The solutions for z³ = i are in polar form:
z₁ = (1, π/6)
z₂ = (1, 5π/6)
z₃ = (1, 3π/2)
1. Converting to polar form:
First, it's helpful to express z and i in polar forms:
z = |z| (cos θ + i sin θ)
i = 1 (cos π/2 + i sin π/2)
where:
|z| is the magnitude of z
θ is the angle of z in the complex plane
Substituting these forms into the equation:
|z|³ (cos 3θ + i sin 3θ) = 1 (cos π/2 + i sin π/2)
2. Simplifying and isolating angle:
Since the magnitude of i is 1 and its angle is π/2, we can focus on the angle portion of the equation:
3θ = π/2 + 2πk for any integer k
Solving for θ:
θ = π/6 + 2πk/3
3. Finding multiple solutions:
For different values of k, we get different solutions for θ, representing multiple cube roots of i:
k = 0: θ = π/6 => z₁ = (1, π/6)
k = 1: θ = 5π/6 => z₂ = (1, 5π/6)
k = 2: θ = 3π/2 => z₃ = (1, 3π/2)
4. Geometric interpretation:
The solutions z₁, z₂, and z₃ correspond to three points on the complex plane spaced equally 2π/3 apart on the unit circle. This reflects the property of a cube root that it rotates the original number by 120 degrees counterclockwise.
Therefore, the solutions for z³ = i are:
z₁ = (1, π/6)
z₂ = (1, 5π/6)
z₃ = (1, 3π/2)
These solutions represent three cube roots of i equally spaced on the unit circle in the complex plane.