Question

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

Answer 1

Answer:

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

Step-by-step 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

Answer 2

Answer:

The correct option is C.

Step-by-step explanation:

The given complex number is

$z=64(\cos[\frac{\pi}{7}]-i\sin[\frac{\pi}{7}])$

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

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

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

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

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

$z^{\frac{1}{3}}=4(\cos[\frac{\pi}{7}\times \frac{1}{3}]-i\sin[\frac{\pi}{7}\times \frac{1}{3}])$

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

Therefore the correct option is C.