If `f(x)=sinx+cosa x` is a periodic function, show that `a` is a rational number

Options

A. `a in Z `
B. `a in N `
C. `a in Q`
D. `a in R `

Answer

C

Explanation

Let T be the period of f(x). Then,
`f(x+T)=f(x)` for all x.
`implies sin(T+x) + cos a (T+x) = sin x + cos ax ` for all `x in R `
Putting x=0 and x=-T respectively , we get
` sinT+ cos a T =1`
and ` -sin T + cos a T=1`
Solving these two equations, we get
`sin T =0 and cos a T =1`
`implies T n pi and a T =2m pi `, where ` m, n in Z`.
`implies (aT)/(T)=(2mpi)/(npi)`
`implies a=(2m)/(n)`, which is a rational number
`=a in Q`