First of all, you can disregard the 9.81 because it's multiplied by every term. So with equation 5 = 10sinx - 3cosx, just put 10sinx - 3cosx in Y1 and 5 in Y2. Now graph the curves and use calc, intersect to find the intersection.
As for the "x=360.*(N44+0.125871) or x=360.*(N44+0.466902)" thing, I would say that x is likely restricted to the domain [0,360), because most such problems are. In radians, that interval is written [0,2*pi). Basically, even though there are an infinite number of answers due to the cyclic nature of the sine and cosine functions, mathematicians generally only consider that interval because beyond that, the structure will repeat with some offset (in this case 44).
I wasn't sure exactly what you were asking, but I think I covered it. Let me know if I missed the point of your post or something.