Find the fourth roots of

Question

− 1/2 + (square root)3/2 i

Write the roots in trigonometric form.

A -
w 1=cos(35°)+isin(35°)
w2 =cos(125°)+isin(125°)
w3 =cos(215°)+isin(215°)
w4 =cos(305°)+isin(305°)

B -
w1 =cos(40°)+isin(40°)
w2 =cos(130°)+isin(130°)
w3 =cos(220°)+isin(220°)
w4 =cos(310°)+isin(3105°)

C - w =cos(20°)+isin(20°) 1
w2 =cos(110°)+isin(110°)
w3 =cos(200°)+isin(200°)
w4 =cos(290°)+isin(290°)

D -
w1 =cos(30°)+isin(30°)
w2 =cos(120°)+isin(120°)
w3 =cos(210°)+isin(210°)
w4 =cos(300°)+isin(300°)

E -
w1 =cos(25°)+isin(25°)
w2 =cos(115°)+isin(115°)
w3 =cos(205°)+isin(205°)
w4 =cos(295°)+isin(295°)

I'm studying for my pre cal exam. Can you please help me with this question?

Answer 1

let z = ( -1/2 + √3/2 i)
= (cos 120° + sin 120 i)

z^(1/4) = cos (120/4) + sin(120/4) i
= cos 30 + sin30 i ------> my primary solution.

there will be 4 roots, and 360/4 = 90°
by adding multiples of 90 to 30° we can get the other 3

z^(1/3)
= cos 30 + sin 30 i
or
= cos 120 + sin 120 i
or
cos 210 + sin 210 i
or
cos 300 + sin 300 i

looks like D

looks like you are reviewing De Moivre's Theorem.
In general , in finding the nth root, there will be n such roots.
Once you have the primary one, divide 360 by n, then repeatedly add 360/n degrees to get the others.
Notice if you add one more at that point, it brings you back to your first solution.
e.g. adding 90° to cos 300 + sin300i will yield
cos 390 + sin 390 i
which is
cos 30 + sin 30 i by coterminal angles, 390 = 360 + 30