Using the De Moivre's Theorem, let us work out for the fourth roots of 81(cos 320° + i sin 320° ).
zⁿ = rⁿ (cos nθ + i sin nθ)
z⁴ = 81(cos 320° + i sin 320° )
z = ∜[81(cos 320° + i sin 320° )]
= ∜[3^4 (cos 4*80° + i sin 4*80°)]
= 3(cos 80° + i sin 80°)
Answer with explanation:
The given expression which is in complex form is :
=81 (Cos 320°+Sin 320°)------------------------------------(1)
For, a Complex number in the form of
Z=r [Cos A + i Sin A], Can be written as
[tex]Z=re^{iA}[/tex]
We have to find four roots of expression (1).
[tex]Z^4=81 (Cos 320^{\circ}+iSin 320^{\circ})\\\\Z=[81\times (Cos(2k\pi + 320^{\circ})+iSin (2k\pi +320^{\circ})]^{\frac{1}{4}}\\\\Z={3^{{4}\times^{\frac{1}{4}}}\times e^{i(\frac{2k\pi +320^{\circ}}{4})}}} \\\\Z=3e^{i(\frac{k\pi}{2}+ 80^{\circ}})\\\\Z_{0}=3(Cos 80^{\circ}+iSin 80^{\circ})\\\\Z_{1}=3(Cos 170^{\circ}+iSin 170^{\circ})\\\\Z_{2}=3(Cos 260^{\circ}+iSin260^{\circ})\\\\Z_{3}=3(Cos 350^{\circ}+iSin 350^{\circ})[/tex]
The four values are obtained for, k=0,1,2,3,.
