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1 Answer

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).

by Top Rated User (1.2m points)

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