Question

Find period of the following functions
(i) `f(x)="sin" x/2+"cos"x/3`
(ii) `f(x)={x}+sinx,` where `{.}` denotes fractional part function
(iii) `f(x)=4cosx.cos3x+2`
(iv) `f(x)="sin"(3x)/2-"cos"x/3-"tan"(2x)/3`

Answer 1

(i) Period of `"sin"x/2` is `4pi` while period of `"cos"x/3` is `6pi`. Hence period of `"sin"x/2+"cos"x/3` is `12pi` {L.C.M of 4 and 6 is 12}
(ii) Period of `sinx=2pi`
Period of `{x}=1`
but L.C.M. of `2pi` and 1 is not possible as their ratio is irrational number it is aperiodic.
(iii) `f(x)=4cosx.cos3x+2`
period of `f(x)` is L.C.M of `(2pi,(2pi)/3)=2pi`
but `2pi` may or may not be fundamental periodic but fundamental period `=(2pi)/n` where `n epsilonN`. Hence cross checking for `n=1,2,3,`....... we find `pi` to be fundamental period `f(pi+x)=4(-cosx)(-cos3x)+2=f(x)`
(iv) Period of `f(x)` is L.C.M of `(2pi)/(3//2),(2pi)/(1//3),(pi)/(2//3)=` L.CM. of `(4pi)/3,6pi,(3pi)/2=12pi`