Solve in set of in

To solve , we look for the sixth roots of unity. This boils down to finding all complex numbers that can be expressed as where is a multiple of . Thus, the roots are given by:

for . This results in roots:

So, the complete set of solutions is .

For an exciting twist, imagine these solutions dancing around the unit circle in the complex plane—each rooting itself at an equidistant angle of . They form a beautiful hexagon, each vertex magically representing one of the roots of unity!

And if you’re curious about how these roots pop up in real life, consider electrical engineering! The sixth roots of unity play a pivotal role in analyzing waveforms and harmonics in AC circuits, helping engineers phase shift signals like pros. Each root corresponds to a specific phase angle, allowing for the control and management of power efficiency—kinda like orchestrating a symphony, balancing every note to create harmony in the circuit!