Question

Реши уравнение! Не сокращай.

Answer 1

$3 \sin(x) + 15 \cos(x) = \sqrt{234} \\ 3( \sin(x) + 5 \cos(x) ) = 3 \sqrt{26} \\ \sin(x) + 5 \cos(x) = \sqrt{26} \\ a \sin(x) + b \cos(x) = \sqrt{ {a}^{2} + {b}^{2} } \sin(x + \arctg( \frac{b}{a} ) ) \\ \sqrt{ {1}^{2} + {5}^{2} } \sin(x + \arctg( \frac{5}{1} ) ) = \sqrt{26} \\ \sqrt{26} \sin(x + \arctg5) = \sqrt{26} \\ \sin(x + \arctg5) = 1 \\ x + \arctg5 = \frac{\pi}{2} + 2\pi n \\ x = \frac{\pi}{2} - \arctg5 + 2\pi n, \: n \in \mathbb Z$