1) sin 3x = 0; 2) cos 4x = 1; 3) tg \frac{x}{2} = -1; 4) 2sin \frac{x}{3} = 1; 5) cos 5x + 1 = 0; 6) 3tg (x + \frac{\pi}{4}) = \sqrt{3}; 7) ctg \frac{2x}{3} = 0; 8) 1-sin \frac{x}{3} = 0; 9) \sqrt{3} - cos (2x - \frac{\pi}{3}) = 0; 10) sin (2x - \frac{\pi}{8}) = \frac{1}{3}; 11) sin^2 x = 1; 12) sin (x - \frac{\pi}{4}) = 0; 13) cos \frac{x}{2} = 0; 14) sin (x - \frac{\pi}{3}) = 1; 15) sin 4x = -1; 16) tg (\frac{\pi}{3} - x) = 0; 17) sin 2x = \frac{1}{\sqrt{2}}; 18) cos x = -\frac{\sqrt{2}}{2}; 19) cos 2x = -\frac{1}{2}; 20) cos(x - \pi) = -1; 21) sin(x - 2\pi) + cos (x - \frac{\pi}{2}) = 1; 22) \frac{1}{2}sin 2x = -\frac{\sqrt{3}}{4}; 23) 3 sin 5x = 0; 24) 1 - cos (\frac{x + \pi}{3}) = \frac{1}{2}; 25) cos 2x = 0,1; 26) tg 3x = 4; 27) tg (\frac{\pi}{3} - x) = -1; 28) cos x \cdot cos 2x = 0; 29) sin (x + \frac{\pi}{4}) \cdot sin (x - \frac{\pi}{4}) = 0; 30) 2 cos^2 x =

Разберем каждое уравнение по порядку:

