Question

from the equation:

x^4 - i = 0
x^4 = i



how do you get: The solutions are the fourth roots of i = cos(pi/2) + isin(pi/2)

I don't understand how you get pi/2? please full and detailed explanation please.

Answer 1

i can also be written e^(i*pi/2), which also equals
cos pi/2 + i sin pi/2

Note that since cos pi/2 = 0 and sin (pi/2) = 1, the equation
i = cos(pi/2) + i sin(pi/2)
IS correct.

The proof that
e(i A) = cos A + i sin A
should be in your textbook. Your problem is related to de Moivre's theorem, as I recall.

one fourth root of e^(i pi/2) is e^(i pi/8), or 0.9238 + 0.3827 i.

There are three other roots you can get by taking the fourth root of e^(i*5 pi/2), e^(i*9 pi/2) and e^(i*13 pi/2). Each of those three terms is i.