Решите уравнение ctg x - 2ctg 2x = (2)/(3)cos x .

Задача №17559: Уравнение

Решите уравнение $ctg x - 2 ctg 2 x = \frac{2}{3} cos x .$

ctg x - 2 ctg 2 x = \frac{2}{3} cos x ⟺ \frac{cos x}{sin x} - \frac{cos 2 x}{sin x cos x} = \frac{2}{3} cos x ⟺ \frac{sin ^{2} x}{sin x cos x} = \frac{2}{3} cos x ⟺

{3 sin x = 2 cos^{2} x sin 2 x \neq = 0 ⟺ {2 sin^{2} x + 3 sin x - 2 = 0 sin 2 x \neq = 0 ⟺

{(2 sin x - 1) (sin x + 2) = 0 sin 2 x \neq = 0 ⟺ sin x = \frac{1}{2} ⟺ x = \frac{π}{6} + 2 k π, \frac{5 π}{6} + 2 k π, k \in Z

Решите уравнение $ctg x - 2 ctg 2 x = \frac{2}{3} cos x .$

ctg x - 2 ctg 2 x = \frac{2}{3} cos x ⟺ \frac{cos x}{sin x} - \frac{cos 2 x}{sin x cos x} = \frac{2}{3} cos x ⟺ \frac{sin ^{2} x}{sin x cos x} = \frac{2}{3} cos x ⟺

{3 sin x = 2 cos^{2} x sin 2 x \neq = 0 ⟺ {2 sin^{2} x + 3 sin x - 2 = 0 sin 2 x \neq = 0 ⟺

{(2 sin x - 1) (sin x + 2) = 0 sin 2 x \neq = 0 ⟺ sin x = \frac{1}{2} ⟺ x = \frac{π}{6} + 2 k π, \frac{5 π}{6} + 2 k π, k \in Z