Question

HELPPPPPP find the fourth roots of 81(cos160+isin160) PLEASE

Answer 1

`81(\cos160^\circ+i\sin160^\circ)=81e^(160^\circ i)`

By DeMoivre's theorem,

`\left(81e^(160^\circ i)\right)^(1/4)=81^(1/4)e^((160+360k)^\circ i/4)`

where
`k=0,1,2,3`.
`81=3^4\implies81^(1/4)=3`, so the 4th roots are

`k=0:\quad3e^(40^\circ i)=3(\cos40^\circ+i\sin40^\circ)`

`k=1:\quad3e^(130^\circ i)=3(\cos130^\circ+i\sin130^\circ)`

`k=2:\quad3e^(220^\circ i)=3(\cos220^\circ+i\sin220^\circ)`

`k=3:\quad3e^(310^\circ i)=3(\cos310^\circ+i\sin310^\circ)`