Question

4 sin^2 (x/2) + 8 cos (x/2) -7 = 0

Answer 1

4-4сos²(x/2)+8cos(x/2)-7=0
cos(x/2)=a
4a²-8a+3=0
D=64-48=16
a1=(8-4)/8=1/2⇒cos(x/2)=1/2⇒x/2=+-π/3+2πn⇒x=+-2π/3+4πn,n∈z
a2=(8+4)/8=1,5⇒cos(x/2)=1,5>1 нет решения

Answer 2

$4sin^2frac{x}{2}+8cosfrac{x}{2}-7=0\\4(1-cos^2frac{x}{2})+8cosfrac{x}{2}-7=0\\t=cosfrac{x}{2}; ,; ; -4t^2+8t-3=0; ,; ; 4t^2-8t+3=0; ,; -1 leq t leq 1\\D/4=16-12=4\\t_1=frac{4+2}{4}=frac{3}{2}; ; ne; podxodit\\t_2= frac{4-2}{4} =frac{1}{2}; ; Rightarrow ; ; cosfrac{x}{2}=frac{1}{2}\\frac{x}{2}=pm arccosfrac{1}{2}+2pi n; ,; nin Z\\frac{x}{2}=pm frac{pi}{3}+2pi n; ,; nin Z\\x=pm frac{2pi }{3}+4pi n; ,; nin Z$