Question

Question #f647b

Answer 1

Using the notation (cos x + i sin x)=cis x, the four fourth roots of 1 are
`cis 0, cis (pi/2), cis (pi) and cis (3pi/2)`. Explicitly, the roots are `1, i, -1 and -i`.

Explanation:

The n values of `1^(1/n)=(cos(2kpi)+i sin(2kpi))^(1/n)`, with k as an integer, are `(cos(2kpi/n)+i sin(2kpi/n)), k=0, 1, 2..,n-1`.
`sin (3pi/2) =sin(pi+pi/2)=-sin(pi/2)=-1`.
Here, n=4.

For checking the answer, use that if a+ib is a root so is a-ib.

We are solving the biquadratic equation `x^4=1`. and complex roots occur in conjugate pairs.