1) 2sinx^2-6cosx+6=0 2) cos2x-2sinxcosx-3sin2x=0 3) sin4xcos3x+sin3xcos4x = - 1 4) 7sin^2x-4sin2x+cos^2x=0 5) 4sinx-3=3cosx

Найдем корни. ∈

1) 2 * sin² x - 6 * cos x + 6 = 0;

2 * (1 - cos² x) - 6 * cos x + 6 = 0;

-2 * cos² x + 2 - 6 * cos x + 6 = 0;

-2 * cos² x - 6 * cos x + 8 = 0;

2 * cos² x + 6 * cos x - 8 = 0;

cos² x + 3 * cos x - 4 = 0;

D = 9 - 4 * 1 * (-4) = 9 + 16 = 25 = 5²;

cos x = (-3 + 5) / 2 = 2/2 = 1;

cos x = 1;

x = 2 * pi * n, n ∈ Z;

cos x = (-3 - 5) / 2 = - 8/2 = - 4;

Нет корней.

Ответ: x = 2 * pi * n, n ∈ Z.

2) cos (2 * x) - 2 * sin x * cos x - 3 * sin (2 * x) = 0;

cos² x - sin² x - 2 * sin x * cos x - 3 * 2 * sin x * cos x = 0;

cos² x - sin² x - 8 * sin x * cos x = 0;

1 - tg² x - 8 * tg x = 0;

tg² x + 8 * tg x - 1 = 0;

D = 64 - 4 * 1 * (-1) = 64 + 4 = 70;

tg x = (-8 + √70) / 2;

x = arctg ((-8 + √70) / 2) + pi * n, n ∈ Z;

tg x = (-8 - √70) / 2;

x = arctg ((-8 - √70) / 2) + pi * n, n ∈ Z.

3) sin (4 * x) * cos (3 * x) + sin (3 * x) * cos (4 * x) = - 1;

sin (4 * x + 3 * x) = - 1;

sin (7 * x) = - 1;

7 * x = - pi/2 + 2 * pi * n, n ∈ Z;

x = - pi/14 + 2 * pi/7 * n, n ∈ Z;

4) 7 * sin² x - 4 * sin (2 * x) + cos² x = 0;

7 * sin² x - 8 * sin x * cos x + cos² x = 0;

7 * tg² x - 8 * tg x + 1 = 0;

D = 64 - 4 * 1 * 7 = 64 - 28 = 36;

tg x = (8 + 6) / 14 = 1;

x = pi/4 + pi * n, ∈ Z;

tg x = (8 - 6) / 14 = 1/7;

x = arctg (1/7) + pi * n, n ∈ Z;

5) 4 * sin x - 3 = 3 * cos x;

16 * sin² x - 24 * sin x + 9 = 9 * cos² x;

16 * sin² x - 24 * sin x + 9 * sin² x + 9 * cos² x - 9 * cos² x = 0;

25 * sin² x - 24 * sin x = 0;

sin x * (25 * sin x - 24) = 0;

sin x = 0;

x = pi * n, n ∈ Z;

25 * sin x = 24;

sin x = 24/25;

x = (-1) ^n * arcsin (24/25) + pi * n, n ∈ Z.