Find the period of the function [sin3x] +
|cos6x| , where [] is greatest integer function and | | denotes modulus function . Answer is 2π/3

Dear Student,

The period of each sin x and cos x is 2π
To find the period of the function given in the question:

Let's say sin 3x+cos 6x=Px+Qx
Now,

Px=sin 3x=sin 3x+2π=sin 3x+2π3
Since, the greatest integer function will give the output of Px=sin 3x as -1, 0, 1 but the period will be similar to sin 3x.
Hence, Period of Px=sin 3x is 2π3
Now,
Qx=cos 6x
Since, modulus always give a positive output, hence the output of cos x will be 0,1, thus the output will repeat after every interval of π
Hence,
Qx=cos 6x=cos 6x+π=cos 6x+π6
Therefore, Period of Qx=cos 6x is π6
Thus, the period of the composite function Px+Qxsin 3x+cos 6x will be the LCM of 2π3 and π6
or,
LCM2π3,π6=LCM2π,πHCF3,6=2π3
Thus, period of sin 3x+cos 6x is 2π3

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