trigonometry sin,cos,tan

Question

solve the following equation giving the solutions in the domain -180<x>180

3sinx + cosx = 2.5

Answer 1

3sinx=2.5-cosx.

Square both sides:

9sin^2x=6.25+cos^2x-5cosx.

9(1-cos^2x)=6.25+cos^2x-5cosx.

9-9cos^2x=6.25+cos^2x-5cosx.

10cos^2x-5cosx-2.75=0.

cos^2x-0.5cosx-0.275=0.

Completing the square:

(cos^2x-0.5cosx+0.0625)-0.0625-0.275=0.

(cosx-0.25)^2=0.3375.

Square root of each side:

cosx-0.25=+0.58095 approx, so cosx=0.25+0.58095, cosx=0.83095 or -0.33095.

x=-33.80°, 33.80°, 109.33°, -109.33°.

Of these tentative solutions, 33.80° and 109.33 are valid.

But, we need to check in the original equation:

3sinx+cosx=