Упр.653 ГДЗ Алимов 10-11 класс (Алгебра)

1. $$\cos\left(\frac{x}{3}+2\right)\geq\frac{1}{2}$$
2. $$\sin\left(\frac{x}{4}-3\right)<-\frac{\sqrt{2}}{2}$$

1) $$\cos\left(\frac{x}{3}+2\right)\ge \frac12$$

Используем формулу:

$$\cos t\ge a \iff -\arccos a+2\pi n\le t\le \arccos a+2\pi n,\quad n\in\mathbb Z.$$

Тогда

$$-\arccos\frac12+2\pi n\le \frac{x}{3}+2\le \arccos\frac12+2\pi n,$$

$$-\frac{\pi}{3}+2\pi n\le \frac{x}{3}+2\le \frac{\pi}{3}+2\pi n,$$

$$-\frac{\pi}{3}-2+2\pi n\le \frac{x}{3}\le \frac{\pi}{3}-2+2\pi n,$$

$$-\pi-6+6\pi n\le x\le \pi-6+6\pi n,\quad n\in\mathbb Z.$$

2) $$\sin\left(\frac{x}{4}-3\right)<-\frac{\sqrt2}{2}$$

Используем формулу:

$$\sin t

При $$a=-\frac{\sqrt2}{2}$$ имеем $$\arcsin\left(-\frac{\sqrt2}{2}\right)=-\frac{\pi}{4}.$$ Тогда

$$-\pi+\frac{\pi}{4}+2\pi n<\frac{x}{4}-3<-\frac{\pi}{4}+2\pi n,$$

$$-\frac{3\pi}{4}+2\pi n<\frac{x}{4}-3<-\frac{\pi}{4}+2\pi n,$$

$$-\frac{3\pi}{4}+3+2\pi n<\frac{x}{4}<-\frac{\pi}{4}+3+2\pi n,$$

$$-3\pi+12+8\pi n<x<-\pi+12+8\pi n,\quad n\in\mathbb Z.$$

Ответ

1) $$-\pi-6+6\pi n\le x\le \pi-6+6\pi n,\quad n\in\mathbb Z.$$

2) $$-3\pi+12+8\pi n<x<-\pi+12+8\pi n,\quad n\in\mathbb Z.$$