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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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