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

Question

asked Feb 26, 2020 10.6k views

2 Answers

Petelids · Answer 1 · 2020-02-28T21:05:53+0000

Answer:

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.

Kira Resari · Answer 2 · 2020-03-03T18:01:32+0000

Answer:

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 ...