Question 1178162
.
<pre>

Your starting equation is


    sin(4x) − sqrt(3)*sin(2x) = 0


Substitute (replace) here     sin(4x) = 2sin(2x(*cos(2x).


    2sin(2x)*cos(2x) - sqrt(3)*sin(2x) = 0.


Factor left side


    2sin(2x)*(cos(2x)-sqrt(3)) = 0.


This equation deploys in two equations


    a)  sin(2x) = 0.


        It has the roots  2x = 0, {{{pi}}},  {{{2pi}}},  {{{3pi}}}  in the interval  [0,{{{4pi}}})

        that create the roots

        x = 0,  {{{pi/2}}},  {{{pi}}},  {{{3pi/2}}}  in the interval  [0,{{{2pi}}}).



    b)  cos(2x) = {{{sqrt(3)/2}}}.

        It has the roots  2x = {{{pi/6}}},  {{{11pi/6}}},  {{{13pi/6}}},  {{{23pi/6}}}  in the interval  [0,{{{4pi}}})

        that create the roots

        x = {{{pi/12}}},  {{{11pi/12}}},  {{{13pi/12}}},  {{{23pi/12}}}  in the interval  [0,{{{2pi}}}).


<U>ANSWER</U>.  There are 8 roots  x = 0,  {{{pi/2}}},  {{{pi}}},  {{{3pi/2}}}

                                {{{pi/12}}},  {{{11pi/12}}},  {{{13pi/12}}},  {{{23pi/12}}}  in the interval  [0,{{{2pi}}}).
</pre>

Solved.



/////////////



Find the differences between my solution and the solution by Edwin.