10 sin^2x+5sinxcosx+cos^2x=3

Найдем корень уравнения.

10 sin^2 x + 5 * sin x * cos x + cos^2 x = 3;

10 sin^2 x + 5 * sin x * cos x + cos^2 x - 3 = 0;

10 * sin^2 x + 5 * sin x * cos x + cos^2 x - 3 * sin^2 x - 3 * cos^2 x = 0;

Приведем подобные.

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

8 * tg^2 x + 5 * tg x - 2 = 0;

Найдем х.

8 * a^2 + 5 * a - 2 = 0;

D = b^2 - 4 * a * c = 25 - 4 * 8 * (-2) = 25 + 64 = 91;

a1 = (-5 + √91) / (2 * 8) = (-5 + √91) / 16;

a2 = (-5 - √91) / 16;

1) tg x = (-5 + √19) / 16;

x = arctg ((-5 + √19) / 16) + пи * n, n ∈ Z;

2) tg x = (-5 - √19) / 16;

x = arctg ((-5 - √19) / 16) + пи * n, n ∈ Z.