How to find the area of region $R$? Region $R$ contains all the points $(x,y)$ such that $x^2+y^2\leq100\;$  and $\:\sin(x+y)\geq0$. Find the area of region $R$.

$\:\:\:\:\:\:\:\:\:$$\sin(x+y)\geq0$
$\:\:\:\:\:\:\:\:\:$$\implies2n\pi\leq x+y\leq(2n+1)\pi$

Thus graph would be probably look like this:

$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:$

Now how to find the area?

For the points in the circle $(a,b)$ if $$\sin(a+b)\leq0$$  then for $(-a,-b)$ $$\sin((-a)+(-b))\geq0$$ 
Thus if our condition does not hold true for a single point we can take a symmetrically opposite point to make our conditions hold. Thus half the are of our circle will satisfy our condition 
(note there will be lines that have $\sin(x+y)=0$ but as they are lines their contribution to the area is negligible, not affecting the probability, in reality it should be slightly greater than 1/2) 
