Question

Тема "тригонометричемкие уравнения сведены к одной функции" розвязать уравнения ​

Answer 1

12

$\sin {}^{2} (x) + 4\sin(x) \cos(x) + 3 \cos {}^{2} (x) = 0 \\ | \div \cos {}^{2} (x) \ne0 \\ \\ {tg}^{2} x + 4tgx + 3 = 0 \\ \\ tgx = t \\ \\ t {}^{2} + 4t + 3 = 0\\ D= 16 - 12 = 4 \\ t_1 = \frac{ - 4 + 2}{2} = - 1 \\ t_2 = - 3 \\ \\ tgx = - 1 \\ x_1 = - \frac{\pi}{4} + \pi \: n \\ tgx = - 3 \\ x_2 = - arctg(3) + \pi \: n$

n принадлежит Z.

13

$2 \sin {}^{2} (x) + 3.5 \sin(2x) - 9\cos {}^{2} (x) = 0 \\ 2 \sin {}^{2} (x) + 7 \sin(x) \cos(x) - 9 \cos {}^{2} (x) = \\ | \div \cos {}^{2} (x) \ne0 \\ 2 {tg}^{2}x + 7 tgx - 9 = 0 \\ \\ tgx = t \\ \\2 t {}^{2} + 7t - 9 = 0 \\ D = 49 + 72 = 121\\ t_1 = \frac{ - 7 + 11}{4} = 1 \\ t_2 = - \frac{18}{4} = - 4.5 \\ \\ tgx = 1 \\ x_1 = \frac{\pi}{4} + \pi \: n \\ \\ tgx = - 4.5 \\ x_2 = - arctg(4.5) + \pi \: n$

n принадлежит Z.