Question

Find the fourth roots of 81(cos 320° + i sin 320° ). Write the answer in trigonometric form.

Answer 1

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 2

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

`Z=re^(iA)`

We have to find four roots of expression (1).

`Z^4=81 (Cos 320^(\circ)+iSin 320^(\circ))\\\\Z=[81* (Cos(2k\pi + 320^(\circ))+iSin (2k\pi +320^(\circ))]^{(1)/(4)}\\\\Z={3^{{4}*^{(1)/(4)}}* e^{i((2k\pi +320^(\circ))/(4))}}} \\\\Z=3e^{i((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))`

The four values are obtained for, k=0,1,2,3,.