Упр.11.30 ГДЗ Мордковича 10 класс профильный уровень (Алгебра)

а) $$t=\pm \frac{\pi}{3}+\pi n,\qquad t=\frac{\pi n}{3}.$$

1) Если $$n=3k\pm 1,$$ то

$$t_2=\frac{\pi(3k\pm 1)}{3}=\pi k\pm \frac{\pi}{3}=\pm \frac{\pi}{3}+\pi k=t_1.$$

2) Все значения чисел:

$$\begin{aligned} n=6k &\Rightarrow t_2=\frac{\pi(6k)}{3}=2\pi k=0,\\ n=6k+1 &\Rightarrow t_2=\frac{\pi(6k+1)}{3}=2\pi k+\frac{\pi}{3}=\frac{\pi}{3},\\ n=6k+2 &\Rightarrow t_2=\frac{\pi(6k+2)}{3}=2\pi k+\frac{2\pi}{3}=\frac{2\pi}{3},\\ n=6k+3 &\Rightarrow t_2=\frac{\pi(6k+3)}{3}=2\pi k+\pi=\pi,\\ n=6k+4 &\Rightarrow t_2=\frac{\pi(6k+4)}{3}=2\pi k+\frac{4\pi}{3}=\frac{4\pi}{3},\\ n=6k+5 &\Rightarrow t_2=\frac{\pi(6k+5)}{3}=2\pi k+\frac{5\pi}{3}=\frac{5\pi}{3}. \end{aligned}$$

Ответ: $$M_1(0),\ M_2\!\left(\frac{\pi}{3}\right),\ M_3\!\left(\frac{2\pi}{3}\right),\ M_4(\pi),\ M_5\!\left(\frac{4\pi}{3}\right),\ M_6\!\left(\frac{5\pi}{3}\right),\quad t=\frac{\pi n}{3}.$$

б) $$t=(-1)^n\cdot \frac{\pi}{4}+\pi n,\qquad t=(-1)^{n+1}\cdot \frac{\pi}{4}+\pi n.$$

1) Все значения чисел:

$$\begin{aligned} n=2k &\Rightarrow t_1=(-1)^{2k}\cdot \frac{\pi}{4}+\pi(2k)=\frac{\pi}{4}+2\pi k=\frac{\pi}{4},\\ &\Rightarrow t_2=(-1)^{2k+1}\cdot \frac{\pi}{4}+\pi(2k)=-\frac{\pi}{4}+2\pi k=\frac{7\pi}{4};\\ n=2k+1 &\Rightarrow t_1=(-1)^{2k+1}\cdot \frac{\pi}{4}+\pi(2k+1)=-\frac{\pi}{4}+2\pi k+\pi=\frac{3\pi}{4},\\ &\Rightarrow t_2=(-1)^{2k+2}\cdot \frac{\pi}{4}+\pi(2k+1)=\frac{\pi}{4}+2\pi k+\pi=\frac{5\pi}{4}. \end{aligned}$$

2) Общая формула чисел:

$$t=\pm \frac{\pi}{4}+\pi n=\frac{\pi}{4}+\frac{\pi(2n)}{2}=\frac{\pi}{4}+\frac{\pi(2n-1)}{2}, \qquad t=\frac{\pi}{4}+\frac{\pi(2n)}{2}.$$

Следовательно,

$$t=\frac{\pi}{4}+\frac{\pi k}{2}.$$

Ответ: $$M_1\!\left(\frac{\pi}{4}\right),\ M_2\!\left(\frac{3\pi}{4}\right),\ M_3\!\left(\frac{5\pi}{4}\right),\quad t=\frac{\pi}{4}+\frac{\pi k}{2}.$$

в) $$t=\pm \frac{2\pi}{3}+2\pi n,\qquad t=2\pi n.$$

1) Все значения чисел:

$$\begin{aligned} t_{11}&=-\frac{2\pi}{3}+2\pi n=2\pi-\frac{2\pi}{3}=\frac{4\pi}{3},\\ t_{12}&=\frac{2\pi}{3}+2\pi n=\frac{2\pi}{3},\\ t_2&=2\pi n=0. \end{aligned}$$

2) Общая формула чисел:

$$t=\pm \frac{2\pi}{3}+2\pi n=\frac{2\pi(3n\pm 1)}{3},\qquad t=2\pi n=\frac{2\pi(3n)}{3}.$$

Значит,

$$t=\frac{2\pi k}{3}.$$

Ответ: $$M_1(0),\ M_2\!\left(\frac{2\pi}{3}\right),\ M_3\!\left(\frac{4\pi}{3}\right),\quad t=\frac{2\pi k}{3}.$$

г) $$t=(-1)^n\cdot \frac{\pi}{6}+\pi n,\qquad t=(-1)^{n+1}\cdot \frac{\pi}{6}+\pi n.$$

1) Все значения чисел:

$$\begin{aligned} n=2k &\Rightarrow t_1=(-1)^{2k}\cdot \frac{\pi}{6}+\pi(2k)=\frac{\pi}{6}+2\pi k=\frac{\pi}{6},\\ &\Rightarrow t_2=(-1)^{2k+1}\cdot \frac{\pi}{6}+\pi(2k)=-\frac{\pi}{6}+2\pi k=\frac{11\pi}{6};\\ n=2k+1 &\Rightarrow t_1=(-1)^{2k+1}\cdot \frac{\pi}{6}+\pi(2k+1)=-\frac{\pi}{6}+2\pi k+\pi=\frac{5\pi}{6},\\ &\Rightarrow t_2=(-1)^{2k+2}\cdot \frac{\pi}{6}+\pi(2k+1)=\frac{\pi}{6}+2\pi k+\pi=\frac{7\pi}{6}. \end{aligned}$$

2) Общая формула чисел:

$$t=\pm \frac{\pi}{6}+\pi n.$$

Ответ: $$M_1\!\left(\frac{\pi}{6}\right),\ M_2\!\left(\frac{5\pi}{6}\right),\ M_3\!\left(\frac{7\pi}{6}\right),\ M_4\!\left(\frac{11\pi}{6}\right),\quad t=\pm \frac{\pi}{6}+\pi n.$$