Find the fifth roots of 243 (cos 300° + i sin 300°).

In looking for the fifth roots, we will use De Moivre's theorem.

The formula for this problem is z^5 = 243 (cos 300 degress + i sin 300 degrees)

Where you'll also need the following dа ta:

300/5 = 60

360/5 = 72

- you'll input these two after the cos and i sin (60+k*72) where k = 0,1,2,3,4

solution:

z^5 = 243 (cos 300 degrees + i sin 300 degrees)

z = 243^1/5 (cos 300 degrees + i sin 300 degrees)

z = 3 (cos (60 + k*72) degrees) + (i sin (60 + k*72) degrees)

so the following are the roots:

3 (cos 60 degrees + i sin 60 degrees)

3 (cos 132 degrees + i sin 132 degrees)

3 (cos 204 degrees + i sin 204 degrees)

3 (cos 276 degrees + i sin 276 degrees)

3 (cos 348 degrees + i sin 348 degrees)