Question

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

Answer 1

Period of `sinx=2 pi =(2pi)/(I)` and period of `cos ax=(2 pi)/(|a|)`
` :. " Period of " sinx +cos ax= LCM " of " (2pi)/(I) " and " (2pi)/(|a|)`
`=(LCM" of " 2pi " and " 2pi)/(HCF " of " 1 and a)`
`=(2pi)/(lambda)`
where ` lambda` is the HCF of 1 and `a,(1)/(lambda) " and " (|a|)/(lambda)` should both be integers.
Suppose `(1)/(lambda)=p " and "(|a|)/(lambda)=q.`
Then, `((|a|)/(lambda))/((1)/(lambda)) =(q)/(p), " where " p, q in Z`
i.e., `|a|=(p)/(q)`
Hence, a is the rational number.