Question

5cosxctgx-5ctgx+2sinx=0

Answer 1

$5, cosxcdot ctgx-5, ctgx+2sinx=0; ; ,; ; ; ODZ:; xne pi n,; nin Z\\5, cosxcdot frac{cosx}{sinx}-5cdot frac{cosx}{sinx}+2sinx=0\\frac{5, cos^2x-5, cosx+2sin^2x}{sinx}=0; ; ,; ; frac{5, cos^2x-5, cosx+2(1-cos^2x)}{sinx}=0; ,\\frac{3, cos^2x-5cosx+2}{sinx}=0; ; Rightarrow ; ; left { {{3, cos^2x-5, cosx+2=0} atop {xne pi n,; nin Z}} right. \\3, cos^2x-5, cosx+2=0; ,\\t=cosx; ,; ; -1leq tleq 1; ; ,; ; 3t^2-5t+2=0; ,; ; D=1; ,; t_1=frac{2}{3}; ,; t_2=1$

$a); ; cosx=frac{2}{3}; ,; ; x=pm arccosfrac{2}{3}+2pi m; ,; min Z\\left { {{x=pm arccos frac{2}{3}+2pi m; ,; min Z} atop {xne pi n; ,; nin Z}} right. ; ; Rightarrow ; ; x=pm arccos frac{2}{3}+2pi m; ,; min Z\\b); ; cosx=1; ,; ; x=2pi k; ,; kin Z; ; Rightarrow ; ; left { {{x=2pi k; ,; kin Z} atop {xne pi n; ,; nin Z}} right. ; ; Rightarrow ; ; xin varnothing \\Otvet:; ; x=pm arccos frac{2}{3}+2pi m; ,; min Z; .$