16cos^4 (x) + sin^2 (x) = 7/4

Найдем корень уравнения.

16 * cos^4 (x) + sin^2 (x) = 7/4;

Умножим на 4.

16 * 4 * cos^4 (x) + 4 * sin^2 (x) = 7/4 * 4;

64 * cos^4 x + 4 * sin^2 x = 7;

64 * cos^4 x + 4 * sin^2 x - 7 = 0;

64 * cos^4 x + 4 * (1 - cos^2 x) - 7 = 0;

64 * cos^4 x + 4 * 1 - 4 * cos^2 x - 7 = 0;

64 * cos^4 x - 4 * cos^2 x - 3 = 0;

Пусть cos^2 x = a, тогда:

64 * a^2 - 4 * a - 3 = 0;

D = (-4) ^2 - 4 * 64 * (-3) = 16 + 16 * 64 = 16 * 65 = 1 040;

a1 = (4 + √1040) / (2 * 64) = (4 + 4 * √65) / (2 * 64) = (1 + √65) / (2 * 16) = (1 + √65) / 32;

a2 = (1 - √65) / 32;

1) cos^2 x = (1 + √65) / 32;

cos x = + -√ ((1 + √65) / 32);

{ x = + -arccos (√ ((1 + √65) / 32) + 2 * pi * n;

x = + -arccos (-√ ((1 + √65) / 32) + 2 * pi * n;

2) cos^2 x = (1 - √65) / 32;

cos x = + -√ ((1 - √65) / 32);

{ x = + -arccos (√ ((1 - √65) / 32) + 2 * pi * n;

x = + -arccos (-√ ((1 - √65) / 32) + 2 * pi * n, n ∈ Z.