Question

Prove that $f\left(x\right)=\sin x$ is periodic function.

Answer 1

If $\sin x$ is periodic, then

$\sin (x+T)=\sin x,\forall x$

$\Rightarrow \sin (x+T)-\sin x=0$

$\Rightarrow 2\cos \left(\frac{x+T+x}{2}\right)\sin \left(\frac{x+T-x}{2}\right)=0$

$\Rightarrow 2\cos \left(\frac{2x+T}{2}\right)\sin \left(\frac{T}{2}\right)=0$

$\Rightarrow \cos (x+\frac{T}{2})=0$ is not possible for all $x$

$\therefore \sin \frac{T}{2}=0$

$\Rightarrow \frac{T}{2}=n\pi$

$\therefore T=2n\pi ,n\in I$ for least positive value of $T$.

$T=2\pi$

Hence $\sin x$ is periodic with period $2\pi$