$$\eqalign{
& 2\sin \left( {{{45}^0} + 3x} \right) + 1 = 0 \cr
& \Leftrightarrow \sin \left( {{{45}^0} + 3x} \right) = - {1 \over 2} \cr
& \Leftrightarrow \left[ \matrix{
{45^0} + 3x = - {30^0} + k{360^0} \hfill \cr
{45^0} + 3x = {180^0} + {30^0} + k{360^0} \hfill \cr} \right. \cr
& \Leftrightarrow \left[ \matrix{
3x = - {75^0} + k{360^0} \hfill \cr
3x = {165^0} + k{360^0} \hfill \cr} \right. \cr
& \Leftrightarrow \left[ \matrix{
x = - {25^0} + k{120^0} \hfill \cr
x = {55^0} + k{120^0} \hfill \cr} \right.\,\,\left( {k \in Z} \right) \cr} $$
