Question

Find all the complex solutions of the equation? `x^4 - i = 0`

Answers are:
`cos22.5^@ + isin22.5^@`,
`cos112.5^@ + isin112.5^@`,
`cos202.5^@ + isin202.5^@`,
`cos292.5^@ + isin292.5^@`

Answer 1

We know the answer has magnitude 1, so our equation becomes `(cos theta + i sin theta)^4 = i` `= cos ( 90^circ + 360^circ k ) + i sin ( 90^circ + 360^circ k)` or by De Moivre `4 theta = 90 ^circ + 360^circ k`, integer `k`, giving the listed answers.

Explanation:

Typically we denote a complex unknown as `z`.

`z^4 = i`

We'll want to do this in polar coordinates. Since `|i|=1` so will its fourth roots, so we don't need a magnitude factor here.

`z = cos theta + i sin theta`

Our equation becomes:

`(cos theta + i sin theta)^ 4 = i`

Most generally, for integer `k`,

`i = cos ( 90^circ + 360^circ k ) + i sin ( 90^circ + 360^circ k)`

By De Moivre's Theorem,

`(cos theta + i sin theta) ^4 = cos (4 theta) + i sin (4theta)`

Matching that up, we see

`4 theta = 90 ^circ + 360^circ k`

`theta = 22.5^circ + 90^circ k`

`z = cos(22.5 ^circ + 90^circ k) + i sin (22.5^circ + 90^circ k)`