Question

If the period of \[\dfrac{{\cos (\sin (nx))}}{{\tan \left( {\dfrac{x}{n}} \right)}}\], \[n \in N\] is \[6\pi\] then \[n = ?\]
A. 3
B. 2
C. 5
D. 1

Answer 1

Hint: The given trigonometric expression defines the period of the function and asking for the value of the variable “n”, here we have to know that period of the function means the distance between the repetition of values for any function, here we have to solve the question be using the value of period in the function because it will satisfy the expression.

Complete step by step solution:
The given question is:
\[\dfrac{{\cos (\sin (nx))}}{{\tan \left( {\dfrac{x}{n}} \right)}},n \in N\] is \[6\pi\] then \[n = ?\]
Here we are going to use the period of the function as the value of “x” and then on solving we can get the value for the variable asked as per question, on solving we get:
\[ \Rightarrow \dfrac{{\cos (\sin (nx))}}{{\tan \left( {\dfrac{x}{n}} \right)}} \]
Substituting \[x = 6\pi \] we get
\[\Rightarrow \dfrac{{\cos (\sin (n6\pi ))}}{{\tan \left( {\dfrac{{6\pi }}{n}} \right)}};\] \[\sin\] value is zero for any integral multiple of $\pi$
\[ \Rightarrow \dfrac{{\cos (0)}}{{\tan \left( {\dfrac{{6\pi }}{n}} \right)}} = \dfrac{1}{{\tan \left( {\dfrac{{6\pi }}{n}} \right)}} \]
Here the denominator should not be equal to zero so that the expression can be valid, so using this condition for our expression and forming the mathematical conditin we can solve for the value of “n”, on solving we get:
\[
   \Rightarrow \tan \left( {\dfrac{{6\pi }}{n}} \right) \ne 0 \\
   \Rightarrow \dfrac{{6\pi }}{n} \ne {\tan ^{ - 1}}0 \ne m\pi (m \in \text{integer}) \\
 \]

\[ \Rightarrow \dfrac{{6\pi }}{{m\pi }} \ne n \]
\[ \Rightarrow n \ne \dfrac{6}{m} \]
Hence $n$ can be any natural number except for $\dfrac{6}{m}$

Here we got the value for the variable “n” which is this variable can have any value of natural numbers except the values of the ratio $\dfrac{6}{m}$, and is our final answer.

Note: The above question is solved by using the period given in the question, bit here we can also go with the graph terminology, here we have to convert every function into the “tan” function and then check the possible values of “n” by seeing the possible values in graph in the given period of the expression.