Страница 411 номер 1383, ГДЗ по алгебре за 10-11 класс Алимов - ответы и решение

1) \(\sin x + \sin{2x} + \sin{3x} + \sin{4x} = 0\)
\(2\sin{\dfrac{5}{2}x}\cos\dfrac{x}{2} + 2\sin{\dfrac{5}{2}x}\cos{\dfrac{3}{2}x} = 0,\)
\(\sin{\dfrac{5}{2}x}\left( \cos\dfrac{x}{2} + \cos{\dfrac{3}{2}x} \right) = 0,\)
\(\sin{\dfrac{5}{2}x} = 0,\)
\(\dfrac{5}{2}x = \pi n,\)
\(x_{1} = \dfrac{2}{5}\pi n, n \in Z;\)

или \(2\cos x\cos\frac{x}{2} = 0\)

\(\cos x = 0,\)
\(x_{2} = \dfrac{\pi}{2} + \pi n, n \in Z;\)
\(\cos\dfrac{x}{2} = 0,\)
\(x_{3} = \pi + 2\pi n, n \in Z;\)

Ответ: \(x_{1} = \frac{2}{5}\pi n\), \(x_{2} = \frac{\pi}{2} + \pi n\), \(x_{3} = \pi + 2\pi n\), \(n \in Z\);

2) \(\cos x + \cos{2x} + \cos{3x} + \cos{4x} = 0\)
\(2\cos{\dfrac{5}{2}x}\cos\dfrac{x}{2} + 2\cos{\dfrac{5}{2}x}\cos{\dfrac{3}{2}x} = 0,\)
\(\cos{\dfrac{5}{2}x}\left( \cos\dfrac{x}{2} + \cos{\dfrac{3}{2}x} \right) = 0,\)
\(\cos{\frac{5}{2}x} = 0\) или \(2\cos x\cos\frac{x}{2} = 0\)
\(x_{1} = \dfrac{\pi}{5} + \dfrac{2}{5}\pi n,\)
\(x_{2} = \dfrac{\pi}{2} + \pi n,\)
\(x_{3} = \pi + 2\pi n, n \in Z.\)