Area

sin(x+y) > 0
→ 2nπ < x + y < (2n+1)π, n is an integer.

Now if we draw both the curves, we get something like this (I haven't marked the intersection points)

The black region is our required area. But no. of black segments = no. of white segments. Their areas are equal (by symmetry). Let it be A.
→ 2A = 100π
→ A = 50π