Question

с тригонометрией 7,8,9,10,11 с подробным решением

Answer 1

7.

$\cos(2\pi - 0.5x) + 3 \cos(\pi - 0.5x) = 1 \\ \cos(0.5x) - 3 \cos(0.5x) = 1 \\ - 2 \cos(0.5x) = 1 \\ \cos(0.5x) = - \frac{1}{2} \\ \frac{x}{2} = \pm \frac{2\pi}{3} + 2\pi \: n \\ x = \pm \frac{4\pi}{3} + 4\pi \: n$

8.

$15 \cos(3\pi - 5x) + 56 \sin( \frac{\pi}{2} - 5x) = 0 \\ - 15 \cos(5x) + 56 \cos(5x) = 0 \\ 41 \cos(5x) = 0 \\ 5x = \frac{\pi}{2} + \pi \: n \\ x = \frac{\pi}{10} + \frac{\pi \: n}{5}$

9.

$ctg(5\pi + x) + 4ctg(4\pi + x) = 1 \\ ctg(x) + 4ctg(x) = 1 \\ 5ctg(x) = 1 \\ ctg(x) = \frac{1}{5} \\ x = arcctg( \frac{1}{5} ) + \pi \: n$

10.

$25 \cos(3\pi + 2x) \sin(5\pi - 2x) = 0 \\ - 25 \cos(2x) \times \sin(2x) = 0 \\ - \frac{25}{2} \times 2 \sin(2x) \cos(2x) = 0 \\ - \frac{25}{2} \sin(4x) = 0 \\ \sin( 4x) = 0 \\ 4x = \pi \: n \\ x = \frac{ \pi \: n}{4}$

11.

$2 \cos {}^{2} ( \frac{3\pi}{2} + 3x ) - 2 \sin {}^{2} ( \frac{5\pi}{2} - 3x ) = \sqrt{3} \\ 2 \sin {}^{2} (3x) - 2 \cos {}^{2} (3x) = \sqrt{3} \\ - 2( \cos {}^{2} (3x) - \sin {}^{2} (3x)) = \sqrt{3} \\ \cos(6x) = - \frac{ \sqrt{3} }{2} \\ 6x = \pm \frac{5\pi}{6} + 2\pi \: n \\ x = \pm \frac{5\pi}{36} + \frac{\pi \: n}{3}$

n принадлежит Z.