1. $$sin(3x) = 0$$
   $$3x = \pi n, n \in Z$$
   $$x = \frac{\pi n}{3}, n \in Z$$
   Ответ: $$x = \frac{\pi n}{3}, n \in Z$$
2. $$cos(4x) = 1$$
   $$4x = 2\pi n, n \in Z$$
   $$x = \frac{\pi n}{2}, n \in Z$$
   Ответ: $$x = \frac{\pi n}{2}, n \in Z$$
3. $$tg(\frac{x}{2}) = -1$$
   $$\frac{x}{2} = -\frac{\pi}{4} + \pi n, n \in Z$$
   $$x = -\frac{\pi}{2} + 2\pi n, n \in Z$$
   Ответ: $$x = -\frac{\pi}{2} + 2\pi n, n \in Z$$
4. $$2sin(\frac{x}{3}) = 1$$
   $$sin(\frac{x}{3}) = \frac{1}{2}$$
   $$\frac{x}{3} = (-1)^n \frac{\pi}{6} + \pi n, n \in Z$$
   $$x = (-1)^n \frac{\pi}{2} + 3\pi n, n \in Z$$
   Ответ: $$x = (-1)^n \frac{\pi}{2} + 3\pi n, n \in Z$$
5. $$cos(5x) + 1 = 0$$
   $$cos(5x) = -1$$
   $$5x = \pi + 2\pi n, n \in Z$$
   $$x = \frac{\pi}{5} + \frac{2\pi n}{5}, n \in Z$$
   Ответ: $$x = \frac{\pi}{5} + \frac{2\pi n}{5}, n \in Z$$
6. $$3tg(x + \frac{\pi}{4}) = \sqrt{3}$$
   $$tg(x + \frac{\pi}{4}) = \frac{\sqrt{3}}{3}$$
   $$x + \frac{\pi}{4} = \frac{\pi}{6} + \pi n, n \in Z$$
   $$x = \frac{\pi}{6} - \frac{\pi}{4} + \pi n, n \in Z$$
   $$x = -\frac{\pi}{12} + \pi n, n \in Z$$
   Ответ: $$x = -\frac{\pi}{12} + \pi n, n \in Z$$
7. $$ctg(\frac{2x}{3}) = 0$$
   $$\frac{2x}{3} = \frac{\pi}{2} + \pi n, n \in Z$$
   $$2x = \frac{3\pi}{2} + 3\pi n, n \in Z$$
   $$x = \frac{3\pi}{4} + \frac{3\pi n}{2}, n \in Z$$
   Ответ: $$x = \frac{3\pi}{4} + \frac{3\pi n}{2}, n \in Z$$
8. $$1 - sin(\frac{x}{3}) = 0$$
   $$sin(\frac{x}{3}) = 1$$
   $$\frac{x}{3} = \frac{\pi}{2} + 2\pi n, n \in Z$$
   $$x = \frac{3\pi}{2} + 6\pi n, n \in Z$$
   Ответ: $$x = \frac{3\pi}{2} + 6\pi n, n \in Z$$
9. $$\sqrt{3} - cos(2x - \frac{\pi}{3}) = 0$$
   $$cos(2x - \frac{\pi}{3}) = \sqrt{3}$$
   Так как $$\sqrt{3} > 1$$, то уравнение не имеет решений.
   Ответ: нет решений
10. $$sin(2x - \frac{\pi}{8}) = \frac{1}{3}$$
    $$2x - \frac{\pi}{8} = (-1)^n arcsin(\frac{1}{3}) + \pi n, n \in Z$$
    $$2x = \frac{\pi}{8} + (-1)^n arcsin(\frac{1}{3}) + \pi n, n \in Z$$
    $$x = \frac{\pi}{16} + (-1)^n \frac{1}{2} arcsin(\frac{1}{3}) + \frac{\pi n}{2}, n \in Z$$
    Ответ: $$x = \frac{\pi}{16} + (-1)^n \frac{1}{2} arcsin(\frac{1}{3}) + \frac{\pi n}{2}, n \in Z$$
11. $$sin^2 x = 1$$
    $$sin x = \pm 1$$
    $$x = \frac{\pi}{2} + \pi n, n \in Z$$
    Ответ: $$x = \frac{\pi}{2} + \pi n, n \in Z$$
12. $$sin(x - \frac{\pi}{4}) = 0$$
    $$x - \frac{\pi}{4} = \pi n, n \in Z$$
    $$x = \frac{\pi}{4} + \pi n, n \in Z$$
    Ответ: $$x = \frac{\pi}{4} + \pi n, n \in Z$$
13. $$cos(\frac{x}{2}) = 0$$
    $$\frac{x}{2} = \frac{\pi}{2} + \pi n, n \in Z$$
    $$x = \pi + 2\pi n, n \in Z$$
    Ответ: $$x = \pi + 2\pi n, n \in Z$$
14. $$sin(x - \frac{\pi}{3}) = 1$$
    $$x - \frac{\pi}{3} = \frac{\pi}{2} + 2\pi n, n \in Z$$
    $$x = \frac{5\pi}{6} + 2\pi n, n \in Z$$
    Ответ: $$x = \frac{5\pi}{6} + 2\pi n, n \in Z$$
15. $$sin(4x) = -1$$
    $$4x = -\frac{\pi}{2} + 2\pi n, n \in Z$$
    $$x = -\frac{\pi}{8} + \frac{\pi n}{2}, n \in Z$$
    Ответ: $$x = -\frac{\pi}{8} + \frac{\pi n}{2}, n \in Z$$
16. $$tg(\frac{\pi}{3} - x) = 0$$
    $$\frac{\pi}{3} - x = \pi n, n \in Z$$
    $$x = \frac{\pi}{3} - \pi n, n \in Z$$
    Ответ: $$x = \frac{\pi}{3} - \pi n, n \in Z$$
17. $$sin(2x) = \frac{1}{\sqrt{2}}$$
    $$2x = (-1)^n \frac{\pi}{4} + \pi n, n \in Z$$
    $$x = (-1)^n \frac{\pi}{8} + \frac{\pi n}{2}, n \in Z$$
    Ответ: $$x = (-1)^n \frac{\pi}{8} + \frac{\pi n}{2}, n \in Z$$
18. $$cos(x) = -\frac{\sqrt{2}}{2}$$
    $$x = \pm \frac{3\pi}{4} + 2\pi n, n \in Z$$
    Ответ: $$x = \pm \frac{3\pi}{4} + 2\pi n, n \in Z$$
19. $$cos(2x) = -\frac{1}{2}$$
    $$2x = \pm \frac{2\pi}{3} + 2\pi n, n \in Z$$
    $$x = \pm \frac{\pi}{3} + \pi n, n \in Z$$
    Ответ: $$x = \pm \frac{\pi}{3} + \pi n, n \in Z$$
20. $$cos(x - \pi) = -1$$
    $$x - \pi = \pi + 2\pi n, n \in Z$$
    $$x = 2\pi + 2\pi n, n \in Z$$
    $$x = 2\pi(1 + n), n \in Z$$
    Ответ: $$x = 2\pi(1 + n), n \in Z$$
21. $$sin(x - 2\pi) + cos(x - \frac{\pi}{2}) = 1$$
    $$sin(x) + sin(x) = 1$$
    $$2sin(x) = 1$$
    $$sin(x) = \frac{1}{2}$$
    $$x = (-1)^n \frac{\pi}{6} + \pi n, n \in Z$$
    Ответ: $$x = (-1)^n \frac{\pi}{6} + \pi n, n \in Z$$
22. $$\frac{1}{2}sin(2x) = -\frac{\sqrt{3}}{4}$$
    $$sin(2x) = -\frac{\sqrt{3}}{2}$$
    $$2x = (-1)^n(-\frac{\pi}{3}) + \pi n, n \in Z$$
    $$x = (-1)^n(-\frac{\pi}{6}) + \frac{\pi n}{2}, n \in Z$$
    Ответ: $$x = (-1)^n(-\frac{\pi}{6}) + \frac{\pi n}{2}, n \in Z$$
23. $$3sin(5x) = 0$$
    $$sin(5x) = 0$$
    $$5x = \pi n, n \in Z$$
    $$x = \frac{\pi n}{5}, n \in Z$$
    Ответ: $$x = \frac{\pi n}{5}, n \in Z$$
24. $$1 - cos(\frac{x + \pi}{3}) = \frac{1}{2}$$
    $$cos(\frac{x + \pi}{3}) = \frac{1}{2}$$
    $$\frac{x + \pi}{3} = \pm \frac{\pi}{3} + 2\pi n, n \in Z$$
    $$x + \pi = \pm \pi + 6\pi n, n \in Z$$
    $$x = -\pi \pm \pi + 6\pi n, n \in Z$$
    
