5sin^2(x)-14sin(x)cos(x)-3cos^2(x)=2 Показать полное решение

[latex]5sin ^{2} x-14sinxcosx-3cos ^{2} x=2[/latex][latex]5sin ^{2} x-14sinxcosx-3cos ^{2} x-2(sin ^{2}x+cos ^{2}x) =0[/latex][latex]5sin ^{2} x-14sinxcosx-3cos ^{2} x-2sin ^{2}x-2cos ^{2}x =0[/latex][latex]3sin ^{2} x-14sinxcosx-5cos ^{2} x =0 |: cos ^{2} x,[/latex]  [latex](cos ^{2} x eq 0)[/latex][latex]3 frac{sin ^{2}x }{cos ^{2}x } -14 frac{sinx}{cosx} -5=0[/latex][latex]3tg ^{2} x-14tgx-5=0[/latex][latex]tgx=y[/latex][latex]3y ^{2} -14y-5=0[/latex]D = (-14)² - 4· 3·(-5) = 196 + 60 = 256[latex] sqrt{D}=16 [/latex][latex] y_{1} = frac{14+16}{6} = frac{30}{6}=5 [/latex][latex] y_{2} = frac{14-16}{6} = frac{-2}{6} =- frac{1}{3} [/latex][latex] left { {{tgx=5} atop {tgx=- frac{1}{3} }} ight. [/latex][latex]x=arctg5+ pi n,[/latex] n∈Z[latex]x=-arctg frac{1}{3} + pi m,[/latex] m∈Z