Short answers, pulled from the story.
The phase represents the fraction of the cycle covered up to a specific point in time. It is an angle-like quantity that varies by one full turn as the variable goes through each period.
Scientists use the symbol phi or lowercase phi to denote this value. The numeric value depends on the choice of where the period starts, often using 0 to 2 pi radians or 0 degrees to 360 degrees.
Two signals are in phase when their phase difference equals zero, meaning they have the same sign and reinforce each other. They are out of phase if the difference is not zero, with opposite signs occurring at a 180-degree shift.
Beating occurs when the phase difference increases linearly with the argument t due to differing frequencies. This causes periodic changes from reinforcement and opposition in the combined signal.
Engineers measure the rate of motion of the test signal relative to a stationary reference to find the offset. Vertical lines mark points where sine signals pass through zero while bars below show width representing the phase difference.