How can I show that this set is bounded?

Question

How can I show that this set is bounded?

Montrez que l'ensemble $$A = \left\{\frac{\sin(xy)}{x^2+y^2}; (x,y)\in\mathbb R^2, (x,y)\ne (0,0)\right\}$$ est borné.

(Source)

this is my try answer

FIRST: I tried to put an R that belongs to A so R=sin(xy)/x²+y²

SECOND: I turnd the equality into inequality and I put the absolute value like that: |R|<=|sin(xy)|/x²+y² and after that I stucked because i didn't find the way how to solve this inequality and my target is to find borders for sin(xy)/x²+y²,Once I find them i can say that's bounded.

Answer 1

$\newcommand{\R}{\mathbb{R}}$ If you know that

1. $|\sin(a)| \leq |a|$ for all $a \in \R$.

2. $2|xy| \leq x^2 + y^2$ for all $x,y\in\R$ because $(|x|-|y|)^2 \geq 0$

$$ \frac{|\sin(xy)|}{x^2 + y^2} \leq \frac{|xy|}{x^2 + y^2} \leq \frac{1}{2} $$ for all $(x,y) \in \R^2\setminus \{(0,0)\}$.