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In physics and mathematics, the phase of a periodic function F {\displaystyle F} of some real variable t {\displaystyle t} is an angle-like quantity representing the fraction of the cycle covered up to t {\displaystyle t} . It is denoted ϕ {\displaystyle \phi } and expressed in such a scale that it varies by one full turn as the variable t {\displaystyle t} goes through each period {\displaystyle F} goes through each complete cycle]. It may be measured in any angular unit such as degrees or radians, thus increasing by 360° or 2 π {\displaystyle 2\pi } as the variable t {\displaystyle t} completes a full period.
This convention is especially appropriate for a sinusoidal function, since its value at any argument t {\displaystyle t} then can be expressed as the sine of the phase ϕ {\displaystyle \phi } , multiplied by some factor.
Usually, whole turns are ignored when expressing the phase; so that ϕ {\displaystyle \phi } is also a periodic function, with the same period as F {\displaystyle F} , that repeatedly scans the same range of angles as t {\displaystyle t} goes through each period. Then, F {\displaystyle F} is said to be "at the same phase" at two argument values t 1 {\displaystyle t_{1}} and t 2 {\displaystyle t_{2}} = ϕ {\displaystyle \phi =\phi } ] if the difference between them is a whole number of periods.
The numeric value of the phase ϕ {\displaystyle \phi } depends on the arbitrary choice of the start of each period, and on the interval of angles that each period is to be mapped to.