Question

Помогите плиз
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Answer 1

Ответ:

1

$4 \cos {}^{2} (x) + 4 \sin(x) - 1 = 0 \\ 4 - 4 \sin {}^{2} (x) + 4\sin(x) - 1 = 0 \\ 4 \sin {}^{2} (x) - 4 \sin(x) - 3 = 0 \\ \\ \sin(x) = t \\ \\4 t {}^{2} - 4 t - 3 = 0 \\ D = 16 + 48 = 64 \\ t_1 = \frac{4 + 8}{8} = \frac{3}{2} = 1.5 \\ t_2 = - \frac{1}{2} \\ \\ \sin(x) = 1.5 \\ 1.5 > 1 \\ \text{корней нет} \\ \\ \sin(x) = - \frac{1}{2} \\ x_1 = - \frac{\pi}{6} + 2\pi \: n \\ x_2 = - \frac{5\pi}{6} + 2\pi \: n$

n принадлежит Z.

2

$3 \sin {}^{2} (3x) - 2.5 \sin(6x) + 1 = 0 \\ 3 \sin {}^{2} (3x) - 5 \sin(3x) \cos(3x) + 1 = 0 \\ 3 \sin {}^{2} (3x) - 5\sin(3x) \cos(3x) + \sin {}^{2} (3x) + \cos {}^{2} (3x) = 0 \\ 4 \sin {}^{2}(x ) - 5\sin(3x) \cos(3x) + \cos {}^{2} (3x) = 0 \\ | \div \cos {}^{2} (3x) \ne0 \\ 4 {tg}^{2} x - 5tgx + 1 = 0 \\ \\ tgx = t \\ 4t {}^{2} - 5 t + 1 = 0 \\ D = 25 - 16 = 9 \\ t_1 = \frac{5 + 3}{8} = 1 \\ t_2 = \frac{1}{4} \\ \\ tgx = 1 \\ x_1 = \frac{\pi}{4} + \pi \: n \\ \\ tgx = \frac{1}{4} \\ x_2 = arctg(0.25) + \pi \: n$

n принадлежит Z.

3

$\sin(9x) + \sin(8x) + \sin(7x) = 0 \\ (\sin(9x) + \sin(7x)) + \sin(8x) = 0 \\ 2 \sin( \frac{9x + 7x}{2} ) \cos( \frac{9x - 7x}{2} ) + \sin(8x) = 0 \\ 2 \sin(8x) \cos(x) + \sin(8x) = 0 \\ \sin(8x) (2\cos(x) + 1) = 0 \\ \\ \sin(8x) = 0 \\ 8x = \pi \: n \\ x_1 = \frac{\pi \: n}{8} \\ \\ \cos(x) = - \frac{1}{2} \\ x_2 = \pm \frac{2\pi}{3} + 2 \pi \: n$

n принадлежит Z.