Решебник по алгебре и начала математического анализа 11 класс Алимов Задание 763

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\[\mathbf{Н}\mathbf{а\ }отрезке\ \lbrack - 2\pi;\ - \pi\rbrack.\]

\[1)\ 1 + 2\cos x \geq 0\]

\[2\cos x \geq - 1\]

\[\cos x \geq - \frac{1}{2}\]

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

\[На\ искомом\ отрезке:\]

\[- 2\pi \leq x_{1} \leq - \frac{4\pi}{3}.\]

\[2)\ 1 - 2\sin x < 0\]

\[- 2\sin x < - 1\]

\[2\sin x > 1\]

\[\sin x > \frac{1}{2}\]

\[\frac{\pi}{6} + 2\pi n < x < \pi - \frac{\pi}{6} + 2\pi n\]

\[\frac{\pi}{6} + 2\pi n < x < \frac{5\pi}{6} + 2\pi n\]

\[На\ искомом\ отрезке:\]

\[- \frac{11\pi}{6} < x_{1} < - \frac{7\pi}{6}.\]

\[3)\ 2 + tg\ x > 0\]

\[tg\ x > - 2\]

\[\text{arctg}( - 2) + \pi n < x < \frac{\pi}{2} + \pi n\]

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

\[На\ искомом\ отрезке:\]

\[- 2\pi \leq x_{1} < - \frac{3\pi}{2}\]

\[- arctg\ 2 - \pi < x_{2} \leq - \pi.\]

\[4)\ 1 - 2\ tg\ x \leq 0\]

\[- 2\ tg\ x \leq - 1\]

\[2\ tg\ x \geq 1\]

\[tg\ x \geq \frac{1}{2}\]

\[\text{arctg}\frac{1}{2} + \pi n \leq x < \frac{\pi}{2} + \pi n.\]

\[На\ искомом\ отрезке:\]

\[\text{arctg}\frac{1}{2} - 2\pi \leq x_{1} < - \frac{3\pi}{2}.\]