HINT (you only need to simplify the final answer):
$$\sin(x)=-5\Longleftrightarrow$$ $$\frac{1}{2}\left(ie^{-ix}-ie^{ix}\right)=-5\Longleftrightarrow$$ $$ie^{-ix}-ie^{ix}=-10\Longleftrightarrow$$
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Substitute $y=-ie^{ix}$:
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$$\frac{1}{(0-i)e^{(0+i)x}}+(0-i)e^{(0+i)x}=-10\Longleftrightarrow$$ $$y+\frac{1}{y}=-10\Longleftrightarrow$$ $$\frac{y^2+1}{y}=-10\Longleftrightarrow$$ $$y^2+1=-10\Longleftrightarrow$$ $$y^2+10y+1=0\Longleftrightarrow$$ $$y^2+10y=-1\Longleftrightarrow$$ $$y^2+10y+25=24\Longleftrightarrow$$ $$(y+5)^2=24\Longleftrightarrow$$ $$y+5=\pm\sqrt{24}\Longleftrightarrow$$ $$y+5=\pm 2\sqrt{6}\Longleftrightarrow$$ $$y=\pm 2\sqrt{6}-5\Longleftrightarrow$$ $$-ie^{ix}=\pm 2\sqrt{6}-5\Longleftrightarrow$$ $$e^{ix}=\frac{\pm 2\sqrt{6}-5}{-i}\Longleftrightarrow$$ $$ix=\ln\left(\frac{\pm 2\sqrt{6}-5}{-i}\right)+2i\pi n\Longleftrightarrow$$ $$x=\frac{\ln\left(\frac{\pm 2\sqrt{6}-5}{-i}\right)+2i\pi n}{i}$$