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Question

Find all the cube root of 4242i


A

[cos + isin ] , ,

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B

, [cos2 \pi + isin 2 \pi ] ,

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C

, ,

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D

, ,

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Solution

The correct option is C

, ,


Let z = 4242i

Write given complex number in polar form

|z| = (42)2+(42)2 = 16

tan(α) = 4242 = 1

α = π4

Since. complex number is in third quadrant

Then, argument θ=(πα) = -(ππ4)= 3π4

Polar form of z = 16 [cos(3π4)+isin(3π4)]

= 16 (cos3π4isin3π4)

Cube root of z = 1613[cos(3π4)+isin(3π4)]13

=1613[cos(2kπ3π4)3+isin(2kπ3π4)3]

When k = 0,1,2

When k = 0

1613 [cos(3π4)3+isin(3π4)3]

= 1613 [cosπ4isinπ4]

When k =1

1613 [cos(2π3π4)3+isin(2π3π4)3]

1613 [cos5π12+isin5π12]

When k = 2

1613 [cos(4π3π4)3+isin(4π3π4)3]

1613 [cos(13π12)+isin13π12]


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