Consider the region $R=\{(x,y):x^{2}+y^{2} \leq 100, \sin (x+y)>0\}.$ What is the area of $R$ ?
If we take a look at it's graph, https://www.desmos.com/calculator/b9jnct7ik0, it easily follows, from the symmetry that the area is half of the area of the circle. But my point, is, to solve this problem, I think approaching it graphically , seems more reasonable. So, I tried sketching the graph without a graphing calculator. The graph of the inequality, $x^{2}+y^{2} \leq 100$ is trivial to sketch. However, the problem comes with the inequality, $\sin(x+y)>0\implies 2k\pi\lt x\lt (2k+1)\pi,$ of which the graph I am neither able to visualise nor able to sketch.
Are there any ways to predict the graph of this inequalities using elementary ways?
If no such method exists, I am interested, in an alternative approach to this problem, which doesn't rely on graphs/graphing calculator.
You got $$\sin(x+y)>0\implies 2k\pi\lt x+y\lt (2k+1)\pi$$ which gives $$y>2k\pi - x$$ and $$y<(2k+1)\pi -x$$ These two are just straight lines and hence fairly easy to draw or visualize.