First slide

Functions (XII)

Question

The value of n ∈ I for which the function $f (x) = \frac{\sin nx}{\sin (\frac{x}{n})} has 4 π$ as its period is

Moderate

A

2
B

3
C

4
D

5

Solution

For n = 2, we have
$f (x) = \frac{\sin 2 x}{\sin (\frac{x}{2})} = \frac{2 \sin xcos x}{\sin (\frac{x}{2})} = \frac{4 \sin x / 2 \cos x / 2 \cos x}{\sin (\frac{x}{2})} = 4 \cos \frac{x}{2} \cos x$
The period of cos x is 2 $π$ and that of cos $\frac{x}{2} is \frac{2 π}{1 / 2} =4 π$
Hence, the period of f (x) is 4 $π$ .

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