Question

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

Answer 1

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).

$Z^4=81 (Cos 320^{\circ}+iSin 320^{\circ})\\\\Z=[81\times (Cos(2k\pi + 320^{\circ})+iSin (2k\pi +320^{\circ})]^{\frac{1}{4}}\\\\Z={3^{{4}\times^{\frac{1}{4}}}\times e^{i(\frac{2k\pi +320^{\circ}}{4})}}} \\\\Z=3e^{i(\frac{k\pi}{2}+ 80^{\circ}})\\\\Z_{0}=3(Cos 80^{\circ}+iSin 80^{\circ})\\\\Z_{1}=3(Cos 170^{\circ}+iSin 170^{\circ})\\\\Z_{2}=3(Cos 260^{\circ}+iSin260^{\circ})\\\\Z_{3}=3(Cos 350^{\circ}+iSin 350^{\circ})$

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

Answer 2

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°)