# JEE Main & Advanced Mathematics Trigonometric Equations Periodic Functions

## Periodic Functions

Category : JEE Main & Advanced

A function is said to be periodic function if its each value is repeated after a definite interval. So a function $f(x)$ will be periodic if a positive real number $T$ exist such that, $f(x+T)=f(x),\,\,\forall x\in$ domain. Here the least positive value of $T$ is called the period of the function. Clearly $f(x)=f(x+T)=f(x+2T)=f(x+3T)=.....$. e.g., $\sin x,\,\cos x,\,\tan x$ are periodic functions with period $2\pi ,\,2\pi$ and $\pi$respectively.

Some Standard Results on Periodic Functions

 Functions Periods ${{\sin }^{n}}x,\,\,{{\cos }^{n}}x,\,\,{{\sec }^{n}}x,\,\,\text{cose}{{\text{c}}^{n}}x$ $\left\{ \begin{matrix} \pi ;\,\,\text{if }n\text{ is even} \\ 2\pi ;\,\,\text{if }n\text{ is odd or fraction} \\ \end{matrix} \right.$ ${{\tan }^{n}}x,\,\,{{\cot }^{n}}x$ $\pi ;\,n$ is even or odd. $\sin (ax+b),\,\cos (ax+b)$ $\sin (ax+b),\,\cos (ax+b)$ $2\pi /a$ $\tan (ax+b),\,\cot (ax+b)$ $\pi /a$ \begin{align} & |\sin x|,\,|\cos x|,\,|\tan x|,\, \\ & |\cot x|,\,\,|\sec x|,\,\,|\text{cosec}\,x| \\ \end{align} $\pi$ $|\sin (ax+b)|,\,|\cos (ax+b)|,$ $\,\sec |ax+b|,\,|\text{cosec }(ax+b)|$ $|\tan (ax+b)|,\,|\cot (ax+b)|$ $\pi /a$ $x-[x]$ 1 Algebraic functions e.g., $\sqrt{x},\,{{x}^{2}},\,{{x}^{3}}+5,....\text{etc}$ Period does not exist

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