Example of a smooth function less than sin function

Question

Please help me to find an example of a smooth non-zero function $f:[0,\infty)\rightarrow\mathbb{R}$ such that $f(t)<\sin(t)$ for $t\neq n\pi$ and $f(n\pi)=0$ for $n\in\mathbb{N}$. I want $f$ in an explicit form.
Thank you.

Answer 1

You can take something like $$ \frac{\sin{x}}{1+a\sin{x}} $$ for some $0<a<1$. This is smooth since it is a nonsingular combination of smooth functions, and $1+a\sin{x}$ is larger than $1$ when $\sin{x}>0$ and smaller than $1$ when $\sin{x}<0$. Hence it is always smaller than $\sin{x}$. And of course it has the same zeros as $\sin{x}$.