Упр.35.3 ГДЗ Мерзляк 10 класс Углубленный уровень (Алгебра)

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

1. $$\tg\left(x-\frac{\pi}{3}\right)\le \sqrt{3}$$

   Так как функция $$\tg x$$ возрастает на каждом промежутке $$\left(-\frac{\pi}{2}+\pi n;\frac{\pi}{2}+\pi n\right)$$, получаем:

   $$-\frac{\pi}{2}+\pi n<x-\frac{\pi}{3}\le \frac{\pi}{3}+\pi n$$
   $$-\frac{\pi}{6}+\pi n<x\le \frac{2\pi}{3}+\pi n,\quad n\in\mathbb Z.$$

2. $$\cos\left(2x-\frac{\pi}{6}\right)>-\frac12$$

   Для косинуса:

   $$-\frac{2\pi}{3}+2\pi n<2x-\frac{\pi}{6}<\frac{2\pi}{3}+2\pi n$$
   $$-\frac{\pi}{2}+2\pi n<2x<\frac{5\pi}{6}+2\pi n$$
   $$-\frac{\pi}{4}+\pi n<x<\frac{5\pi}{12}+\pi n,\quad n\in\mathbb Z.$$

3. $$\ctg\left(\frac{\pi}{4}-x\right)>\frac{1}{\sqrt{3}}$$

   Преобразуем неравенство:

   $$\ctg\left(x-\frac{\pi}{4}\right)<-\frac{\sqrt{3}}{3}$$

   Тогда

   $$\frac{2\pi}{3}+\pi n<x-\frac{\pi}{4}<\pi+\pi n$$
   $$\frac{11\pi}{12}+\pi n<x<\frac{5\pi}{4}+\pi n,\quad n\in\mathbb Z.$$

4. $$2\sin\left(\frac{\pi}{6}-3x\right)\le \sqrt{3}$$
   $$\sin\left(3x-\frac{\pi}{6}\right)\ge -\frac{\sqrt{3}}{2}$$

   Для синуса:

   $$-\frac{\pi}{3}+2\pi n\le 3x-\frac{\pi}{6}\le \frac{4\pi}{3}+2\pi n$$
   $$-\frac{\pi}{6}+2\pi n\le 3x\le \frac{3\pi}{2}+2\pi n$$
   $$-\frac{\pi}{18}+\frac{2\pi n}{3}\le x\le \frac{\pi}{2}+\frac{2\pi n}{3},\quad n\in\mathbb Z.$$

5. $$\cos\left(\frac{x}{2}+\frac{\pi}{4}\right)\le -\frac{\sqrt{2}}{2}$$

   Тогда

   $$\frac{3\pi}{4}+2\pi n\le \frac{x}{2}+\frac{\pi}{4}\le \frac{5\pi}{4}+2\pi n$$
   $$\frac{\pi}{2}+2\pi n\le \frac{x}{2}\le \pi+2\pi n$$
   $$\pi+4\pi n\le x\le 2\pi+4\pi n,\quad n\in\mathbb Z.$$

6. $$\sin(1-2x)<-\frac{\sqrt{2}}{2}$$
   $$\sin(2x-1)>\frac{\sqrt{2}}{2}$$

   Для синуса:

   $$\frac{\pi}{4}+2\pi n<2x-1<\frac{3\pi}{4}+2\pi n$$
   $$\frac{\pi}{4}+1+2\pi n<2x<\frac{3\pi}{4}+1+2\pi n$$
   $$\frac{\pi}{8}+\frac12+\pi n<x<\frac{3\pi}{8}+\frac12+\pi n,\quad n\in\mathbb Z.$$

Ответ

1) $$x\in\left(-\frac{\pi}{6}+\pi n;\frac{2\pi}{3}+\pi n\right],\ n\in\mathbb Z;$$

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

3) $$x\in\left(\frac{11\pi}{12}+\pi n;\frac{5\pi}{4}+\pi n\right),\ n\in\mathbb Z;$$

4) $$x\in\left[-\frac{\pi}{18}+\frac{2\pi n}{3};\frac{\pi}{2}+\frac{2\pi n}{3}\right],\ n\in\mathbb Z;$$

5) $$x\in\left[\pi+4\pi n;2\pi+4\pi n\right],\ n\in\mathbb Z;$$

6) $$x\in\left(\frac{\pi}{8}+\frac12+\pi n;\frac{3\pi}{8}+\frac12+\pi n\right),\ n\in\mathbb Z.$$