Question

Пожалуйста, даю 30 балов!
Решить уравнение
5cos14x+cos7x-4=0

Answer 1

$5\cos14x+\cos7x-4 = 0\\\cos14x = \cos^2 7x - \sin^2 7x = \cos^2 7x - 1 + \cos^2 7x = 2\cos^2 7x - 1\\10\cos^2 7x - 5 + \cos 7x - 4 = 10\cos^2 7x +\cos7x - 9 = 0\\D = 1 + 4\cdot9\cdot10 = 361\\\cos_1 7x = \frac{-1 + 19}{20} = \frac{9}{10}\\\cos_2 7x = \frac{-1 - 19}{20} = -1\\\cos_1 7x = \frac9{10} \Rightarrow x = \frac17\cdot\arccos\frac9{10}\\\cos_2 7x = -1 \Rightarrow x = \frac17\cdot(\pi + \pi k)$