Question 172642
{{{(cos(4X))^2 + (sqrt(3)/2) * sin(4X) = 0}}}
Ok so the first thing to remember is that {{{(cos(x))^2=1-(sin(x))^2}}}
which means that {{{(cos(4X))^2=1-(sin(4X))^2}}}

So we now rewrite your equation as {{{1-(sin(4X))^2 + (sqrt(3)/2) * sin(4X) = 0}}}
which is the same as {{{-(sin(4X))^2 + (sqrt(3)/2) * sin(4X) + 1 = 0}}}

Now set {{{y=sin(4x)}}} this also means that {{{y^2=(sin(4X))^2}}}

Your new equation is {{{-y^2 + (sqrt(3)/2)y + 1 = 0}}}

This can now be solved using the quadratic formula a=-1, b={{{(sqrt(3)/2)}}}, c=1

The formula is {{{y = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}}

So you have {{{x = (-(sqrt(3)/2) +- sqrt( ((sqrt(3)/2))^2-4*(-1)*1 ))/(2*(-1)) }}}

*[invoke quadratic "y", -1, (sqrt(3)/2), 1 ]

The solver will give you the solutions to the equation
y1=-0.656712033992949
y2=1.52273743777739
But there is one more step, to make the problem clearer we set {{{y=sin(4x)}}}, so we actually need a solution for {{{x}}}

To do this we replace they {{{y}}}'s we got and solve for {{{x}}}

So we have
{{{sin(4x)=-0.656712033992949}}}
{{{sin(4x)=1.52273743777739}}}

Next step
{{{4x=arcsin(-0.656712033992949)}}} use a calculator to compute this
{{{4x=arcsin(1.52273743777739)}}} when you try to evaluate this you will get an error which means that there is no solution to that part. Only the -0.656712033992949 will yield a result

Then {{{x=(arcsin(-0.656712033992949))/4}}} use a calculator for this two.

Enjoy :)