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Alternating voltage generators are essential components in electrical engineering. They create alternating current (AC) by converting mechanical energy into electrical energy, using the principle of electromagnetic induction. In simple terms, as the generator's rotor spins within a magnetic field, it produces an alternating voltage or e.m.f (electromotive force). These devices are crucial in supplying power across various applications ranging from industrial machinery to household electronics.

This alternating nature means the voltage periodically reverses direction, following a sine wave pattern. Understanding this pattern is important because it helps in analyzing how these currents interact when more than one source is connected together, like in the case of the problem we are examining.

Multiple generators can be connected in parallel, and when they are, their outputs can add together creating a single resultant voltage. The phase of output from each generator is crucial and can lead to situations where they either aid or oppose one another, owing much to the concept of phase difference.

Phase difference refers to the difference in phase angle between the voltages produced by alternating voltage generators. It is a significant concept in AC circuits since it affects how two waveforms align in time when combined.

Phase is measured in degrees or radians, and one complete cycle of a waveform corresponds to 360 degrees or 2π radians. A phase difference arises when there is a horizontal shift in the waveforms. For instance, in our exercise, two generators have a phase difference of π/3 radians (or 60 degrees).

The phase difference determines the constructive or destructive interference of the waves. Constructive interference occurs when the waves are in phase, leading to a higher resultant amplitude. In contrast, destructive interference takes place when the waves are out of phase, which can reduce or cancel the resultant amplitude.

Understanding phase difference is critical because it informs how the outputs from different power sources can be efficiently managed and optimized.

The sine addition formula is a trigonometric identity used to simplify calculations involving the sine of the sum of two angles. It’s expressed as: \[\sin(a + b) = \sin a \cos b + \cos a \sin b\] This identity is particularly useful in problems involving alternating voltages.

For the problem at hand, we used this formula to manage the phase shift in the second generator’s output relative to the first. By applying the sine addition formula, the expression for the second e.m.f can be expanded to every individual trigonometric component, making calculations simpler.

This step is essential as it allows us to break complex expressions into simpler components that are much easier to manipulate mathematically.

Once simplified, these components can be added accurately, letting us find the resultant voltage when the two e.m.f.s are combined, considering all phase differences and amplitudes. This shows how the sine addition formula is fundamental in solving AC circuit problems efficiently.