Question

Решить 3 sin x + корень из 3 * cos x = 2

Answer 1

$3sinx+\sqrt3cosx=2\, |:\sqrt{12},\\\\\frac{3}{\sqrt{12}}sinx+\frac{\sqrt3}{\sqrt{12}}cosx=\frac{2}{\sqrt12}\\\\(\frac{3}{\sqrt12})^2+(\frac{\sqrt3}{\sqrt12})^2=1\; \; \to cos \alpha =\frac{3}{\sqrt12},\; \; sin \alpha =\frac{\sqrt3}{\sqrt12},\; \to \; tg \alpha =\frac{\sqrt3}{3}\\\\sinx\cdot cos \alpha +cosx\cdot sin \alpha \alpha =sin(x+ \alpha )\\\\sin(x+ \alpha )=\frac{2}{\sqrt12}=\frac{1}{\sqrt3}\\\\x+ \alpha =(-1)^{n}arcsin\frac{1}{\sqrt3}+\pi n,\; n\in Z\\\\x= -\alpha +(-1)^{n}arcsin\frac{1}{\sqrt3}+\pi n,\; n\in Z\, ,\; gde\; \; tg \alpha =\frac{\sqrt3}{3}$