How do you find the complex cube roots of 1 plus 0i?

Answer 1

z = 1 + 0i

So

|rz| = 1 and az = 0 radians.

which allows you to write z = rz*cos(az) + i*sin(az)

Then, if y = z1/3 then

|y| = |z1/3| = |11/3| = 1

and

ay is the angle in [0, 360) such that 3*ay = 0 mod(2*pi)

that is, ay = 0, 2pi/3 and 4pi/3

And therefore,

Root 1 = cos(0) + i*sin(0)

Root 2 = cos(2pi/3) + i*sin(2pi/3) and

Root 3 = cos(4pi/3) + i*sin(4pi/3).