If `(a costheta_1,asintheta_1),(acostheta_2,a sintheta_2)`, and `(acostheta_3a sintheta_3)` represent the vertces of an equilateral triangle inscribed

Question

If `(a costheta_1,asintheta_1),(acostheta_2,a sintheta_2)`, and `(acostheta_3a sintheta_3)` represent the vertces of an equilateral triangle inscribed

If `(a costheta_1,asintheta_1),(acostheta_2,a sintheta_2)`, and `(acostheta_3a sintheta_3)` represent the vertces of an equilateral triangle inscribed in a circle. Then.
A. `costheta_1+costheta_2+costheta+3=0`
B. `sintheta_1+sintheta_2+sin theta_3=0`
C. `tantheta_1+tantheta_2+tantheta_3=0`
D. `cottheta_1+cottheta_2+cottheta_3=0`

Monjur Alahi

4 views

2025-01-04 04:42

1 Answers

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Salim Ahmed Kawsar · Accepted Answer · 2023-02-05T15:29:48+06:00

Correct Answer - A::B
Vertices `(a cos theta_1,a sintheta_1),(acostheta_2,a sintheta_2)`, and origin is the circumcenter (centroid) of circumcircle. Therefore, the coordinates of the centroid are
`((a(costheta_1+costheta_2+costheta_3))/(3),(a(sintheta_1,+sintheta_2+sintheta_3))/(3))`
But as the centroid is the origin (0,0) we have `cos theta_1+costheta_2+costheta_3=0`
and `sin theta_1+sintheta_2+sintheta_3=0`