Question

List all values of x such that 2 ((sin (x))^2)+((sqrt (3))sin(x))=(2sin (x)+sqrt (3))

Answer 1

`\bf 2[sin(x)]^2+√(3)sin(x)=2sin(x)+√(3) \\\\\\ 2[sin(x)]^2+√(3)sin(x)-2sin(x)-√(3)=0 \\\\\\\ [2sin^2(x)-2sin(x)]~+~[√(3)sin(x)-√(3)]=0 \\\\\\ \stackrel{common~factor}{2sin(x)}[sin(x)-1]~~+~~\stackrel{common~factor}{√(3)}[sin(x)-1]=0 \\\\\\`


`\bf \stackrel{common~factor}{[sin(x)-1]}~[2sin(x)+√(3)]=0\qquad \begin{cases} sin(x)-1=0\\ sin(x)=1\\ \measuredangle x=(\pi )/(2)\\ ----------\\ 2sin(x)+√(3)=0\\ 2sin(x)=-√(3)\\ sin(x)=-(√(3))/(2)\\\\ \measuredangle x=(4\pi )/(3)~,~(5\pi )/(3) \end{cases}`