1. Решите уравнения: a) sin(3x + π/6)= -√3/2 б) cos(x/5 + π/3)= √2/2 в) lg(4x + π/3)= -√3 г) sin x/3 = 6/5 д) cos(5x + π/4)= 0 e) tg(x/4 + π/6)= 1 ж) 2tg²x+3tgx-2=0 з) cos²x-2 cos x sin x = 0

1. Решите уравнения:

a) $$sin(3x + \frac{\pi}{6}) = -\frac{\sqrt{3}}{2}$$

$$3x + \frac{\pi}{6} = -\frac{\pi}{3} + 2\pi n, n \in Z$$ или $$3x + \frac{\pi}{6} = \frac{4\pi}{3} + 2\pi n, n \in Z$$

$$3x = -\frac{\pi}{2} + 2\pi n, n \in Z$$ или $$3x = \frac{7\pi}{6} + 2\pi n, n \in Z$$

$$x = -\frac{\pi}{6} + \frac{2\pi n}{3}, n \in Z$$ или $$x = \frac{7\pi}{18} + \frac{2\pi n}{3}, n \in Z$$

б) $$cos(\frac{x}{5} + \frac{\pi}{3}) = \frac{\sqrt{2}}{2}$$

$$\frac{x}{5} + \frac{\pi}{3} = \frac{\pi}{4} + 2\pi n, n \in Z$$ или $$\frac{x}{5} + \frac{\pi}{3} = -\frac{\pi}{4} + 2\pi n, n \in Z$$

$$\frac{x}{5} = -\frac{\pi}{12} + 2\pi n, n \in Z$$ или $$\frac{x}{5} = -\frac{7\pi}{12} + 2\pi n, n \in Z$$

$$x = -\frac{5\pi}{12} + 10\pi n, n \in Z$$ или $$x = -\frac{35\pi}{12} + 10\pi n, n \in Z$$

в) $$tg(4x + \frac{\pi}{3}) = -\sqrt{3}$$

$$4x + \frac{\pi}{3} = -\frac{\pi}{3} + \pi n, n \in Z$$

$$4x = -\frac{2\pi}{3} + \pi n, n \in Z$$

$$x = -\frac{\pi}{6} + \frac{\pi n}{4}, n \in Z$$

г) $$sin(\frac{x}{3}) = \frac{6}{5}$$

Так как $$|\frac{6}{5}| > 1$$, то уравнение не имеет решений.

д) $$cos(5x + \frac{\pi}{4}) = 0$$

$$5x + \frac{\pi}{4} = \frac{\pi}{2} + \pi n, n \in Z$$

$$5x = \frac{\pi}{4} + \pi n, n \in Z$$

$$x = \frac{\pi}{20} + \frac{\pi n}{5}, n \in Z$$

e) $$tg(\frac{x}{4} + \frac{\pi}{6}) = 1$$

$$\frac{x}{4} + \frac{\pi}{6} = \frac{\pi}{4} + \pi n, n \in Z$$

$$\frac{x}{4} = \frac{\pi}{12} + \pi n, n \in Z$$

$$x = \frac{\pi}{3} + 4\pi n, n \in Z$$

ж) $$2tg^2(x) + 3tg(x) - 2 = 0$$

Пусть $$t = tg(x)$$, тогда:

$$2t^2 + 3t - 2 = 0$$

$$D = 9 + 16 = 25$$

$$t_1 = \frac{-3 + 5}{4} = \frac{1}{2}$$

$$t_2 = \frac{-3 - 5}{4} = -2$$

$$tg(x) = \frac{1}{2}$$ или $$tg(x) = -2$$

$$x = arctg(\frac{1}{2}) + \pi n, n \in Z$$ или $$x = arctg(-2) + \pi n, n \in Z$$

$$x = arctg(\frac{1}{2}) + \pi n, n \in Z$$ или $$x = -arctg(2) + \pi n, n \in Z$$

з) $$cos^2(x) - 2cos(x)sin(x) = 0$$

$$cos(x)(cos(x) - 2sin(x)) = 0$$

$$cos(x) = 0$$ или $$cos(x) = 2sin(x)$$

$$x = \frac{\pi}{2} + \pi n, n \in Z$$ или $$ctg(x) = 2$$

$$x = \frac{\pi}{2} + \pi n, n \in Z$$ или $$x = arcctg(2) + \pi n, n \in Z$$

Ответ:

a) $$x = -\frac{\pi}{6} + \frac{2\pi n}{3}, n \in Z$$ или $$x = \frac{7\pi}{18} + \frac{2\pi n}{3}, n \in Z$$

б) $$x = -\frac{5\pi}{12} + 10\pi n, n \in Z$$ или $$x = -\frac{35\pi}{12} + 10\pi n, n \in Z$$

в) $$x = -\frac{\pi}{6} + \frac{\pi n}{4}, n \in Z$$

г) Решений нет

д) $$x = \frac{\pi}{20} + \frac{\pi n}{5}, n \in Z$$

e) $$x = \frac{\pi}{3} + 4\pi n, n \in Z$$

ж) $$x = arctg(\frac{1}{2}) + \pi n, n \in Z$$ или $$x = -arctg(2) + \pi n, n \in Z$$

з) $$x = \frac{\pi}{2} + \pi n, n \in Z$$ или $$x = arcctg(2) + \pi n, n \in Z$$