Question

Which of the following is a third root of the given complex number.

Answer 1

The correct option is C.

Step-by-step explanation:

The given complex number is

`z=64(\cos[(\pi)/(7)]-i\sin[(\pi)/(7)])`

We need to find the third root of the given complex number.

`z^{(1)/(3)}=(64(\cos[(\pi)/(7)]-i\sin[(\pi)/(7)]))^{(1)/(3)}`

`z^{(1)/(3)}=64^{(1)/(3)}(\cos[(\pi)/(7)]-i\sin[(\pi)/(7)])^{(1)/(3)}`

`z^{(1)/(3)}=4(\cos[(\pi)/(7)]-i\sin[(\pi)/(7)])^{(1)/(3)}`

De moivre's theorem:

`(\cos\theta+i\sin\theta)^n=\cos n\theta+i\sin n\theta`

where, n is an integer.

Using de moivre's theorem, we get

`z^{(1)/(3)}=4(\cos[(\pi)/(7)* (1)/(3)]-i\sin[(\pi)/(7)* (1)/(3)])`

`z^{(1)/(3)}=4(\cos[(\pi)/(21)]-i\sin[(\pi)/(21)])`

Therefore the correct option is C.

Answer 2

• 4(cos(15π/21) +i·sin(15π/21))
• 4(cos(π/21) +i·sin(π/21))

Explanation:

The third root of the number 64∠(π/7) will be ...

64^(1/3) ∠((π/7 +2nπ)/3) . . . . for n = 0, 1, 2 (angles repeat after that)

= 4 ∠((π/21)(1+14n)) . . . . . . . . for n = 0, 1, 2

= 4∠(π/21), 4∠(15π/21), 4∠(29π/21)

Of these three third roots, only the first two are listed among the answer choices. In your preferred form, they are ...

• 4(cos(π/21) +i·sin(π/21)) . . . . . . . . . matches 3rd choice
• 4(cos(15π/21) +i·sin(15π/21)) . . . . . matches 1st choice