Question

Which of the following expressions is the 4th root of unity?
a. -i
b.-1
c. i
d. all are correct

Answer 1

@mustafa2014

Answer 2

@phi

Answer 3

1 can be written as 1(cos(0) + i sin(0) )

to find the 4th roots, divide the angle by 4
and find the 4th root of the leading 1... which is just 1
1 ( cos(0/4) + i sin(0/4) which is what we started with
however there are 4 roots spaced equally around the circle

2pi/4 or pi/2 spacing
so the other three roots are at angles 0+pi/2 0+ 2 pi/2, and 0 + 3 pi /2

Answer 4

you can list all 4 roots as
cos(0) + i sin(0) = 1 + i*0 = 1
cos(pi/2) + i sin(pi/2) = 0 + i*1 = i
cos( pi ) + i sin(pi) = -1 + i*0 = -1
cos(3pi/2) + i sin (3pi/2) = 0 + i*-1 = -i

of course, you could have just checked the choices they gave you
for example: (a) -i: -i*-i * -i*-i = i^2 * i^2 = -1 * -1 = 1 works!
(b) -1 * -1 * -1 * -1 = 1*1 = 1 works!
( c ) i*i * i*i = -1 * -1 = 1 works!