Regarding three-phase power: The electrical outlets in your home have two power-carrying prongs. The third prong is just for safety (it connects to the ground), and should never carry power unless something is going horribly wrong. In that case, the power should hopefully prefer to reach the ground through that third wire than through your human body. Anyhow, the voltage in those two power-carrying wires is constantl…
There aren't any mathematical tricks, rather how the real world (generator) translates to math. The generator at the power company has 3 stators that are on equal distance between each other on a circle. As the rotor rotates, it induces AC voltage at each of them. Because the rotor rotates at a stable speed, the AC voltage follows a sine function. Because the stators are placed 2/3pi apart, their sine waves have a ph…
Incidentally, the closely spaced multiple wires within each conductor of an electrical transmission line, typically held apart by spacers, act to reduce the inductance of the transmission line, which reduces losses. The little "dumbbells" which are on each side of each tower, are vibration dampers to reduce metal fatigue.