1) 2cos5x + √3=0; 2) 8sinx + 5 = 2cos2x; 3) cos² x/3 - 5sinx/3·cosx/3 = 3; 4) (2sinx - 1) ·sinx = sin2x-cosx; 5) cos (π+x) - sin (π/2 + x) - sin2x=0; 6) 5sin2x - 2cosx = 0; 7) cos2x - cos6x = 7sin²x2x; 8) √2sin10x + sin2x = cos2x.

Question

Егор Бирюков

Математика

1 июня, 21:27

1) 2cos5x + √3=0; 2) 8sinx + 5 = 2cos2x; 3) cos² x/3 - 5sinx/3·cosx/3 = 3; 4) (2sinx - 1) ·sinx = sin2x-cosx; 5) cos (π+x) - sin (π/2 + x) - sin2x=0; 6) 5sin2x - 2cosx = 0; 7) cos2x - cos6x = 7sin²x2x; 8) √2sin10x + sin2x = cos2x.

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Александр Зубков · Answer 1

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) cos² x/3 - 5 * sin x/3 * cos x/3 = 3;

cos² x/3 - 5 * sin x/3 * cos x/3 - 3 * cos² x/3 - 3 * sin² x/3 = 0;

-3 * sin² x/3 - 5 * sin x/3 * cos x/3 - 2 * cos² x/3 = 0;

3 * tg² 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 = ½;

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 (π + x) - sin (π/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 = 7sin² (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.