\(sin^3x+2sinx+3cosx=0\)
\(\Leftrightarrow sin^3x-sinx+3sinx+3cosx=0\)
\(\Leftrightarrow sinx\left(sin^2x-1\right)+3\left(sinx+cosx\right)=0\)
\(\Leftrightarrow-sinx.cosx+3\left(sinx+cosx\right)=0\)
\(\Leftrightarrow\dfrac{1-\left(sinx+cosx\right)^2}{2}+3\left(sinx+cosx\right)=0\)
\(\Leftrightarrow\left(sinx+cosx\right)^2-6\left(sinx+cosx\right)-1=0\)
\(\Leftrightarrow\left(sinx+cosx\right)^2-6\left(sinx+cosx\right)-1=0\)
\(\Leftrightarrow\left[{}\begin{matrix}sinx+cosx=3+\sqrt{10}\\sinx+cosx=3-\sqrt{10}\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}sin\left(x+\dfrac{\pi}{4}\right)=\dfrac{3+\sqrt{10}}{\sqrt{2}}\left(l\right)\\sin\left(x+\dfrac{\pi}{4}\right)=\dfrac{3-\sqrt{10}}{\sqrt{2}}\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x+\dfrac{\pi}{4}=arcsin\left(\dfrac{3-\sqrt{10}}{\sqrt{2}}\right)+k2\pi\\x+\dfrac{\pi}{4}=\pi-arcsin\left(\dfrac{3-\sqrt{10}}{\sqrt{2}}\right)+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=-\dfrac{\pi}{4}+arcsin\left(\dfrac{3-\sqrt{10}}{\sqrt{2}}\right)+k2\pi\\x=\dfrac{3\pi}{4}-arcsin\left(\dfrac{3-\sqrt{10}}{\sqrt{2}}\right)+k2\pi\end{matrix}\right.\)