Find all solutions for the complex numbers:
a) z^4 = 9i -46
b) 8*sqrt3 / z^4 +8 =+i
best convert to polar form
z^4 = 46.87 cis 2.95
so, z = 2.62 cis 0.738 + k*pi/2 for k=0..3
then convert back to rectangular form.
using de Moivre's Theorem:
let z^4 = -46 + 9i
changing -46 + 9i to rectangular form
r = √46^2+9^2 =√2197
tanØ = 9/-46
Ø = 168.9298°
z^4 = √2197(cos168.9298° + i sin 168.9298°)
z = .....
so there will be 4 roots,
z = √2197^(1/4) (cos (168.9.. +k(360°)/4 + i sin(168.9..+k(360°)/4 )
where k = 0, 1, 2, 3
if k = 0
z = √2197^(1/4) (cos42.2324 + i sin42.2324)
= 2.61655(.740424 , .67214 i )
= (1.937355 , 1.758687 i )
if k = 1
z = √2197(1/4) (cos 132.2324° + i sin 132.2324°)
= ( -1.758687 , 1.937355 i)
k = 2
I will let you finish the other two.
testing for k = 1
if z = ( -1.758687 , 1.937355 i)
r = appr 2.61655
Ø = 132.2324
then z^4 = 2.6155^4 (cos 4(132.2324) , i sin 4(132.2324))
= 46.872...(-.98139.. , .19201..)
= -46 + 9i , YEAAAAHHH
a) To find all solutions for the complex number equation z^4 = 9i - 46, we can follow these steps:
1. Rewrite 9i - 46 in polar form: 9i - 46 = (46 cos θ + 46i sin θ), where θ is the argument.
2. Convert 9i - 46 to polar form using the formula r(cos θ + i sin θ), where r is the magnitude and θ is the argument:
r = √(46^2 + 9^2) = √(2117),
cos θ = 46/√(2117),
sin θ = 9/√(2117).
3. Express z in polar form: z = r^(1/4)(cos(θ/4 + kπ/2) + i sin(θ/4 + kπ/2)), where k is an integer from 0 to 3.
4. Substitute the values of r, θ, and k into the polar form expression for z and simplify to get the 4 solutions.
b) To find all solutions for the complex number equation (8√3 / (z^4 + 8)) = i, we can follow these steps:
1. Multiply both sides of the equation by (z^4 + 8) to isolate z:
8√3 = i(z^4 + 8).
2. Rewrite i in polar form as i = cos(π/2) + i sin(π/2).
3. Express z^4 + 8 in polar form: z^4 + 8 = r^4(cos(4θ) + i sin(4θ)).
4. Equate the real and imaginary parts of both sides to find the values of r and θ:
Equating the real parts: 8√3 = r^4 cos(4θ).
Equating the imaginary parts: 0 = r^4 sin(4θ).
5. Solve the two equations simultaneously to find the values of r and θ.
6. Once r and θ are known, express z in polar form: z = r^(1/4)(cos(θ/4 + kπ/2) + i sin(θ/4 + kπ/2)), where k is an integer from 0 to 3.
7. Substitute the values of r, θ, and k into the polar form expression for z and simplify to get the 4 solutions.