I think its essentially asking to find the three cube roots of unity of z=1+i. Convert z=1+i into mod-arg form then using De Moivre's Theorem would he what I'd do.

3. Note that 1/3arg(z) can also take other values namely:

So other roots will be:

4. Convert cis roots to cartesian which will be quite laborious in this instance because the arguments aren't nice so compound/double angle formulae for cos and sin will be required.

I think its essentially asking to find the three cube roots of unity of z=1+i. Convert z=1+i into mod-arg form then using De Moivre's Theorem would he what I'd do.