Вычислить 2 cos 2 П/5+cos4 п/5+cos6 П/5

1. Применим формулу для суммы косинусов:

x = 2cos (2π/5) + cos (4π/5) + cos (6π/5); x = 2cos (2π/5) + 2cos (π) cos (π/5); x = 2 (cos (2π/5) - cos (π/5)). (1)

2. Возведем в квадрат уравнение (1), с учетом того, что x < 0:

x^2 = 4 (cos (2π/5) - cos (π/5)) ^2; x^2 = 4 (cos^2 (2π/5) - 2cos (2π/5) cos (π/5) + cos^2 (π/5)); x^2 = 2 (1 + cos (4π/5)) - 4 (cos (3π/5) + cos (π/5)) + 2 (1 + cos (2π/5)); x^2 = 2 - 2cos (π/5) + 4 (cos (2π/5) - cos (π/5)) + 2 + 2cos (2π/5); x^2 = 4 + 6 (cos (2π/5) - cos (π/5)); x^2 = 4 + 3x; x^2 - 3x - 4 = 0; D = 3^2 + 4 * 4 = 9 + 16 = 25; x = (3 ± √25) / 2 = (3 ± 5) / 2; x1 = (3 - 5) / 2 = - 2/2 = - 1; x2 = (3 + 5) / 2 = 8/2 = 4, не подходит.

Ответ: - 1.