HELP ME PLEASE!! PLEASE - id13408551, natalymorales96

Ответ:

googoomylizard

27.05.2023

Mathematics

The cube roots are

5(cos96+isin96), 5(cos216+isin216) and 5(cos336+isin336)

Step-by-step explanation:

125(cos 288° + i sin 288°) can be written as 5³(cos 288° + i sin 288°)

5³(cos 288° + i sin 288°) $=5^3e^{i288^0}$

Complex number

$e^{i\theta }=e^{i(\theta +2\pi) }=e^{i(\theta +4\pi) }$

$5^3e^{i288^0}=5^3e^{i(288^0+360^0)}=5^3e^{i(288^0+720^0)}$

Finding cube root

$\left (5^3e^{i288^0} \right )^{\frac{1}{3}}=5e^\frac{i288^0}{3}=5e^{i96^0}=5(cos96+isin96)\\\\\left (5^3e^{i(288^0+360^0)} \right )^{\frac{1}{3}}=5e^\frac{i(288^0+360^0)}{3}=5e^{i216^0}=5(cos216+isin216)\\\\\left (5^3e^{i(288^0+720^0)} \right )^{\frac{1}{3}}=5e^\frac{i(288^0+720^0)}{3}=5e^{i3366^0}=5(cos336+isin336)$

So the cube roots are

5(cos96+isin96), 5(cos216+isin216) and 5(cos336+isin336)

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