The value of n∈Z for which the function f(x)=sinnxsin(x/n) has 4π | Filo

Class 12

Math

Trigonometry

Trigonometric Functions

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150

The value of n∈Z for which the function f(x)=sinnxsin(x/n) has 4π as its period, is

1. 2
2. 3
3. 4
4. 5
Solution: Forn=2,f(x)=sin2xsin(x2)=4sin(x2)cos(x2)cosxsin(x2)                   =4cos(x2)cosx The period of cosx=2π and that of cosx2 is 4π, so period of sin2xsin(x2) is 4π. For n=3,sin{3(x+4π)}sin{(x+4π)3}=sin3xsin(x3+4π3)≠sin3xsin(x3) So, 4π is not the period for n=3. Similarly, we can see that 4π is not the period of sinnxsin(xn) for n=4 and 5 also.
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