Question

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Answer 1

$1.;3sin^2x-10sinx+7=0, quad sinx=t\3t^2-10t+7=0\frac{D}{4}:(frac{10}{2})^2-7*3=4\t_1,_2=frac{5pm2}{3}, quad t_2=frac{7}{3}, ; ; t_2=1;\\sinx_1 neq frac{7}{3},quad sinxin [-1;+1];\\sinx_2=1\x=frac{pi}{2}+2pi n, ; nin Z;\\2.;8sin^2x+10cosx-1=0\8(1-cos^2x)+10cosx-1=0|*(-1)\cos^2x-10cosx-7=0, quad cosx=t\t^2-10t-7=0\frac{D}{4}:(frac{10}{2})^2+7=32\t_1,_2=10pm4sqrt2, quad t_1=10+4sqrt2 textgreater 1, ; ; t_2=10-sqrt2 textgreater 1, \cosxin[-1;1];$

$3.;4sin^2x+13sinxcosx+10cos^2x=0|:cos^2x,\cosx neq 0,;x neq frac{pi}{2}+pi n,;nin Z;\4tg^2x+13tgx+10=0, quad tgx=t\4t^2+13t+10=0\D:169-160=9\t_1,_2=frac{-13pm3}{8}, quad t_1=-frac{5}{4}, ; t_2=-2;\tgx_1=-frac{5}{4}\x_1=-arctgfrac{5}{4}+pi n, ; nin Z;\\tgx_2=-2\x_2=-arctg2+pi n, ; nin Z;$

$4.;3tgx-3ctgx+8=0\3tgx-frac{3}{tgx}+8=0|*tgx,; tgx neq 0,;x neq pi n, ; nin Z;\3tg^2x+8tgx-3=0, quad tgx=t\3t^2+8t-3=0\frac{D}{4}:(frac{8}{2})^2+3*3=25\t_1,_2=frac{-4pm5}{3}, quad t_1=frac{1}{3}, ; t_2=-3;\\tgx_1=frac{1}{3}\x_1=arctgfrac{1}{3}+pi n, ; nin Z;\\tgx_2=-3\x_2=-arctg3+pi n, ; nin Z.$

$5.;sin2x+4cos^2x=1\2sinxcosx+4cos^2x-(sin^2x+cos^2x)=0\3cos^2x+2sinxcosx-sin^2x=0|:sin^2x,;sinx neq 0,;x neq pi n,;nin Z;\3ctg^2x+2ctgx-1=0, quad ctgx=t\3t^2+2t-1=0\frac{D}{2}:(frac{2}{2})^2+3*1=4;\t_1,_2=frac{-1pm2}{3},quad t_1=frac{1}{3}, ; t_2=-1;\\ctgx_1=frac{1}{3}\x_1=arcctgfrac{1}{3};\\ctgx_2=-1\x_2=frac{3pi}{4}+pi n, ; nin Z;$

$6.; 10cos^2x-9sin2x=4cos2x-4\10cos^2x-9sin2x-4(cos2x-1)=0\10cos^2x-9*2sinxcosx-4(-2sin^2x)=0\10cos^2x-18sinxcosx+8sin^2x=0|:2sin^2x\sinx neq 0,;x neq pi n,;nin Z;\cos^2x-9sinxcosx=0\4tg^2x-9tgx+5=0, quad tgx=t;\D:81-80=1\t_1,_2=frac{9pm1}{8}, quad t_1=frac{5}{4},;t_2=1;\\tgx_1=frac{5}{4}\x_1=arctgfrac{5}{4}+pi n, ; nin Z;\\tgx_2=1\x_2=frac{pi}{4}+pi n, ; nin Z.$