Question

Решите уравнение пожалуйста ​

Answer 1

ОДЗ :

$\left \{ {{-Ctgx>0} \atop {Sinx\neq 0}} \right.\\\\\left \{ {{Ctgx<0} \atop {x\neq\pi n,n\in Z}} \right.\Rightarrow - \frac{\pi }{2}+\pi n<x<\pi n,n\in Z$

$\frac{3Cos2x+5Cosx-1}{\sqrt{-Ctgx} }=0\\\\3Cos2x+5Cosx-1=0\\\\3(2Cos^{2}x-1)+5Cosx-1=0\\\\6Cos^{2}x-3+5Cosx-1=0\\\\6Cos^{2}x+5Cosx-4=0\\\\Cosx=m, -1 \leq m \leq 1\\\\6m^{2}+5m-4=0\\\\D=5^{2}-4*6*(-4)=25+96=121=11^{2}\\\\m_{1} =\frac{-5+11}{12}=\frac{6}{12}=\frac{1}{2}\\\\m_{2}=\frac{-5-11}{12}=-\frac{16}{12}<-1-neyd$

$Cosx=\frac{1}{2}\\\\\left[\begin{array}{ccc}x_{1} =-\frac{\pi }{3}+2\pi n ,n\in Z\\x_{2}=\frac{\pi }{3}+2\pi n,n\in Z \end{array}\right\\\\x_{2}-neyd\\\\Otvet:\boxed{-\frac{\pi }{3}+2\pi n,n\in Z}$