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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