Question

решите уравнение sin (5x-pi/3) = sin x

Answer 1

Ответ:

$\sin(5x - \frac{\pi}{3} ) = \sin(x) \\ \sin(5x - \frac{\pi}{3} ) - \sin(x) = 0 \\ 2 \sin( \frac{5x - \frac{\pi}{3} - x }{2} ) \cos( \frac{5x - \frac{\pi}{3} + x}{2} ) = 0 \\ 2 \sin(2x - \frac{\pi}{6} ) \cos(3x - \frac{\pi}{6} ) = 0 \\ \\ \sin(2x - \frac{\pi}{6} ) = 0 \\ 2x - \frac{\pi}{6} = \pi \: n \\ 2x = \frac{\pi}{6} + \pi \: n \\ x1 = \frac{\pi}{12 } + \frac{\pi \: n}{2} \\ \\ \cos( 3x - \frac{\pi}{6}) = 0 \\ 3x - \frac{\pi}{6} = \frac{\pi}{2} + \pi \: n \\ 3x = \frac{2\pi}{3} + \pi \: n \\ x2 = \frac{2\pi}{9} + \frac{\pi \: n}{3}$

n принадлежит Z.