Question

Find the fifth roots of 243(cos 260° + i sin 260°).

Answer 1

By DeMoivre's theorem, you have
\[\large243(\cos260^\circ+i\sin260^\circ)=\left(3(\cos52^\circ+i\sin52^\circ)\right)^5=\left(3e^{i(52^\circ)}\right)^5\]
For any complex number \(z=re^{i\theta}\), the \(n\)th roots are given as
\[z^{1/n}=r^{1/n}e^{(\theta+2\pi k)/n}=r^{1/n}e^{(\theta+180 k^\circ)/n}\]
for \(k=0,1,\ldots,n-1\).

Answer 2

Typo: should be \(360k^\circ\) in place of \(180k^\circ\).