Count them all!

Clearly $(x,y) = (0, 0)$ is one solution. The range of the sine curve is contained in $[-99, 99]$ , which occur at the abscissae:

$x = {0.5, 2.5, 4.5, ... , 98.5}$ for maximum of 99.

$x = {-0.5, -2.5, -4.5, ... , -98.5}$ for minimum of -99

Each of these sine peaks contains two points of intersection with the line $y = x$ . The curves will no longer intersect each other in $(-\infty, -99] \cup [99, +\infty)$ . So we ultimately end up with: $50 \cdot 2$ RHS points + $50 \cdot 2$ LHS points - 1 point (the repeated origin $(0,0)$ ) = $\boxed{199}$ total intersection points.