Question

1)sin x+2cos x=0 2)4cos x-sin x=0 3)4cos^2 x+0,5 sin^2 x+3sin^2 x=3 4)3sin^2x+4sinx*cosx+ 5 cos^2 x=2 ; з

Answer 1

$sinx+2cosx=0|:cosx\\tgx+2=0\\tgx=-2\\x=-arctg2+\pi n, \; n\in Z.\\\\\\4cosx-sinx=0|:(-cosx)\\tgx-4=0\\tgx=4\\x=arctg4+\pi n, \; n\in Z.\\\\\\4cos^2x+0,5sin^2x+3sin^2x=3\\4cos^2x+3,5sin^2x=3(sin^2x+cos^2x)\\4cos^2x+3,5sin^2x-3sin^2x-3cos^2x=0\\cos^2x+0,5sin^2x=0\\1-sin^2x+0,5sin^2x=0\\1-\frac{1}{2}sin^2x=0\\sin^2x=2\\sinx=\pm \sqrt2\\\\sinx=\sqrt2\\x=(-1)^narcsin \sqrt2 +\pi n, \; n\in Z;\\\\sinx=-\sqrt2\\x=-arcsin\sqrt2 +\pi n, \; n\in Z.$

$3sin^2x+4sinxcosx+5cos^2x=2\\3sin^2x+4sinxcosx+5cos^2x=2(sin^2x+cos^2x)\\3sin^2x+4sinxcosx+5cos^2x-2sin^2x-2cos^2x=0\\sin^2x+4sinxcosx+3cos^2x=0|cos^2x\\tg^2x+4tgx+3=0\\tgx=u\\u^2+4u+3=0\\D:16-12=4\\u=\frac{-4\pm 2}{2}=-4\pm 1\\\\u_1-3\\tgx=-3\\x=-arctg3+\pi n, \; n\in Z;\\\\u_2=-5\\tgx=-5\\x=-arctg5+\pi n, \; n\in Z.$