# Periodic Functions

**Periodic Functions:**

Periodic functions are functions which repeat: f (t P)= f (t) for all *t*. For example, if f (t) is the amount of time between sunrise and sunset at a certain latitude, as a function of time *t*, and *P *is the length of the year, then f (t P) = f (t) for all *t*, since the Earth and Sun are in the same position after one full revolution of the Earth around the Sun. We state this explicitly as the following definition: a function *f *(*t*) is **periodic **with **period ***P *> 0 if,

f (t P)= f (t) for all t.

**Example-1**

f (x) = cos x

f (x 2π) = cos (x 2π)

= cos x cos2π - sin x sin 2π

= cos x

= f (x), Hence cosx is periodic of period 2π.

**Example-2**

f (x) = sin 4x

Hence sin nx is periodic of period π/2, observe that 2π is also a period of sin 4x.

**Useful identification:**

**1. **sin (ax b) = sin ax cosb cos ax sinb

**2**. cos (ax B) = cos ax cosb sin ax sinb

**Functionality:
**

**1**. Any function can be considered periodic with period zero, this period is trivial and is not considered as a period.

**2**. If p is a period of f , then np is a period for any integer n.

**3**. If p is a period then p/2 is not necessarily a period.

**Fundamental Period:
**

The most interesting period for a periodic function is the smallest positive period , this period is called the Fundamental Period.

**1**. sin x is 2π

**2**. sin 3x is 2π/3