[Math] How to prove the function $\sin(x)$ is not onto

Question

\begin{align}
f:\mathbb{R}\rightarrow\mathbb{R},\:f\left(x\right)=\sin\left(x\right)\tag{1}
\end{align}
I know that it is not onto because for all values of $y$ past $[-1,1]$ there is no $x$.

Graphically it makes sense but I'm finding it hard to do an actual proof for this function.

Answer 1

Since $\sin^2x+\cos^2x=1$, we see that $\sin^2x=1-\cos^2x\le 1$. Therefore $$ -1\le |\sin x|\le 1 $$ for all $x\in\mathbb{R}$. In particular, there's no $x\in\mathbb{R}$ such that $\sin x=42$.

_{Why $42$? Well, that's obvious! ;-)}