Is the space generated by $f(x)=\sin(x)$ a vector space? In my textbook it says that set of all the functions on a given interval is a vector space.
Does that mean that space generated by any function is a vector space?
If not then explain what it means with example.
If yes then how can one say that space generated by $\sin(x)$ is  a vector space because it's not closed under multiplication
$\sin(90)=1$
so $1$ belongs to vector space
$10*\sin(90)=10$ should also belong to vector space but it does not.
The vector space generated by $\sin{x}$ over $\mathbb{R}$ in the sense you mention would be
$$ \{ k\sin{x} : k \in \mathbb{R} \}. $$
It is isomorphic as a vector space to $\mathbb{R}$ by the linear map $k \sin{x} \mapsto k $.

What the book should actually mean is that
$$ \{ f : (f: [a,b] \to \mathbb{R} ) \} $$
is a vector space over $\mathbb{R}$. This is easy to check: clearly if $f,g$ are such functions, and $a,b \in \mathbb{R}$, then
$$ a f + b g $$
is another such function.
It means that the set of functions $f(x)$ where $f(x)$ has domain and range $a$ is a vector space with vector addition defined as the composition of functions. $f(x)= \sin (x)$ is not a vector spaces for the reason you gave.