Question

Помогите решить уравнения
а)sinx-2sinxcosx+4cosx-2=0
b)3sin^2x=2sinxcosx+cos^2x
c)5sin^2x-2sinxcosx+cost^2x=4

Answer 1

$sinx-2sinxcosx+4cosx-2=0\sinx(1-2cosx)-2(1-cosx)=0\(sinx-2)(1-2cosx)=0\\sinx-2=0\sinx neq 2\sinxin [-1;1]; \\1-2cosx=0\cosx=frac{1}{2}\x=pm frac{pi}{3}+2pi n, ; n in Z;$


$3sin^2x=2sinxcosx+cos^2x\3sin^2x-2sinxcosx-cos^2x=0|:cos^2x\3tg^2x-2tgx-1=0\tgx=u\3u^2-2u-1=0\D:4+12=16\u=frac{2pm 4}{6}\\u_1=1\tgx=1\x=frac{pi}{4}+pi n. ; nin Z;\\u_2=-frac{1}{3}\tgx=-frac{1}{3}\x=-arctgfrac{1}{3}+pi n, ; nin Z\\ cosx neq 0\x neq frac{pi}{2}+pi k, ; kin Z$


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