Решебник по алгебре и начала математического анализа 11 класс Колягин Задание 118

\[\lbrack - 2\pi;\ - \pi\rbrack.\]

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

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

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

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

\[Ответ:\ \ \left\lbrack - 2\pi;\ - \frac{4\pi}{3} \right\rbrack.\]

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

\[2\sin x > 1\]

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

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

\[Ответ:\ \ \left( - \frac{11\pi}{6};\ - \frac{7\pi}{6} \right).\]

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

\[tg\ x > - 2\]

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

\[Ответ:\ \ \]

\[\left\lbrack - 2\pi;\ - \frac{3\pi}{2} \right) \cup ( - arctg\ 2 - \pi;\ - \pi\rbrack.\]

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

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

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

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

\[Ответ:\ \ \left\lbrack \text{arctg}\frac{1}{2} - 2\pi;\ - \frac{3\pi}{2} \right).\]