Saying r=θ doesn't really make much sense, so I shall ignore that part.
If we have z^5=(1+i), we will first change to the exponential form. Here θ^5=π/4 and r^5=sqrt(2), this is fairly obvious and easy to obtain.
Now, to obtain the roots it is fairly simple to obtain the modulus, r=2^(1/10), once again this is very simple to obtain.
For the angles, we look for the 2nd smallest angle, for that we need θ/5 + 2*π/5. Notice that we will obtain all roots by adding n*(2*π/5), where n=1,2,3,4 since 5*n*(2*π/5)=n*2*π and will only be an additional full turn.
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Recall:
z = a + ib ← this is a complex number
m = √(a² + b²) ← this is its modulus
tan(α) = b/a → then you can deduce α ← this is the argument
z⁵ = 1 + i
The modulus is of z⁵ is: m = √[(1)² + (1)²] → m = √2 = 2^(1/2)
The argument is of z⁵ is such as: tan(α) = 1/1 = 1 → α = π/4
According this result, you can deduce that the modulus of z is: → m^(1/5) = [2^(1/2)]^(1/5) = 2^(1/10)
According this result, you can deduce that the argument of z is: θ = α/5 = (π/4)/5 = π/20
…and you can write:
z₁ = 2^(1/10) * [cos(π/20) + i.sin(π/20)] → to get the second root, you add (2π/5)
z₂ = 2^(1/10) * [cos{(π/20) + (2π/5)} + i.sin{(π/20) + (2π/5)}] → to get the third root, you add (2π/5)
z₃ = 2^(1/10) * [cos{(π/20) + (4π/5)} + i.sin{(π/20) + (4π/5)}] → to get the fourth root, you add (2π/5)
z₄ = 2^(1/10) * [cos{(π/20) + (6π/5)} + i.sin{(π/20) + (6π/5)}] → to get the fifth root, you add (2π/5)
z₅ = 2^(1/10) * [cos{(π/20) + (8π/5)} + i.sin{(π/20) + (8π/5)}]
Resume:
z₁ = 2^(1/10) * [cos(π/20) + i.sin(π/20)]
z₂ = 2^(1/10) * [cos(9π/20) + i.sin(9π/20)]
z₃ = 2^(1/10) * [cos(17π/20) + i.sin(17π/20)]
z₄ = 2^(1/10) * [cos(25π/20) + i.sin(25π/20)]
z₅ = 2^(1/10) * [cos(33π/20) + i.sin(33π/20)]
You can simplify z₄ & z₅
z₄ = 2^(1/10) * [cos(5π/4) + i.sin(5π/4)]
z₅ = 2^(1/10) * [cos(33π/20) + i.sin(33π/20)]
z₅ = 2^(1/10) * [cos{(20π + 13π)/20} + i.sin{(20π + 13π)/20)}]
z₅ = 2^(1/10) * [cos{π + (13π/20)} + i.sin{π + (13π/20)}]
z₅ = 2^(1/10) * [- cos(13π/20) - i.sin(13π/20)]
z₅ = - 2^(1/10) * [cos(13π/20) + i.sin(13π/20)]
Saying r=θ doesn't really make much sense, so I shall ignore that part.
If we have z^5=(1+i), we will first change to the exponential form. Here θ^5=π/4 and r^5=sqrt(2), this is fairly obvious and easy to obtain.
Now, to obtain the roots it is fairly simple to obtain the modulus, r=2^(1/10), once again this is very simple to obtain.
For the angles, we look for the 2nd smallest angle, for that we need θ/5 + 2*π/5. Notice that we will obtain all roots by adding n*(2*π/5), where n=1,2,3,4 since 5*n*(2*π/5)=n*2*π and will only be an additional full turn.