Question 1102259
.
The solution by @Lightning_Fast is partly WRONG and partly INCOMPLETE, so NEITHER PART OF HIS SOLUTION IS USEFUL.


Below please find the CORRECT solution.


<pre>
sin(2x) = {{{-sqrt(2)/2}}}  ======================>


This equation has two solutions for 2x:


1)  2x = {{{5pi/4}}}   or   2)  2x = {{{7pi/4}}}.


Case 1.  2x = {{{5pi/4}}}  has TWO solutions for the angle x:  

                                                   a)  x = {{{5pi/8}}}  and

                                                   b)  x = {{{5pi/8+ pi}}} = {{{13pi/8}}}.



Case 2.  2x = {{{7pi/4}}}  has TWO solutions for the angle x:  

                                                   a)  x = {{{7pi/8}}}  and

                                                   b)  x = {{{7pi/8+ pi}}} = {{{15pi/8}}}.


<U>Answer</U>.  The given equation has four solutions:  {{{5pi/8}}},  {{{13pi/8}}},  {{{7pi/8}}}  and  {{{15pi/8}}}.
</pre>

Solved.



The plot below CONFIRMS existing of four solutions in the interval [0, {{{2pi}}}):



{{{graph( 330, 330, -0.5, 6.5, -1.5, 1.5,
          sin(2x), -sqrt(2)/2
)}}}


Plot y = sin(2x) (red) and y = {{{-sqrt(2)/2}}} (green)