If, for some value p, f(x+p)=f(x) for all x, the real function f is periodic and has period p. For example, cos x is periodic with period 2π, since cos(x+2π)=cos x for all x; or, using degrees, cos x° is periodic with period 360, since cos(x+360)°=cos x° for all x. Some authors restrict the use of the term ‘period’ to the smallest positive value of p with this property.
In mechanics, any phenomenon that repeats regularly may be called periodic, and the time taken before the phenomenon repeats itself is then called the period. The motion may be said to consist of repeated cycles, the period being the time taken for the execution of one cycle. Suppose that x=A sin(ωt+α), where A (>0), ω and α are constants. This may, for example, give the displacement x of a particle, moving in a straight line, at time t. The particle is thus oscillating about the origin. The period is the time taken for one complete oscillation and is equal to 2π/ω.