From the first equation:
3sinA=2-5cosA; 3sqrt(1-cos^2A)=(2-5cosA); squaring each side: 9-9cos^2A=4-20cosA+25cos^2A; 34cos^2A-20cosA-5=0. Use the quadratic formula for finding cosA=(20+sqrt(400+680))/68=0.7774 or -0.1892.
cosA=0.7774 or -0.1892, A=38.98 or 100.90 degrees. Only the second answer satisfies the original equation.
So 5sinA-3cosA=5.4772 approx.
However, 5.4772 is sqrt(30), as can be seen below:
3sinA+5cosA=2; 9sin^2A+30sinAcosA+25cos^2A=4 (squaring both sides);
9-9cos^2A+30sinAcosA+25-25sin^2A=4;
30-9cos^2A-25sin^2A+30sinAcosA=0;
30=9cos^2A-30sinAcos+25sin^2A=(3cosA-5sinA)^2=(5sinA-3cosA)^2;
5sinA-3cosA=sqrt(30). But is this + or -? We know from earlier that it is +sqrt(30).