Question

Find the fourth roots of 256(cos 280° + i sin 280°).

Answer 1

take the fourth root of 256, divide 280 by 4
that is the first one anyways

Answer 2

use DeMoivre's theorem

Answer 3

let x be the required root then
x^4 = 256(cos 280° + i sin 280°)
x^4 = 256(cos (2*k*pi+280° )+ i sin (2*k*pi+280° ))

Answer 4

doesn't the 4th root of 256 = 4

Answer 5

so
x = ( (256)^(1/4 )) (cos (2*k*pi+280° )+ i sin (2*k*pi+280° ))^(1/4)
x = ( (256)^(1/4 )) (cos (2*k*pi+280° )/4+ i sin (2*k*pi+280° )/4) [k=0,1,2,3]

Answer 6

alright

Answer 7

i hope u will be able to continue further...

Answer 8

thanks for the help, i'll refer back to here if i get stuck. Thanks!

Answer 9

z = 256(cos280 + i sin280) = 256[cos(280+360k) + i sin(280+360k)]
z^(1/4) = 4[cos(70+90k) + i sin(70+90k)} : k = 0, 1, 2, 3

z0 = 4[cos(70) + i sin(70)]
z1 = 4[cos(160) + i sin(160)]
z2 = 4[cos(250) + i sin(250)]
z3 = 4[cos(340) + i sin(340)]

No, cos(453) + i sin(453) is not an acceptable answer.
453 is coterminal with your result for k = 0. cos(453) = cos(93) and sin(453) = sin(93). All the angles in the polar representation should be reduced to the coterminal angle in the interval 0 ≤ Θ < 2π