Question

Thr period of the function : $f\left(x\right)=3\sin 2\left\{3x\right\}+5\cos 3\left\{2x\right\}$ is (where $\left\{\right\}$ denotes fraction part function)

Answer 1

We know that, period of $\left\{x\right\}$ is $1$ .
So the period of function $\left\{3x\right\}$ is $\frac{1}{3}$.
Hence, period of $3\sin 2\left\{3x\right\}$ is $\frac{1}{3}$.
and period of $5\cos 3\left\{2x\right\}$ is $\frac{1}{2}.$

$\therefore$ Period of $f\left(x\right)=LCM$ of $\frac{1}{2}$ and $\frac{1}{3}$ is :
$\Rightarrow LCM=\frac{LCM\text{of}(1,1)}{HCF\text{of}(2,3)}=\frac{1}{1}=1$