find all the 3rd roots z13 of z6cispi answer in trig formSo

find all the 3rd roots, z^1/3, of z=6cispi ( answer in trig form)

Solution

z = 6 cis pi can be expressed as

6 (cos pi + i sin pi )

z^1/3 = 6 (cos pi + i sin pi )^1/3

applying de moivres theorem

= 6^1/3 ( cos (1/3 pi )+ i sin (1/3 pi ))

1st root = 6^1/3 ( cos pi/3 + i sin pi/3 )

2nd root = 6^1/3 ( cos pi + i sin pi )

3rd root = 6^1/3 ( cos 5pi/3 + i sin 5pi/3 )