If $(\text{a\hspace{0.17em}}\cos {\theta }_1,\text{a}\sin {\theta }_1),(\text{a\hspace{0.17em}}\cos {\theta }_2,\text{a}\sin {\theta }_2),(\text{a\hspace{0.17em}}\cos {\theta }_3,\text{a}\sin {\theta }_3)$  represent the vertices of an equilateral triangle inscribed in a circle, then

Since origin is centre of given circle and in an equilateral triangle, centroid and circumcenter are same point
$\Rightarrow \frac{\text{a}}{\text{3}}(\sum \cos {\theta }_i)=\frac{\text{a}}{\text{3}}(\sum \sin {\theta }_i)=0$