Решебник по алгебре 11 класс Никольский Параграф 7. Равносильность уравнений и неравенств Задание 20

\[\boxed{\mathbf{20.}}\]

\[\textbf{а)}\cos{2x} + 3\sin^{2}x + 2\sin x < 4\]

\[1 - 2\sin^{2}x + 3\sin^{2}x +\]

\[+ 2\sin x - 4 < 0\]

\[1 + \sin^{2}x + 2\sin x - 4 < 0\]

\[\sin^{2}x + 2\sin^{2}x - 3 < 0\]

\[t = \sin x:\]

\[t^{2} + 2t - 3 = 0\]

\[D_{1} = 1 + 3 = 4\]

\[t_{1} = - 1 + 2 = 1;\]

\[t_{2} = - 1 - 2 = - 3.\]

\[- 3 < t < 1\]

\[- 3 < \sin x < 1\]

\[\sin x < 1\]

\[\sin x = 1\]

\[x = \frac{\pi}{2} + 2\pi n.\]

\[- \frac{3\pi}{2} + 2\pi n < x < \frac{\pi}{2} + 2\pi n.\]

\[\textbf{б)}\cos{2x} - \cos^{2}x - 2\cos x < - 2\]

\[2\cos^{2}x - 1 - \cos^{2}x -\]

\[- 2\cos x + 2 < 0\]

\[\cos^{2}x - 2\cos x + 1 < 0\]

\[\left( \cos x - 1 \right)^{2} < 0\]

\[нет\ решений.\]