SOLUTION: solve the equation on the interval [0,2pi] sin2x= -root3 sinx

 = {{-sqrt(3)*sin(x)}}}  --->  (Use the formula sin(2x) = 2*sin(x)*cos(x). You will get )

 =   --->

 = 0  --->  (factor left side)

xin(x)*(2*cos(x)+sqrt(3)) = 0 .


Then the last equation deploys in two independent equations.


1.  sin(x) = 0  with the solutions x = 0 and x = ,   and


2.  2*cos(x) + sqrt(3) = 0  --->  cos(x) =  with the solutions

    x =  and  x = .


Answer.  The solutions to the given equations are x= 0, x= , x =  and x= .