Question #f647b

1 Answer
A. S. Adikesavan
Apr 4, 2016

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.

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