ANSWER:
[tex]x=\frac{5\pi}{16},\frac{13\pi}{16},\frac{21\pi}{16},\frac{29\pi}{16},\frac{7\pi}{16},\frac{15\pi}{16},\frac{23\pi}{16},\frac{31\pi}{16}[/tex]
STEP-BY-STEP EXPLANATION:
We have the following equation:
[tex]\sin \: 4x=-\frac{\sqrt{2}}{2}[/tex]
We solve for x:
[tex]\begin{gathered} 4x=\arcsin \mleft(-\frac{\sqrt{2}}{2}\mright) \\ 4x=\frac{5\pi}{4}+2\pi n,\frac{7\pi}{4}+2\pi n \\ x=\frac{5\pi}{16}+\frac{1}{2}\pi n,\frac{7\pi}{16}+\frac{1}{2}\pi n \\ x_1=\frac{5\pi}{16} \\ x_2=\frac{5\pi}{16}+\frac{\pi}{2}=\frac{13\pi}{16} \\ x_3=\frac{5\pi}{16}+2\cdot\frac{\pi}{2}=\frac{21\pi}{16} \\ x_4=\frac{5\pi}{16}+3\cdot\frac{\pi}{2}=\frac{29\pi}{16} \\ x_5=\frac{7\pi}{16} \\ x_6=\frac{7\pi}{16}+\frac{\pi}{2}=\frac{15\pi}{16} \\ x_7=\frac{7\pi}{16}+2\cdot\frac{\pi}{2}=\frac{23\pi}{16} \\ x_8=\frac{7\pi}{16}+3\cdot\frac{\pi}{2}=\frac{31\pi}{16} \end{gathered}[/tex]
