Страница 411 номер 1375, ГДЗ по алгебре за 10-11 класс Алимов - ответы и решение

1) \(\sin^{3}x + \cos^{3}x = 0\)
\(\left( \sin x + \cos x \right)\left( \sin^{2}x - \sin x\cos x + \cos^{2}x \right) = 0,\)
\(\sin x + \cos x = 0\) или \(1 - \sin x\cos x = 0\)
\(\operatorname{tg} x = - 1\)
\(x = - \dfrac{\pi}{4} + \pi n, n \in Z;\)

или \(\operatorname{tg}^{2}x - \operatorname{tg} x + 1 = 0\)

\(D < 0\) – корней нет;

2) \(2\sin^{2}x + \sin^{2}{2x} = 2\)
\(2\sin^{2}x + 4\sin^{2}x\left( 1 - \sin^{2}x \right) = 2,\)
\(2\sin^{2}x + 4\sin^{2}x - 4\sin^{4}x = 2,\)
\(2\sin^{4}x - 3\sin^{2}x + 1 = 0,\)
\(\sin^{2}x = z, 2z^{2} - 3z + 1 = 0\)
\(z_{1{,}2} = \dfrac{3 \pm \sqrt{9 - 8}}{4},\)
\(z_{1} = 1, z_{2} = \dfrac{1}{2}\)
\(\sin x = \pm 1\) или \(\sin x = \pm \frac{1}{\sqrt{2}}\)
\(x_{1} = \dfrac{\pi}{2} + \pi n,\)
\(x_{2} = \dfrac{\pi}{4} + \dfrac{\pi}{2}n, n \in Z;\)

3) \(8\sin x\cos{2x}\cos x = \sqrt{3}\)
\(4\sin{2x}\cos{2x} = \sqrt{3},\)
\(2\sin{4x} = \sqrt{3},\)
\(\sin{4x} = \dfrac{\sqrt{3}}{2},\)
\(4x = (-1)^{n} \cdot \dfrac{\pi}{3} + \pi n,\)
\(x = (-1)^{n} \cdot \dfrac{\pi}{12} + \dfrac{\pi}{4}n, n \in Z;\)

4) \(4\sin x\cos x\cos{2x} = \cos{4x}\)
\(2\sin{2x}\cos{2x} = \cos{4x},\)
\(\sin{4x} = \cos{4x},\)
\(\operatorname{tg} {4x} = 1,\)
\(4x = \dfrac{\pi}{4} + \pi n,\)
\(x = \dfrac{\pi}{16} + \dfrac{\pi}{4}n, n \in Z.\)