    1. $$x = -\pi + \pi + 6\pi n, n \in Z$$
       $$x = 6\pi n, n \in Z$$
    2. $$x = -\pi - \pi + 6\pi n, n \in Z$$
       $$x = -2\pi + 6\pi n, n \in Z$$
    Ответ: $$x = 6\pi n, n \in Z$$; $$x = -2\pi + 6\pi n, n \in Z$$
25. $$cos(2x) = 0.1$$
    $$2x = \pm arccos(0.1) + 2 \pi n, n \in Z$$
    $$x = \pm \frac{arccos(0.1)}{2} + \pi n, n \in Z$$
    Ответ: $$x = \pm \frac{arccos(0.1)}{2} + \pi n, n \in Z$$
26. $$tg(3x) = 4$$
    $$3x = arctg(4) + \pi n, n \in Z$$
    $$x = \frac{arctg(4)}{3} + \frac{\pi n}{3}, n \in Z$$
    Ответ: $$x = \frac{arctg(4)}{3} + \frac{\pi n}{3}, n \in Z$$
27. $$tg(\frac{\pi}{3} - x) = -1$$
    $$\frac{\pi}{3} - x = -\frac{\pi}{4} + \pi n, n \in Z$$
    $$x = \frac{\pi}{3} + \frac{\pi}{4} - \pi n, n \in Z$$
    $$x = \frac{7\pi}{12} - \pi n, n \in Z$$
    Ответ: $$x = \frac{7\pi}{12} - \pi n, n \in Z$$
28. $$cos(x) \cdot cos(2x) = 0$$
    $$cos(x) = 0$$ или $$cos(2x) = 0$$
    
    1. $$cos(x) = 0$$
       $$x = \frac{\pi}{2} + \pi n, n \in Z$$
    2. $$cos(2x) = 0$$
       $$2x = \frac{\pi}{2} + \pi n, n \in Z$$
       $$x = \frac{\pi}{4} + \frac{\pi n}{2}, n \in Z$$
    Ответ: $$x = \frac{\pi}{2} + \pi n, n \in Z$$; $$x = \frac{\pi}{4} + \frac{\pi n}{2}, n \in Z$$
29. $$sin(x + \frac{\pi}{4}) \cdot sin(x - \frac{\pi}{4}) = 0$$
    $$sin(x + \frac{\pi}{4}) = 0$$ или $$sin(x - \frac{\pi}{4}) = 0$$
    
    1. $$sin(x + \frac{\pi}{4}) = 0$$
       $$x + \frac{\pi}{4} = \pi n, n \in Z$$
       $$x = -\frac{\pi}{4} + \pi n, n \in Z$$
    2. $$sin(x - \frac{\pi}{4}) = 0$$
       $$x - \frac{\pi}{4} = \pi n, n \in Z$$
       $$x = \frac{\pi}{4} + \pi n, n \in Z$$
    Ответ: $$x = -\frac{\pi}{4} + \pi n, n \in Z$$; $$x = \frac{\pi}{4} + \pi n, n \in Z$$
30. $$2cos^2 x =$$ Не хватает правой части уравнения.