20(3 балла). Решить уравнение: 2 sin²x + 7 cosx + 2 = 0

Заменим sin²x на 1 - cos²x: 2(1 - cos²x) + 7 cosx + 2 = 0.

2 - 2cos²x + 7 cosx + 2 = 0 => -2cos²x + 7 cosx + 4 = 0.

Пусть t = cosx. -2t² + 7t + 4 = 0 => 2t² - 7t - 4 = 0.

t = (7 ± sqrt(49 - 4*2*(-4))) / 4 = (7 ± sqrt(49 + 32)) / 4 = (7 ± sqrt(81)) / 4 = (7 ± 9) / 4.

t1 = 16/4 = 4 (не подходит, т.к. cosx <= 1). t2 = -2/4 = -1/2.

cosx = -1/2. x = ± 2π/3 + 2πk, k ∈ Z.