$$\eqalign{
& \sin \left( {3x + {\pi \over 4}} \right) + \cos x = 0 \cr
& \Leftrightarrow \sin \left( {3x + {\pi \over 4}} \right) = - \cos x \cr
& \Leftrightarrow \sin \left( {3x + {\pi \over 4}} \right) = - \sin \left( {{\pi \over 2} - x} \right) \cr
& \Leftrightarrow \sin \left( {3x + {\pi \over 4}} \right) = \sin \left( {x - {\pi \over 2}} \right) \cr
& \Leftrightarrow \left[ \matrix{
3x + {\pi \over 4} = x - {\pi \over 2} + k2\pi \hfill \cr
3x + {\pi \over 4} = \pi - x + {\pi \over 2} + k2\pi \hfill \cr} \right. \cr
& \Leftrightarrow \left[ \matrix{
2x = - {{3\pi } \over 4} + k2\pi \hfill \cr
4x = {{5\pi } \over 4} + k2\pi \hfill \cr} \right. \cr
& \Leftrightarrow \left[ \matrix{
x = - {{3\pi } \over 8} + k\pi \hfill \cr
x = {{5\pi } \over {14}} + {{k\pi } \over 2} \hfill \cr} \right.\,\,\left( {k \in Z} \right) \cr} $$
