Задача 76423 2cos 2 * (x - pi/3) + 8cos(x - pi/3) =...

65fb1da8e3fa0101c8ad820b

3/20/2024 5:34:05 PM

2cos 2 * (x - pi/3) + 8cos(x - pi/3) = 3. ПЛИЗ

626fab9a354ab65fb2f43abb

3/20/2024 6:38:40 PM

★

2cos (2(x - π/3)) + 8cos (x - π/3) = 3
Сделаем замену: t = x - π/3
2cos (2t) + 8cos (t) - 3 = 0
2(2cos^2 (t) - 1) + 8cos (t) - 3 = 0
4cos^2 (t) + 8cos (t) - 5 = 0
Получили квадратное уравнение относительно cos (t)
D/4 = 4^2 - 4(-5) = 16 + 20 = 36 = 6^2
cos (t) = (-4 - 6)/4 = -10/4 < -1 - решений нет.
cos (t) = (-4 + 6)/4 = 2/4 = 1/2
t1 = -π/3 + 2π*n, n ∈ Z
x1 = t1 + π/3 = -π/3 + 2π*n + π/3 = 2π*n, n ∈ Z
t2 = π/3 + 2π*n, n ∈ Z
x2 = t2 + π/3 = π/3 + 2π*n + π/3 = 2π/3 + 2π*n, n ∈ Z

Ответ: x1 = 2π*n, x2 = 2π/3 + 2π*n, n ∈ Z