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The given value is$\sin 40°+\sin 130°+\sin 220°+\sin 310°$

The reference angle for $\sin 130°is50°$

The reference angle for $\sin 220°is40°$

The reference angle for $\sin 310°is50°$

If$\mathrm{Î\pm }$is any reference angle for$\mathrm{θ}$then,

$\sin \mathrm{θ}=Â\pm \sin \mathrm{Î\pm }$

so,$\sin 130°=Â\pm \sin 50°,\sin 220°=Â\pm \sin 40°,and\sin 310°=Â\pm \sin 50°$

Now check in which quadrant the given angles lie

$\sin 40°$ lies in quadrant-I,where is positive

$\sin 130°$ lies in quadrant-II,where is positive

$\sin 220°$lies in quadrant-III,where is negative

$\sin 310°$ lies in quadrant-IV,where is negative

Replace the given angle in the expression with the respective reference angle

$$\begin{gathered} \sin 40°+\sin 130°+\sin 220°+\sin 310°=\sin 40°+\sin 50°+(-\sin 40°)+(-\sin 50°) \\ =(\sin 40°-\sin 40°)+(\sin 50°-\sin 50°) \\ =0 \end{gathered}$$

Therefore,the value of$\sin 40°+\sin 130°+\sin 220°+\sin 310°=0$