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 \neq 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} }} \right. [/latex] [latex]x=arctg5+ \pi n,[/latex] n∈Z [latex]x=-arctg \frac{1}{3} + \pi m,[/latex] m∈Z