Question
Find the roots of z^3 = 1 in polar form
Answers
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To find the roots of z^3 = 1 in polar form, we first note that 1 can also be written as 1 cis(0). We can then write the cube roots of 1 as:
z = (1 cis(0))^(1/3)
z = 1^(1/3) cis(0/3) = 1 cis(0)
z = (1 cis(0 + 2π))^(1/3)
z = 1^(1/3) cis(2π/3) = cos(2π/3) + i sin(2π/3) = -1/2 + i√3/2
z = (1 cis(0 + 4π/3))^(1/3)
z = 1^(1/3) cis(4π/9) = cos(4π/9) + i sin(4π/9) = √3/2 + i/2
Therefore, the roots of z^3 = 1 in polar form are:
z = 1 cis(0), -1/2 + i√3/2, √3/2 + i/2
z = (1 cis(0))^(1/3)
z = 1^(1/3) cis(0/3) = 1 cis(0)
z = (1 cis(0 + 2π))^(1/3)
z = 1^(1/3) cis(2π/3) = cos(2π/3) + i sin(2π/3) = -1/2 + i√3/2
z = (1 cis(0 + 4π/3))^(1/3)
z = 1^(1/3) cis(4π/9) = cos(4π/9) + i sin(4π/9) = √3/2 + i/2
Therefore, the roots of z^3 = 1 in polar form are:
z = 1 cis(0), -1/2 + i√3/2, √3/2 + i/2