173

$2\sin\ \frac{x}{3}+\sqrt{2}=0$
$\sin\ \frac{x}{3}=-\frac{\sqrt{2}}{2}$

$\sin\ \frac{x}{3}=-\frac{1}{\sqrt{2}}$

$\frac{5\pi}{4}$

kaikki ratkaisut ovat:

$\frac{x}{3}=\frac{5\pi}{4}+n\cdot2\pi$
$x=\frac{15\pi}{4}+n\cdot6\pi$

tai

$\frac{x}{3}=\pi-\frac{5\pi}{4}+n\cdot2\pi$
$x=-\frac{3\pi}{4}+n\cdot6\pi$

halutaan ratkaisut välillä ]-12π,12π[

eli

$\begin{matrix} n&x=\frac{15\pi}{4}+n\cdot6\pi&x=-\frac{3\pi}{4}+n\cdot6\pi\\ 0&\frac{15\pi}{4}&-\frac{3\pi}{4}\\ 1&\frac{39\pi}{4}&\frac{21\pi}{4}\\ -1&-\frac{9\pi}{4}&-\frac{27\pi}{4}\\ 2&\frac{63\pi}{4}\left(hyl.\right)&\ \frac{45\pi}{4}\\ -2&-\frac{33\pi}{4}&x=-\frac{51\pi}{4}\ \left(hyl.\right)\\ -3&-\frac{57\pi}{4}\left(hyl.\right)& \end{matrix}$

ratkaisuista halutulle välille kuuluvat $-\frac{33\pi}{4}{,}\ -\frac{27\pi}{4}{,}\ -\frac{9\pi}{4}{,}\ -\frac{3\pi}{4}{,}\ \frac{15\pi}{4}{,}\ \frac{21\pi}{4}{,}\ \frac{35\pi}{4}\ tai\ \frac{45\pi}{4}$