Question

If the period of the function $ f(x)=\dfrac{\sin \left\{ \left( \sin (nx) \right) \right\}}{\tan \left( \dfrac{x}{n} \right)},n\in N $ , is $ 6\pi $ , then $ n $ is equal to
(a) 3
(b) 2
(c) 1
(d) None of these

Answer 1

Hint: Use the period of $ \sin x $ and $ \tan x. $ So if there is function $ f(ax) $ then its period is $ \dfrac{T}{a} $ ,
For a function $ f\left( \dfrac{x}{a} \right) $ the period is $ aT $ . Use these properties. So find the period of $ f(x)=\dfrac{\sin \left\{ \left( \sin (nx) \right) \right\}}{\tan \left( \dfrac{x}{n} \right)} $ and put it equal to $ 6\pi $ you will get the value of $ n $ .

Complete step-by-step answer:
In question it is given that the period of function $ f(x)=\dfrac{\sin \left\{ \sin (nx) \right\}}{\tan \left( \dfrac{x}{n} \right)} $ is $ 6\pi $ .
So we have to find the value of $ n $ .
Now first period of function,
The period of a periodic function is the interval between two “matching” points on the graph. In other words, it's the distance along the $ x $ -axis that the function has to travel before it starts to repeat its pattern. The basic sine and cosine functions have a period of $ 2\pi $ , while tangent has a period of $ \pi $ .
The time interval between two waves is known as a Period whereas a function that repeats its values at regular intervals or periods is known as a Periodic Function. Or you can say, Periodic function is a function that repeats its values after every particular interval.
The period of the function is this particular interval mentioned above.
A function $ f $ will be periodic with period $ m $ , so if we have
 $ f(a+m)=f(a), $ for every $ m>0 $
It Shows that the function $ f(a) $ possesses the same values after an interval of $ m $ . One can say that after every interval of $ m $ the function $ f $ repeats all its values.
Now we know if there is a function $ f(x) $ which has a period function of $ T $ ,
 $ f(x+T)=f(x) $ where $ T $ is period,
and $ f(x+b) $ has period $ T $ since we are adding in $ x $ that will shift $ x $ -axis .
e.g. $ \sin (x+5) $ has period $ \pi $ .
similar to this
 $ f(ax+b)=f\left( a(x+\dfrac{b}{a}) \right)=f(az) $ where $ z=\left( x+\dfrac{b}{a} \right) $
if $ a=1 $ ,
then $ f(az)=f(z)=\left( x+b \right) $ so the period is $ T $ ,
but now we have $ x=az $ so the period is $ Ta $ .
So if $ f(x) $ has period $ T $ ,
So we get,
So if there is function $ f(ax) $ then its period is $ \dfrac{T}{a} $ ,
For function $ f\left( \dfrac{x}{a} \right) $ the period is $ aT $ ,
So we know the period function of $ \sin \theta $ ,
So $ \sin \theta $ has period of $ 2\pi $ ,
So we are given $ \sin (nx) $ ,
So the period function for $ \sin (\sin (nx)) $ is $ \dfrac{2\pi }{n} $ ,
And we know period of $ \tan x $ ,
So period of $ \tan x $ is $ \pi $ ,
So we get the period of $ \tan \left( \dfrac{x}{n} \right) $ as $ \pi n $ ,
So if there is function $ \dfrac{p(x)}{g(x)} $ , and if period of $ p(x) $ is $ \dfrac{a}{b} $ and the period of $ g(x) $ is $ \dfrac{c}{d} $ ,
So we write period of $ \dfrac{p(x)}{g(x)} $ as ratio of L.C.M of $ (a,c) $ to H.C.F of $ (b,d) $ ,
So now applying the above to the problem, we get,
So we want to find the $ \dfrac{\sin \left\{ \sin (nx) \right\}}{\tan \left( \dfrac{x}{n} \right)} $ ,
So let $ p(x)=\sin \left\{ \sin (nx) \right\} $ and $ g(x)=\tan \left( \dfrac{x}{n} \right) $ ,
So period of $ p(x)\to \dfrac{2\pi }{n} $ and $ g(x)\to \dfrac{n\pi }{1} $ ,
So overall period of $ f(x)=\dfrac{\sin \left\{ \sin (nx) \right\}}{\tan \left( \dfrac{x}{n} \right)}, $ We get,
So period of $ f(x)=\dfrac{LCMof(2\pi ,n\pi )}{HCFof(n,1)} $
So period of $ f(x)=6\pi $ ………….. (Given in question)
So we get,
 $ 6\pi =\dfrac{2n\pi }{1} $
So simplifying we get the value of $ n $ as $ 3 $ .
Hence, option (a) is correct.

Note: So you should know that the period of all trigonometric identities such as $ \sin x,\tan x $ etc.
You should know how we have to calculate the period of function. Don’t jumble yourself while taking LCM and HCF. So if there is function $ \dfrac{p(x)}{g(x)} $ , and if period of $ p(x) $ is $ \dfrac{a}{b} $ and the period of $ g(x) $ is $ \dfrac{c}{d} $ , So we write period of $ \dfrac{p(x)}{g(x)} $ as ratio of L.C.M of $ (a,c) $ to H.C.F of $ (b,d) $ .