Question

If the function $f\left(x\right)=\sin x+\cos px$ is periodic, then $p$ must be a

• A. Rational Number
• B. Imaginary number
• C. Irrational Number
• D. Purely imaginary number

Answer 1

The correct option is A Rational Number

$f\left(x\right)=\sin x+\cos \left(px\right)$
$f(x+T)=\sin (x+T)+\cos (px+pT)$
Now $f\left(0\right)=1.\sin \left(T\right)+\cos pT=1\Rightarrow 2\sin T=0\Rightarrow T=n\pi$
Now $\sin n\pi +\cos pn\pi =1$ $\cos pn\pi =2m\pi$$p=\frac{2m}{n}$ i.e. Rational.