Question

Решить заранее решите неравенство: е) ж) з)

Answer 1

е) $\text{cos}(2x) \leqslant -\dfrac{1}{2}$

$t_{1} + 2\pi n \leqslant t \leqslant t_{2} + 2\pi n, \ n \in Z$

$t = 2x\\t_{1} = \text{arccos}\bigg(-\dfrac{1}{2} \bigg) = \dfrac{2\pi}{3}\\t_{2} = 2\pi - \dfrac{2\pi}{3} = \dfrac{4\pi}{3}$

$\dfrac{2\pi}{3} + 2\pi n \leqslant 2x \leqslant \dfrac{4\pi}{3} + 2\pi n, \ n \in Z\\\\\dfrac{2\pi}{6} + \dfrac{2\pi n}{2} \leqslant \dfrac{2x}{2} \leqslant \dfrac{4\pi}{6} + \dfrac{2\pi n}{2}, \ n \in Z\\\\\dfrac{\pi}{3} + \pi n \leqslant x \leqslant \dfrac{2\pi}{3} + \pi n, \ n \in Z\\\\\text{OTBET:} \ x \in \bigg[\dfrac{\pi}{3} + \pi n; \dfrac{2\pi}{3} + \pi n \bigg], \ n \in Z$

ж) $\text{tg}(5x) \leqslant -1$

$-\dfrac{\pi}{2} + \pi n < t \leqslant t' + \pi n, \ n \in Z$

$t = 5x\\t' = \text{arctg(-1)} = -\dfrac{\pi}{4}$

$-\dfrac{\pi}{2} + \pi n < 5x \leqslant -\dfrac{\pi}{4} + \pi n, \ n \in Z\\\\-\dfrac{\pi}{10} + \dfrac{\pi n}{5} < \dfrac{5x}{5}\leqslant -\dfrac{\pi}{20} + \dfrac{\pi n}{5}, \ n \in Z\\\\-\dfrac{\pi}{10} + \dfrac{\pi n}{5} < x \leqslant -\dfrac{\pi}{20} + \dfrac{\pi n}{5}, \ n \in Z\\\\\text{OTBET:} \ x \in \bigg(-\dfrac{\pi}{10} + \dfrac{\pi n}{5}; -\dfrac{\pi}{20} + \dfrac{\pi n}{5} \bigg], \ n \in Z$

з) $\text{ctg}(4x) \geqslant -1$

$\pi n < t \leqslant t' + \pi n, \ n \in Z\\\\t = 4x\\t' = \text{arcctg}(-1) = \pi - \text{arcctg}(1) = \pi - \dfrac{\pi}{4} = \dfrac{3\pi}{4}$

$\pi n < 4x \leqslant \dfrac{3\pi}{4} + \pi n, \ n \in Z\\\\\dfrac{\pi n}{4} < \dfrac{4x}{4} \leqslant \dfrac{3\pi}{16} + \dfrac{\pi n}{4}, \ n \in Z\\\\\dfrac{\pi n}{4} < x \leqslant \dfrac{3\pi}{16} + \dfrac{\pi n}{4}, \ n \in Z\\\\\text{OTBET:} \ x \in \bigg(\dfrac{\pi n}{4}; \dfrac{3\pi}{16} + \dfrac{\pi n}{4} \bigg], \ n \in Z$