Упр.370 Повторение ГДЗ Мерзляк 11 класс Базовый уровень (Алгебра)

1) sin(3x) > v2/2; 3) sin(x+?/4)?v3/2; 5) tg(x/4+?/3)?v3/3;
2) cos(x/2)?1/2; 4) cos(2x-?/6)?-1/2; 6) ctg(2x/3-?/5)?-1.

1. $$\sin 3x>\frac{\sqrt2}{2}$$
   
   Для синуса:
   $$\frac{\pi}{4}+2\pi n<3x<\frac{3\pi}{4}+2\pi n$$ $$\frac{\pi}{12}+\frac{2\pi n}{3}
2. $$\cos\frac{x}{2}\le \frac12$$
   
   Для косинуса:
   $$\frac{\pi}{3}+2\pi n\le \frac{x}{2}\le \frac{5\pi}{3}+2\pi n$$
   $$\frac{2\pi}{3}+4\pi n\le x\le \frac{10\pi}{3}+4\pi n,\quad n\in\mathbb Z.$$
3. $$\sin\left(x+\frac{\pi}{4}\right)\le \frac{\sqrt3}{2}$$
   
   Для синуса:
   $$-\frac{\pi}{3}+2\pi n\le x+\frac{\pi}{4}\le \frac{\pi}{3}+2\pi n$$
   $$-\frac{7\pi}{12}+2\pi n\le x\le \frac{\pi}{12}+2\pi n,\quad n\in\mathbb Z.$$
4. $$\cos\left(2x-\frac{\pi}{6}\right)\ge -\frac12$$
   
   Для косинуса:
   $$2\pi n-\frac{2\pi}{3}\le 2x-\frac{\pi}{6}\le \frac{2\pi}{3}+2\pi n$$
   $$-\frac{\pi}{2}+\pi n\le 2x\le \frac{5\pi}{6}+\pi n$$
   $$-\frac{\pi}{4}+\frac{\pi n}{2}\le x\le \frac{5\pi}{12}+\frac{\pi n}{2},\quad n\in\mathbb Z.$$
5. $$\tg\left(\frac{x}{4}+\frac{\pi}{3}\right)\le \frac{\sqrt3}{3}$$
   
   Так как $$\frac{\sqrt3}{3}=\tg\frac{\pi}{6},$$ то
   $$-\frac{\pi}{2}+\pi n<\frac{x}{4}+\frac{\pi}{3}\le \frac{\pi}{6}+\pi n$$ $$-\frac{5\pi}{6}+\pi n<\frac{x}{4}\le -\frac{\pi}{6}+\pi n$$ $$-\frac{10\pi}{3}+4\pi n
6. $$\ctg\left(\frac{2x}{3}-\frac{\pi}{5}\right)\ge -1$$
   
   Для котангенса:
   $$\pi n<\frac{2x}{3}-\frac{\pi}{5}\le \frac{3\pi}{4}+\pi n$$ $$\frac{\pi}{5}+\pi n<\frac{2x}{3}\le \frac{19\pi}{20}+\pi n$$ $$\frac{3\pi}{10}+\frac{3\pi n}{2}

Ответ

1) $$x\in\left(\frac{\pi}{12}+\frac{2\pi n}{3},\frac{\pi}{4}+\frac{2\pi n}{3}\right),\ n\in\mathbb Z;$$

2) $$x\in\left[\frac{2\pi}{3}+4\pi n,\frac{10\pi}{3}+4\pi n\right],\ n\in\mathbb Z;$$

3) $$x\in\left[-\frac{7\pi}{12}+2\pi n,\frac{\pi}{12}+2\pi n\right],\ n\in\mathbb Z;$$

4) $$x\in\left[-\frac{\pi}{4}+\frac{\pi n}{2},\frac{5\pi}{12}+\frac{\pi n}{2}\right],\ n\in\mathbb Z;$$

5) $$x\in\left(-\frac{10\pi}{3}+4\pi n,-\frac{2\pi}{3}+4\pi n\right],\ n\in\mathbb Z;$$

6) $$x\in\left(\frac{3\pi}{10}+\frac{3\pi n}{2},\frac{57\pi}{40}+\frac{3\pi n}{2}\right],\ n\in\mathbb Z.$$