For what integral value of n, is 3π the period of the function

cos(nx) sin(5x/n)

For a function to be periodic f(x+T) = f(x)

i.e.     cos[n(x+T)]sin[5(x+T)/n]=cos(nx)sin(5x/n)

but period of a function is a positive real no. independent of x.

=>cos(nT)sin(5T/n)=0

CASE1

If cos(nT)=0    => nT=[(2K+1)PI/2]     =>3PI(n) =[(2K+1)PI/2]      =>(2K+1)=6n

=>3n-k=1/2      which is not possible as n and k both are integral values

CASE 2

sin(5T/n)=0            =>(5T/n) =(k)pi        =>15/k=n

Now k is an integral value and thus n can attain 8 integral values   ±1 ,±3  ,±5  ,±15