Решите уравнение cos^2 x+sqrt(3)sin^2 x=(1+sqrt(3))(cos x-cos xsin x+sin x).

Задача №17637: Уравнение

Решите уравнение

cos^{2} x + 3 sin^{2} x = (1 + 3) (cos x - cos x sin x + sin x) .

cos^{2} x + 3 sin^{2} x = (1 + 3) (cos x - cos x sin x + sin x) ⟺

⟺ cos^{2} x + (1 + 3) cos x sin x + 3 sin^{2} x = (1 + 3) (cos x + sin x) ⟺

⟺ (cos x + sin x) (cos x + 3 sin x) = (1 + 3) (cos x + sin x) ⟺

⟺ (cos x + sin x) (\frac{1}{2} cos x + \frac{3}{2} sin x - \frac{1 + 3}{2}) = 0 ⟺

⟺ cos (x - \frac{π}{4}) (cos (x - \frac{π}{3}) - \frac{1 + 3}{2}) = 0 ⟺

⟺ cos (x - \frac{π}{4}) = 0 ⟺ x - \frac{π}{4} = \frac{π}{2} + k π, k \in Z ⟺ x = \frac{3 π}{4} + k π, k \in Z .

Решите уравнение

cos^{2} x + 3 sin^{2} x = (1 + 3) (cos x - cos x sin x + sin x) .

cos^{2} x + 3 sin^{2} x = (1 + 3) (cos x - cos x sin x + sin x) ⟺

⟺ cos^{2} x + (1 + 3) cos x sin x + 3 sin^{2} x = (1 + 3) (cos x + sin x) ⟺

⟺ (cos x + sin x) (cos x + 3 sin x) = (1 + 3) (cos x + sin x) ⟺

⟺ (cos x + sin x) (\frac{1}{2} cos x + \frac{3}{2} sin x - \frac{1 + 3}{2}) = 0 ⟺

⟺ cos (x - \frac{π}{4}) (cos (x - \frac{π}{3}) - \frac{1 + 3}{2}) = 0 ⟺

⟺ cos (x - \frac{π}{4}) = 0 ⟺ x - \frac{π}{4} = \frac{π}{2} + k π, k \in Z ⟺ x = \frac{3 π}{4} + k π, k \in Z .