1) 2cos5x + 3=0; 2) 8sinx + 5= 2cos2x; 3) cos

1) 2 * cos (5 * x) + 3 = 0;

2 * cos (5 * x) = -3;

cos (5 * x) = -3/2;

5 * x = +-arccos (-3/2) + 2 * pi * n, n Z;

5 * x = +-5 * pi/6 + 2 * pi * n, n Z;

x = +-pi/6 + 2 * pi/5 * n, n Z.

2) 8 * sin x + 5 = 2 * cos (2 * x); 

2 * cos^2 x 2 * sin^2 x 8 * sin x 5 * sin^2 x 5 * cos^2 x = 0; 

-7 * sin^2 x 8 * sin x 3 * cos^2 x = 0;

7 * sin^2 x + 8 * sin x + 3 * cos^2 x = 0;

7 * tg^2 x + 8 * tg x + 3 = 0;

D = 64 4 * 7 * 3 = 64 84 = -20;

Нет корней.

3) cos x/3 5 * sin x/3 * cos x/3 = 3; 

cos x/3 5 * sin x/3 * cos x/3 3 * cos x/3 3 * sin x/3 = 0;

-3 * sin x/3 5 * sin x/3 * cos x/3 2 * cos x/3 = 0;

3 * tg x/3 + 5 * tg x + 2 = 0;

D = 25 4 * 3 * 2 = 1;

1) tg x/3 = (-5 + 1)/6 = -4/6 = -2/3;

x/3 = arctg (-2/3) + pi * n, n Z; 

x = 3 * arctg (-2/3) + 3 * pi * n, n Z;

2) tg x/3 = (-5 1)/6 = -1;

x/3 = arctg (-1) + pi * n, n Z;

x/3 = 3 * pi/4 + pi * n, n Z;

x = 9 * pi/4 + 3 * pi * n, n Z. 

4) (2 * sin x - 1) * sin x = sin (2 * x) cos x;  

(2 * sin x 1) * sin x cos x * (2 * sin x - 1) = 0;

(2 * sin x 1) * (sin x cos x) = 0;

sin x = ;

sin^2 x 2 * sin x * cos x + cos^2 x = 0;

x = (-1)^n * пи/6 + пи * n;

sin (2 * x) = 1;

x = (-1)^n * пи/6 + пи * n;

2 * x = пи/2 + пи * n;

x = (-1)^n * пи/6 + пи * n;

x = пи/4 + пи/2 * n.

5) cos ( + x)  - sin (/2 + x) sin (2 * x) = 0;  

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

2 * cos x + 2 * sin x * cos x = 0;

2 * cos x * (1 + sin x) = 0;

cos x = 0;

sin x = -1;

x = пи * n;

x = -пи/2 + 2 * пи * n.

6) 5 * sin2x 2 * cos x = 0;   

10 * sin x * cos x 2 * cos x = 0;

2 * cos x * (5 * sin x 1) = 0;

cos x = 0;

sin x = 1/5;

x = пи * n;

x = (-1)^n * arcsin (1/5) + пи * n.

7) cos2x - cos6x = 7sin  (2 * x);

-2 * sin (4 * x) * sin (-2 * x) 7 * sin^2 (2 * x) = 0;

sin (2 * x) * (7 * sin (4 * x) 7 * sin (2 * x) = 0;

sin (2 * x) = 0;

2 * x = пи * n;

x = пи/2 * n.

8) sin10x + sin2x = cos2x;

2 * sin (6 * x) * cos (4 * x) cos (2 * x) = 0;

2 * sin (6 * x) * (cos^2 (2 * x) sin^2 (2 * x) cos (2 * x) = 0;

Нет корней.