does presence of constant affect perodicity of a perodic function like sinx?

Question

As we know sinx is a perodic function with 2 pi period but if it is lets say 2\3+sinx is it perodic As 2/3 has no period?

Answer 1

Yes, it's periodic, because if you write $$ f(x) = \frac{2}{3} + \sin(x) $$ then it's true that $f(x+ 2\pi) = f(x)$, which means it's periodic with a period of $2\pi$. (It's also periodic with period $4\pi, 6\pi, ...$, but it happens that $2\pi$ is the smallest period.)

Sometimes it's easiest, for problems like this, to go back to the definition and just check things out, as I've done here.