Решите уравнение
[tex]2 \sqrt{3} \times \sin \frac{x}{2} - \cos x + 2.5 = 0[/tex]
[tex]2 \sqrt{3} \times \sin \frac{x}{2} - \cos x + 2.5 = 0[/tex]
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Verified answer
2√3sin(x/2) - cosx + 2.5 = 0sin(x/2) = ±√(1 - cosx)/2
±2√3√(1 - cosx)/2 = cosx - 2.5
±2√3(1 - cosx)/2 = cosx - 2.5
Возводим обе части в квадрат
6(1 - cosx) = (cosx - 2.5)²
6 - 6cosx = cos²x - 5cosx + 6.25
cos²x + cosx + 1/4 = 0
Обозначим cosx = t
t² + t + 1/4 = 0
D = 1 - 4*1/4 = 1 - 1 = 0
t = -1/2
cosx = -1/2
x = ±2π/3 + 2πn
Проверим корень 2π/3.
Получаем 2√3*√3/2 + 1/2 + 2.5 ≠ 0
Подставим корень -2π/3.
Получим -2√3*√3/2 + 1/2 + 2.5 =
1/2 - 3 + 2.5 = -2.5 + 2.5 = 0.
Ответ: x = -2π/3 + 2πn, n ∈ Z.