Find the periods (if periodic) of the following functions ([.] denotes the greatest function):

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Solution

$f (x) = e^{l o g (s i n x) + t a n^{3} x - c o s e c (3 x - 5)}$
$f (x) = e^{l o g (s i n x)} + t a n^{3} (x) - c o s e c (3 x - 5)$
$\to$ If $l o g (s i n x) i s l o g_{10} (s i n x) t h e n e^{l o g_{10} (s i n x)}$ will be non periodic.
$f (x) \to$ non-periodic
$\to$ If $l o g (s i n x) i s l n (s i n x)$ , then $e^{l n (s i n x) = s i n x}$
$= f (x) = s i n x + t a n^{3} x - c o s e c (3 x - 5)$
$↓$ $↓$ $↓$
Period $= 2 π$ Period $= 3 π$ Period $= \frac{2 π}{3}$
$\frac{2 π}{3 π} = \frac{2}{3} \to$ Rational
$\frac{2 π}{\frac{2 π}{3}} = 3 \to$ Rational
$∴$ Period = LCM $(2 π, 3 π, \frac{2 π}{3})$
$∴$ Period $= 6 π$

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