1) 5cos в квадрате x + 6sinx-6=0 2) 2tg в квадрате x+tgx-2=0 3) sinx-3cosx=0 4) корень из 3*sinx*cosx+3cosв квадрате x=0 5) sin в квадрате * 3x/4-корень из 2/2 = sinx-cox в квадрате * 3x/4+1 6) 2cos в квадрате x/2+сорень из 3cosx/2=0 7) 2sin в квадрате 2x-5sin2x*cos2x+2cos в квадрате 2x=0 8) 5sin в квадрате x-14sinx*cosx-3cos в квпдрате x=2

1) 5cos^2 x + 6sinx - 6 = 0;

5 (1 - sin^2x) + 6sinx - 6 = 0;

5 - 5sin^2x + 6sinx - 6 = 0;

-5sin^2x + 6sinx - 1 = 0;

5sin^2x - 6sinx + 1 = 0;

5t^2 - 6t + 1 = 0;

D = 36 - 4 * 5 = 16;

t = (6 + - 4) / 10;

t1 = 1; t2 = 0,2;

sinx = 1 = > x = pi/2 + 2pin, n ∈ Z;

sinx = 0,2 = > x = (-1) ^narcsin0,2 + pin, n ∈ Z;

2) 2tg^2 x + tgx - 2 = 0;

2t + t - 2 = 0;

D = 1 + 4 * 2 * 2 = 17;

t = (-1 + - sqrt17) / 4;

tgx = (-1 + sqrt17) / 2 = > x = arctg (-1 + sqrt17) / 2 + pin, n ∈ Z;

tgx = (-1 - sqrt17) / 2 = > x = arctg (-1 - sqrt17) / 2 + pin, n ∈ Z;

3) sinx - 3cosx = 0;

tgx - 3 = 0;

tgx = 3;

x = arctg3 + pin, n ∈ Z;

4) sqrt3 * sinx * cosx + 3cos^2 x = 0;

cosx (sqrt3sinx + 3cosx) = 0;

cosx = 0 = > x = pi/2 + pin, n ∈ Z;

sqrt3sinx + 3cosx = 0;

sqrt3tgx + 3 = 0;

sqrt3tgx = - 3;

tgx = - 3/sqrt3;

tgx = - sqrt3 = > x = 2pi/3 + pin, n ∈ Z;

5) sin^2 (3x/4) - sqrt2/2 = sinx - cos^2 (3x/4) + 1;

1 - sqrt2/2 = sinx + 1;

sinx = - sqrt2/2 = > x = (-1) ^n5pi/4 + pin, n ∈ Z;

6) 2cos^2 x/2 + sqrt3cosx/2 = 0;

Пусть х/2 = a;

2cos^2a + sqrt3cosa = 0;

cosa (2cosa + sqrt3) = 0;

cosa = 0 = > a = pi/2 + pin = > x = pi + 2pin, n ∈ Z;

2cosa + sqrt3 = 0;

2cosa = - sqrt3;

cosa = - sqrt3/2 = > a = + -5pi/6 + 2pin = > x = + -5pi/3 + 4pin, n ∈ Z;

7) 2sin^2 2x - 5sin2x * cos2x + 2cos^2 2x = 0;

2tg^2 2x - 5tg2x + 2 = 0;

2t - 5t + 2 = 0;

D = 25 - 4 * 4 = 9;

t = (5 + - 3) / 4;

t1 = 1/2, t2 = 2;

tg2x = 1/2 = > 2x = arctg1/2 + pin = > x = (arctg1/2) / 2 + pi/2n, n ∈ Z;

tg2x = 2 = > 2x = arctg2 + pin = > x = (arctg2) / 2 + pin/2, n ∈ Z;

8) 5sin^2 x - 14sinx * cosx - 3cos^2 x = 2;

5sin^2x - 14sinxcosx - 3cos^2x = 2sin^2x + 2cos^2x;

5sin^2x - 14sinxcosx - 3cos^2x - 2sin^2x - 2cos^2x = 0;

3sin^2x - 14sinxcosx - 5cos^2x = 0;

3tg^2x - 14tgx - 5 = 0;

3t^2 - 14t - 5 = 0;

D = 14^2 + 4 * 3 * 5 = 256;

t = (14 + - 16) / 6;

t1 = 5, t2 = - 1/3;

x = arctg5 + pin, n ∈ Z;

x = arctg (-1/3) + pin, n ∈ Z.