Find the period of $f (x) = [s i n 3 x] + | c o s 6 x |$ where $[.]$ is a greatest integer function and $|$ is a modulus function.

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Solution

$f (x) = [s i n 3 x] + | c o s 6 x |$

$f (x) = f_{1} (x) + f_{2} (x)$

$f_{1} (x) = [sin (3 x)]$

$= 0, 0 \leq x \leq \frac{π}{3}$

$= - 1, \frac{π}{3} \leq x \leq \frac{2 π}{3}$

$T i m e p e r i o d o f f_{1} (x) :$

$\frac{2 π}{3} = \frac{4 π}{6}$

$f_{2} (x) = | cos (6 x) |$

$T i m e p e r i o d o f f_{2} (x) :$ $\frac{π}{6}$

$T i m e p e r i o d o f f (x) :$ $\frac{(L C M o f t i m e p e r i o d o f f_{1} (x) & f_{2} (x))}{(H C F o f t i m e p e r i o d o f f_{1} (x) & f_{2} (x))}$ $= \frac{4 π}{6} = \frac{2 π}{3}$

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