3sin^2x-7sinxcosx+4cos^2x=0

[latex]3sin^2x - 7sinxcosx + 4cos^2x = 0 [/latex] Разделим на [latex]cos^2x[/latex]. [latex]3tg^2x - 7tgx + 4 = 0[/latex] Пусть [latex]t = tgx.[/latex] [latex]3t^2 - 7t + 4 = 0 \\ D = 49 - 4 \cdot 4 \cdot 3 = 1 \\ t_1 = \frac{7 + 1}{6} = \frac{4}{3} \\ t_2 = \frac{7 - 1}{6} = 1[/latex] Обратная замена: [latex]tgx = \frac{4}{3} \\ x = \boxed{arctg \frac{4}{3} + \pi n, \ n \in Z} \\ tgx = 1 \\ \boxed{x = \frac{ \pi }{4} + \pi n, \ n \in Z}[/latex]