# 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 $2\mathrm{\pi }$
To find the period of the function given in the question:

Let's say =${P}_{x}+{Q}_{x}$
Now,

Since, the greatest integer function will give the output of  as -1, 0, 1 but the period will be similar to sin 3x.
Hence, Period of  is $\frac{2\mathrm{\pi }}{3}$
Now,

Since, modulus always give a positive output, hence the output of  will be 0,1, thus the output will repeat after every interval of $\mathrm{\pi }$
Hence,

Therefore, Period of  is $\frac{\mathrm{\pi }}{6}$
Thus, the period of the composite function ${P}_{x}+{Q}_{x}$ will be the LCM of $\frac{2\mathrm{\pi }}{3}$ and $\frac{\mathrm{\pi }}{6}$
or,
$LCM\left(\frac{2\mathrm{\pi }}{3},\frac{\mathrm{\pi }}{6}\right)=\frac{LCM\left(2\mathrm{\pi },\mathrm{\pi }\right)}{HCF\left(3,6\right)}=\frac{2\mathrm{\pi }}{3}$
Thus, period of  is $\frac{2\mathrm{\pi }}{3}$