Properties of trigonometrical functions

## Periodic function

A function f(x) is said to be periodic function with period 'T' if f(x + T) = f(x).

The least positive value of T is called the fundamental period of the function.

Note that 'T' is a real number.

All the trigonometrical functions are periodic functions.

For example: sin(x + 2π) = sin x; sin(x – 4π) = sin x; sin(x + 6π) = sin x etc.

In general,

sin(x + 2nπ) = sin x, n ∈ z

Here T = .... – 8π, – 6π, – 4π, – 2π, 0, 2π, 4π, 6π, 8π, ...

But the fundamental period is least +ve real number.

∴ T = 2π is the fundamental period

Hence, we can say that sine function is periodic function with fundamental period 2π.

The fundamental periods of other trigonometric functions are given below:

Trigonometric Function |
Fundamental Period |

sin x, cosec x |
2π |

cos x, sec x |
2π |

tan x, cot x |
π |

**Note:**

(i) If f(x) is periodic function with period 'T' then 1/f(x), √f(x) are also periodic functions with period 'T'

(ii) If f(x) = {x} then period of f(x) is '1' where {x} is fractional part of 'x'.

(iii) Constant function is a periodic function with no fundamental period.

(iv) Period of sin^{n}x & cos^{n} x → π if n is even integer

Period of sin^{n} x & cos^{n} x → 2π if n is odd integer

Period of tan^{n} x → π for every integer

(v) If period of f(x) is T, then periods of c + f(x), cf(x), f(x + c) are T itself

(vi) If period of f(x) is T, then period of f(ax + b) = T/| a |

(vii) If period of f(x) is T_{1} and period of g(x) is T_{2}, then period of f(x) ± g(x) is L.C.M. {T_{1}, T_{2}}.

**Examples of periodic motion**

If an object repeats its motion along a certain path, about a certain point, in a fixed interval of time(T), the motion of such an object is known as periodic motion. Few common examples (with reference to adjacent images) are listed below: |

(i) motion of a pendulum |

(ii) motion of a spring |

(iii) The vibration of a guitar string |

(iv) The rotation of the earth over its axis |

(v) The revolving of the earth around the sun |

(vi) The revolving of the sun around the center of the Galaxy |