Question

решить тригонометрию. Даю 60б​

Answer 1

1.

$- 2 \sin(5x) - 1 = 0 \\ \sin(5x) = - \frac{1}{2} \\ \\ 5x1 = - \frac{\pi}{6} + 2\pi \: n \\ x1 = - \frac{\pi}{30} + \frac{2\pi \: n}{5} \\ \\ 5x2 = - \frac{5\pi}{6} + 2\pi \: n \\ x2 = - \frac{\pi}{6} + \frac{2\pi \: n}{5}$

2.

$\cos( - \frac{x}{6} ) = - \frac{ \sqrt{3} }{2} \\ - \frac{x}{6} = \pm \frac{5\pi}{6} + 2\pi \: n \: \: \: | \times 6 \\ x = \pm5\pi + 12\pi \: n$

3.

$tg(8x - 1) = \sqrt{3} \\ 8x - 1 = \frac{\pi}{3} + \pi \: n \\ 8x = \frac{\pi}{3} + 1 + \pi \: n \\ x = \frac{\pi}{24} + \frac{1}{8} + \frac{\pi \: n}{8}$

4.

$\sqrt{2} \sin(1 - \frac{3x}{7} ) = - 1 \\ \sin(1 - \frac{3x}{7} ) = - \frac{ \sqrt{2} }{2} \\ \\ 1 - \frac{3x}{7} = - \frac{\pi}{4} + 2\pi \: n \\ - \frac{3}{7} x = - \frac{\pi}{4} - 1 + 2\pi \: n \: \: \: | \times ( - \frac{7}{3} ) \\ x1 = \frac{7\pi}{12} + \frac{7}{3} - \frac{14\pi \: n}{3} \\ \\ 1 - \frac{3x}{7} = - \frac{3\pi}{4} + 2\pi \: n \\ - \frac{3x}{7} = - \frac{3\pi}{4} - 1 + 2\pi \: n \\ x2 = \frac{7\pi}{4} + \frac{7}{3} + \frac{14\pi}{3}$

5.

$ctg( - 2x) = - \frac{ \sqrt{3} }{3} \\ - 2x = - \frac{\pi}{3} + \pi \: n \\ x = \frac{\pi}{6} + \frac{\pi \: n}{2}$

6.

$\sin( \frac{\pi}{2} - 2x ) - 2 \cos(6\pi - 2x) = \frac{ \sqrt{2} }{2} \\ \cos(2x) - 2 \cos(2x) = \frac{ \sqrt{2} }{2} \\ - \cos(2x) = \frac{ \sqrt{2} }{2} \\ \cos(2x) = - \frac{ \sqrt{2} }{2} \\ 2x = \pm \frac{3\pi}{4} + 2\pi \: n \\ x = \pm \frac{3\pi}{8} + \pi \: n$

7.

$\sin(4\pi - 0.5x) + 3 \sin(5\pi - 0.5x) = - \sqrt{2} \\ - \sin(0.5x) + 3 \sin(0.5x) = - \sqrt{2} \\ 2 \sin(0.5x) = - \sqrt{2} \\ \sin( \frac{x}{2} ) = - \frac{ \sqrt{2} }{2} \\ \\ \frac{x}{2} = - \frac{\pi}{4} + 2\pi \: n \\ x1 = - \frac{\pi}{2} + 4 \pi \: n \\ \\ \frac{x}{2} = - \frac{3\pi}{4} + 2 \pi \: n \\ x2 = - \frac{3\pi}{2} + 4\pi \: n$

8.

$2 \cos(\pi - \frac{x}{4} ) + 3 \sin( \frac{5\pi}{2} - \frac{x}{4} ) = 0 \\ - 2 \cos( \frac{x}{4} ) + 3 \cos( \frac{x}{4} ) = 0 \\ \cos( \frac{x}{4} ) = 0 \\ \frac{x}{4} = \frac{\pi}{2} + \pi \: n \\ x = 2\pi + 4\pi \: n$

9.

$ctg( \frac{3\pi}{2} + x) - 2tg(3\pi + x) = - 1 \\ - tg(x) - 2tg(x) = - 1 \\ - 3tg(x) = - 1 \\ tg(x) = \frac{1}{3} \\ x = arctg( \frac{1}{3}) + \pi \: n$

10.

$4 \cos(7\pi + 4x) \sin(\pi - 4x) = 0 \\ - 4 \cos(4x) \times \sin(4x) = 0 \\ - 2 \sin(8x) = 0 \\ \sin(8x) = 0 \\ 8x = \pi \: n \\ x = \frac{\pi \: n}{8}$

11.

$2 \cos {}^{2} ( \frac{\pi}{2} + 5x) - 2 \sin { }^{2} ( \frac{3\pi}{2} - 5x) = 1 \\ 2 \sin {}^{2} (5x) - 2 \cos {}^{2} (5x) = 1 \\ - 2( \cos {}^{2} (5x) - \sin {}^{2} (5x)) = 1 \\ \cos(10x) = - \frac{1}{2} \\ 10x = \pm \frac{2\pi}{ 3} + 2 \pi \: n \\ x = \pm \frac{\pi}{15} + \frac{\pi \: n}{5}$

n принадлежит Z.