Question

$\sin( \frac{x}{3} ) \cos( \frac{\pi}{5} ) - \cos( \frac{x}{3} ) \sin( \frac{\pi}{5} ) = \frac{ \sqrt{2} }{2}$

Answer 1

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

Ответ: $\frac{3\pi}{5}+(-1)^k*\frac{3\pi}{4}+3\pi k,\ k\in Z$