3cos^2(x) - sin(2x)=0.5 - Все ответы

$3cos^2x - sin2x = 0,5 \\ \\ 3cos^2x - 2sinxcosx = 0,5(sin^2x + cos^2x) \\ \\ 3cos^2x - 0,5cos^2x - 2sinxcosx - 0,5sin^2x = 0 \\ \\ 2,5cos^2x - 2sinxcosx - 0,5sin^2x = 0 \\ \\ sin^2x + 4sinxcosx - 5 cos^2x = 0 \\ \\ \dfrac{sin^2x}{cos^2x} + 4 \dfrac{sinxcosx}{cos^2x} - 5\dfrac{cos^2x}{cos^2x} = 0 \\ \\ tg^2x + 4tgx - 5 = 0$
Пусть $t = tgx \ (t \neq 0)$
$t^2 + 4t - 5 = 0 \\ \\ t_1 + t_2 = -4 \\ t_1 \cdot t_2 = -5 \\ \\ t_1 = 1 \\ t_2 = -5$
Обратная замена:
$tgx = 1 \\ \\ x = \dfrac{ \pi }{4} + \pi n, \ n \in Z \\ \\ tgx = -5 \\ \\ x = arctg(-5) + \pi k, \ k \in Z \\ \\ \boxed{OTBET: x = \dfrac{ \pi }{4} + \pi n, \ n \in Z; \ \ x = arctg(-5) + \pi k, \ k \in Z}$