Question

решите уравнения
1)sin^2x+√3sinxcosx=0
2)2cos^2x-cosx-1=0
​

Answer 1

Ответ:

1.

$\sin {}^{2} (x) + \sqrt{3} \sin(x) \cos(x) = 0 \\ \sin(x) (\sin(x) + \sqrt{3} \cos(x)) = 0 \\ \\ \sin(x) = 0 \\ x_1 = \pi \: n \\ \\ \sin(x) + \sqrt{3} \cos(x) = 0 \\ | \div \cos(x) \ne 0\\ tgx + \sqrt{3} = 0 \\ tgx = - \sqrt{3} \\ x_2 = - \frac{\pi}{3} + \pi \: n \\ \\n \in \:Z$

2.

$2 \cos {}^{2} (x) - \cos(x) - 1 = 0 \\ \\ \cos(x) = t \\ \\ 2t {}^{2} - t - 1 = 0 \\ D = 1 + 8 = 9 \\ t_1 = \frac{1 + 3}{4} = 1 \\ t_2 = - \frac{1}{2} \\ \\ \cos(x) = 1 \\ x_1 = 2\pi \: n \\ \\ \cos(x) = - \frac{1}{2} \\ x_2 = \pm \frac{2\pi}{3} + 2\pi \: n \\ \\ n\in \:Z$