Trigonometric Equations Question 128
Question: The value of $ n\in Z $ for which the function $ f(x)=\frac{\sin nx}{\sin (x/n)} $ has $ 4\pi $ as its period, is
Options:
A) 2
B) 3
C) 4
D) 5
Show Answer
Answer:
Correct Answer: A
Solution:
- For $ n=2,f(x)=\frac{\sin ,2x}{\sin ,( \frac{x}{2} )}=\frac{4,\sin ,( \frac{x}{2} ),\cos ,( \frac{x}{2} ),\cos x}{\sin ,( \frac{x}{2} )} $ $ =4,\cos ,( \frac{x}{2} ),\cos x $ The period of $ \cos x=2\pi $ and that of $ \cos \frac{x}{2} $ is $ 4\pi , $ so period of $ \frac{\sin 2x}{\sin ,( \frac{x}{2} )} $ is $ 4\pi . $ For $ n=3,\frac{\sin ,{ 3,(x+4\pi ) }}{\sin ,{ \frac{(x+4\pi )}{3} }}=\frac{\sin ,3x}{\sin ,( \frac{x}{3}+\frac{4\pi }{3} )}\ne \frac{\sin ,3x}{\sin ,( \frac{x}{3} )} $ So, $ 4\pi $ is not the period for $ n=3. $ Similarly, we can see that $ 4\pi $ is not the period of $ \frac{\sin ,nx}{\sin ( \frac{x}{n} )} $ for $ n=4 $ and 5 also.
BETA
