First slide

Functions (XII)

Question

If the period of $\frac{\cos (\sin (n x))}{\tan (\frac{x}{n})}, n \in N, is 6 π, then n =$

Moderate

A

3
B

2
C

6
D

1

Solution

The period of $\cos (\sin n x)$ is $\frac{π}{n}$ and the period of $\tan (\frac{x}{n})$ is $π n$ .
$\begin{array}{l} Thus, 6 π = LCM (\frac{π}{n}, π n) \\ or 6 π = \frac{π}{n} λ_{1} or n = \frac{λ_{1}}{6}, \\ and 6 π = λ_{2} π n or n = \frac{6}{λ 2}, λ_{1}, λ_{2} \in I^{+} \end{array}$
From $n = \frac{6}{λ_{2}}, n = 6, 3, 2, 1 .$
Clearly, for n = 6, we get the period of f(x) to be $6 π$

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