Упр.725 ГДЗ Алимов 10-11 класс (Алгебра)

1) sinx > 1/2;
2) sinx < =корень 2/2; 3) sinx > =-1/2;
4) sinx < - корень 3/2.

1. $$\sin x>\frac12$$

   На отрезке $$[0;3\pi]$$ имеем:

   $$\sin x>\frac12 \iff x\in\left(\frac{\pi}{6};\frac{5\pi}{6}\right)\cup\left(\frac{13\pi}{6};\frac{17\pi}{6}\right).$$

2. $$\sin x\le \frac{\sqrt2}{2}$$

   На отрезке $$[0;3\pi]$$ получаем:

   $$\sin x\le \frac{\sqrt2}{2} \iff x\in\left[0;\frac{\pi}{4}\right]\cup\left[\frac{3\pi}{4};\frac{9\pi}{4}\right]\cup\left[\frac{11\pi}{4};3\pi\right].$$

3. $$\sin x\ge -\frac12$$

   На отрезке $$[0;3\pi]$$:

   $$\sin x\ge -\frac12 \iff x\in\left[0;\frac{7\pi}{6}\right]\cup\left[\frac{11\pi}{6};3\pi\right].$$

4. $$\sin x<-\frac{\sqrt3}{2}$$

   На отрезке $$[0;3\pi]$$:

   $$\sin x<-\frac{\sqrt3}{2} \iff x\in\left(\frac{4\pi}{3};\frac{5\pi}{3}\right).$$

Ответ

1) $$x\in\left(\frac{\pi}{6};\frac{5\pi}{6}\right)\cup\left(\frac{13\pi}{6};\frac{17\pi}{6}\right)$$
2) $$x\in\left[0;\frac{\pi}{4}\right]\cup\left[\frac{3\pi}{4};\frac{9\pi}{4}\right]\cup\left[\frac{11\pi}{4};3\pi\right]$$
3) $$x\in\left[0;\frac{7\pi}{6}\right]\cup\left[\frac{11\pi}{6};3\pi\right]$$
4) $$x\in\left(\frac{4\pi}{3};\frac{5\pi}{3}\right)$$