Question

Find the period of
(i) `f(x)=sin pi x +{x//3}`, where {.} represents the fractional part.
(ii) `f(x)=|sin 7x|-"cos"^(4)(3x)/(4)+"tan"(2x)/(3)`

Answer 1

(i) `f(x)=sin pi x +{x//3}`, where {.} represents the fractional part
Period of ` sin pi x " is " (2pi)/(pi)=2`
Period of `{x//3} " is " (1)/(1//3)=3`
Therefore, period of f(x) is L.C.M. of `(2,3)=6`
(ii) `f(x)=|sin 7x|-"cos"^(4)(3x)/(4)+"tan"(2x)/(3)`
Period of `|sin 7x| " is " (pi)/(7)`
Period of `"cos"^(4)(3x)/(4) " is " (pi)/(3//4)=(4pi)/(3)`
Period of `tan(2x)/(3) " is " (pi)/(2//3)=(3pi)/(2)`
Therefore, period of f(x) is L.C.M. of
`((pi)/(7),(4pi)/(3),(3pi)/(2))=pi xx (L.C.M. of (1,4,3))/(H.C.F. of (7,3,2)`
`=12 pi`