Question

Решите уравнение
2(sinx/3 cospi/5-cosx/3 sinpi/5)=корень из 2​

Answer 1

$2( \sin( \frac{x}{3} ) \cos( \frac{\pi}{5} ) - \cos( \frac{x}{3} ) \sin( \frac{\pi}{5} ) ) = \sqrt{2} \\ 2( \sin( \frac{x}{3} - \frac{\pi}{5} ) ) = \sqrt{2} \\ \sin( \frac{x}{3} - \frac{\pi}{5} ) = \frac{ \sqrt{2} }{2} \\ \frac{x}{3} - \frac{\pi}{5} = ( - 1) {}^{n} \arcsin( \frac{ \sqrt{2} }{2} ) + \pi n \\ \frac{x}{3} = ( - 1) {}^{n} \frac{\pi}{4} + \frac{\pi}{5} + \pi n \\ x = ( - 1) {}^{n} \frac{3\pi}{4} + \frac{3\pi}{5} + 3\pi n, \: n \in \mathbb Z$