2 вариант 1. cosx /3 = √3 /2 2. 2sin2x-1=0 3. tgx /3 -√3 =0 4. 3 ctgx /4 +√3 =0 5. sin (π/2 - α)=0 6. cos (3π/2 + α)=-1 7. tg (π + α) + √3 = 0 8. ctg (2π – α) – √3/3 = 0 9. sin (π + α) - 1 = 0 10. cos x/4 = -√2/2 11. tgx /3 +1=0 12. ctg2x +√3 =0 13. 2sin5x- √3 =0 14. sin5x=-1 15. tg (π/2 – 3x)= √2/2

Давай разберем эти уравнения по порядку! Начнем с первого: 1. \[\cos{\frac{x}{3}} = \frac{\sqrt{3}}{2}\] \(\frac{x}{3} = \pm \frac{\pi}{6} + 2\pi k, k \in \mathbb{Z}\) \(x = \pm \frac{\pi}{2} + 6\pi k, k \in \mathbb{Z}\) 2. \[2\sin{2x} - 1 = 0\] \[\sin{2x} = \frac{1}{2}\] \[2x = (-1)^n \frac{\pi}{6} + \pi n, n \in \mathbb{Z}\] \[x = (-1)^n \frac{\pi}{12} + \frac{\pi n}{2}, n \in \mathbb{Z}\] 3. \[\tan{\frac{x}{3}} - \sqrt{3} = 0\] \[\tan{\frac{x}{3}} = \sqrt{3}\] \[\frac{x}{3} = \frac{\pi}{3} + \pi k, k \in \mathbb{Z}\] \[x = \pi + 3\pi k, k \in \mathbb{Z}\] 4. \[3\cot{\frac{x}{4}} + \sqrt{3} = 0\] \[\cot{\frac{x}{4}} = -\frac{\sqrt{3}}{3}\] \[\frac{x}{4} = -\frac{\pi}{3} + \pi k, k \in \mathbb{Z}\] \[x = -\frac{4\pi}{3} + 4\pi k, k \in \mathbb{Z}\] 5. \[\sin{(\frac{\pi}{2} - \alpha)} = 0\] \[\frac{\pi}{2} - \alpha = \pi k, k \in \mathbb{Z}\] \[\alpha = \frac{\pi}{2} - \pi k, k \in \mathbb{Z}\] 6. \[\cos{(\frac{3\pi}{2} + \alpha)} = -1\] \[\frac{3\pi}{2} + \alpha = \pi + 2\pi k, k \in \mathbb{Z}\] \[\alpha = -\frac{\pi}{2} + 2\pi k, k \in \mathbb{Z}\] 7. \[\tan{(\pi + \alpha)} + \sqrt{3} = 0\] \[\tan{\alpha} = -\sqrt{3}\] \[\alpha = -\frac{\pi}{3} + \pi k, k \in \mathbb{Z}\] 8. \[\cot{(2\pi - \alpha)} - \frac{\sqrt{3}}{3} = 0\] \[\cot{(-\alpha)} = \frac{\sqrt{3}}{3}\] \[-\alpha = \frac{\pi}{3} + \pi k, k \in \mathbb{Z}\] \[\alpha = -\frac{\pi}{3} - \pi k, k \in \mathbb{Z}\] 9. \[\sin{(\pi + \alpha)} - 1 = 0\] \[\sin{(\pi + \alpha)} = 1\] \[\pi + \alpha = \frac{\pi}{2} + 2\pi k, k \in \mathbb{Z}\] \[\alpha = -\frac{\pi}{2} + 2\pi k, k \in \mathbb{Z}\] 10. \[\cos{\frac{x}{4}} = -\frac{\sqrt{2}}{2}\] \[\frac{x}{4} = \pm \frac{3\pi}{4} + 2\pi k, k \in \mathbb{Z}\] \[x = \pm 3\pi + 8\pi k, k \in \mathbb{Z}\] 11. \[ \tan{\frac{x}{3}} + 1 = 0\] \[\tan{\frac{x}{3}} = -1\] \[\frac{x}{3} = -\frac{\pi}{4} + \pi k, k \in \mathbb{Z}\] \[x = -\frac{3\pi}{4} + 3\pi k, k \in \mathbb{Z}\] 12. \[ \cot{2x} + \sqrt{3} = 0\] \[\cot{2x} = -\sqrt{3}\] \[2x = \frac{5\pi}{6} + \pi k, k \in \mathbb{Z}\] \[x = \frac{5\pi}{12} + \frac{\pi k}{2}, k \in \mathbb{Z}\] 13. \[ 2\sin{5x} - \sqrt{3} = 0\] \[\sin{5x} = \frac{\sqrt{3}}{2}\] \[5x = (-1)^n \frac{\pi}{3} + \pi n, n \in \mathbb{Z}\] \[x = (-1)^n \frac{\pi}{15} + \frac{\pi n}{5}, n \in \mathbb{Z}\] 14. \[ \sin{5x} = -1\] \[5x = -\frac{\pi}{2} + 2\pi k, k \in \mathbb{Z}\] \[x = -\frac{\pi}{10} + \frac{2\pi k}{5}, k \in \mathbb{Z}\] 15. \[ \tan{(\frac{\pi}{2} - 3x)} = \frac{\sqrt{2}}{2}\] \[\cot{3x} = \frac{\sqrt{2}}{2}\] \[3x = \arctan{\sqrt{2}} + \pi k, k \in \mathbb{Z}\] \[x = \frac{1}{3} \arctan{\sqrt{2}} + \frac{\pi k}{3}, k \in \mathbb{Z}\]

Ответ: Решения уравнений выше.

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