Question

is sine function is onto function for its natural domain?

Answer 1

A function f from A to B is called onto if for all b in B there is an a in A such that f (a) = b. In other words, all elements in B have to be used.

Now, the natural domain of sine function is R (the set of all real numbers).

Now, the sine function has range only [-1, 1]. And these values occur once each when $x\in [0,2\mathrm{\pi })$. (That is, sine x is one-one in $[0,2\mathrm{\pi })$). Beyond these values of x, sin x repeats values. So, sine function is not onto in R, which is its natural domain.