Using formula zk = r(cos 𝜃 + 2𝜋k/n + i sin 𝜃 + 2𝜋k/n) To find the answer.

Question

Use the formula  z_k = r(cos 𝜃 + 2𝜋k/n + i sin 𝜃 + 2𝜋k/n)

 to find all solutions of the equation. (Enter your answers in trigonometric form. Let 0 ≤ 𝜃 < 2𝜋.)

x⁴ + i = 0

z0=___

z1=__

z2=___

z3=__

z4=__

I have gotten some pretty weird answer and I have already tried putting 3 different answers for each blank space.. apparently I'm pretty bad at this. Could anyone show me how to do this right?

Answer 1

The formula you want to use is this:

and it looks like the equation you want to solve is

So you have n = 4 and -i = r (cos θ + i sin θ).

r = |-i| = 1. That means we don't have to worry about multiplying by the nth root of r.

cos θ = 0 and sin θ = -1, so θ = 3π/2.

Then z₀ = cos (3π/8) + i sin (3π/8).

For the next value of k, we have to add 2π/4 = π/2, so

z₁ = cos (3π/8 + π/2) + i sin (3π/8 + π/2) = cos (7π/8) + i sin (7π/8)

z₂ = cos (7π/8 + π/2) + i sin (7π/8 + π/2) = cos (11π/8) + i sin (11π/8)

z₃ = cos (11π/8 + π/2) + i sin (11π/8 + π/2) = cos (15π/8) + i sin (15π/8)

And if you're trying to solve a fourth-degree equation, z₄ will be the same as z₀.

Let me know if anything's unclear.

Answer 2

I feel that more information is needed. The only value given is "z". I'd also be lost.

Is π 3.14 usa or 180 rad?