Question

2sin^2(x-п/3)+sin(x-п/3)-1=0

Answer 1

$2Sin^{2}(x-frac{pi }{3})+Sin(x-frac{pi }{3})-1=0\\Sin(x-frac{pi }{3})=m, -1 leq m leq1\\2m^{2}+m-1=0\\D=1^{2}-4*2*(-1)=1+8=9=3^{2}\\m_{1}=frac{-1+3}{4}=frac{1}{2}\\m_{2} =frac{-1-3}{4}=-1$

$Sin(x-frac{pi }{3} )=frac{1}{2}\\1)x-frac{pi }{3}=arcSinfrac{1}{2}+2pi n,nin z\\x-frac{pi }{3}=frac{pi }{6}+2pi n,nin z\\x=frac{pi }{6}+frac{pi }{3}+2pi n,nin z\\x_{1}=frac{pi }{2}+2pi n,nin z\\2)x-frac{pi }{3}=pi-arcSinfrac{1}{2}+2pi n,nin z\\x-frac{pi }{3}=pi-frac{pi }{6} +2pi n,nin z\\x-frac{pi }{3}=frac{5pi }{6}+2pi n,nin z\\x=frac{5pi }{6}+frac{pi }{3}+2pi n,nin z$

$x_{2}=frac{7pi }{6}+2pi n,nin z\\Sin(x-frac{pi }{3})=-1\\3)x-frac{pi }{3}=-frac{pi }{2}+2pi n,nin z\\x=frac{pi }{3}-frac{pi }{2}+2pi n,nin z\\x_{3}=-frac{pi }{6}+2pi n,nin z$