Question 537174
{{{0<=theta<=2pi}}} --> {{{0<=4theta<=8pi=4*full*turns}}} 
{{{2sqrt(3) sin (4theta) = 3 }}}-->{{{ sin (4theta) = 3/(2sqrt(3))=sqrt(3)/2 }}}
Teachers like to give students a chart, or better yet a printed circle, with the angle measurements and exact values for sine, cosine, tangent for certain notable angles, so that number may ring a bell. You do not need to memorize them, or have the chart because they all derive from Pythagoras theorem and triangles that are half of a square or half or an equilateral triangle. However, having such a chart helps keep your calculations straight.
You should know that {{{4theta=pi/3}}} (60 degrees) is one answer (sine and cosine of 30, 45, and 60 degrees are well known values).
{{{4theta=(2/3)pi}}} is another solution (180-60=120 degrees, the supplementary angle has the same sine). You find more (coterminal) solutions for {{{4theta}}} in the next turn, and the next, and the next> You can find the coterminal {{{4theta}}} solutions by adding to the two solutions above {{{2pi}}} (360 degrees), {{{4pi}}}, and {{{6pi}}} to get a total of eight values for {{{4theta}}}:
{{{(1/3)pi}}}, {{{(2/3)pi}}}, {{{(7/3)pi}}}, {{{(8/3)pi}}}, {{{(13/3)pi}}}, {{{(14/3)pi}}}, {{{(19/3)pi}}}, and {{{(20/3)pi}}}.
Dividing by 4, we get the values for {{{theta}}}
{{{(1/12)pi}}}, {{{(2/12)pi=(1/6)pi}}}, {{{(7/12)pi}}}, {{{(8/12)pi=(2/3)pi}}}, {{{(13/12)pi}}}, {{{(14/12)pi=(7/6)pi}}}, {{{(19/12)pi}}}, and {{{(20/12)pi=(5/3)pi}}}.