Given that $\cos A + \cos B + \cos C = 0$ and $\sin A + \sin B + \sin C = 0$.

Question

If $\cos A + \cos B + \cos C = 0$ and $\sin A + \sin B + \sin C = 0$.

The value of $ \sin^3A+\sin^3B+\sin^3C$

What I can see here is that as $\sin A + \sin B + \sin C = 0$ hence $ \sin^3A+\sin^3B+\sin^3C=3\sin A \sin B\sin C$ but I am not able to achieve a constant value. Please give some hint.

@Ananya, Find the necessary & the sufficient condition : http://math.stackexchange.com/questions/1397066/clarification-regarding-a-question — lab bhattacharjee, Apr 24 '16 at 04:35

Batominovski · Answer 1 · 2016-04-24T10:55:13.073

1

Hint: If the centroid of a triangle coincides with it circumcenter, the triangle is equilateral.

Answer: $\sin^3(A)+\sin^3(B)+\sin^3(C)=3\sin^3(A)-\frac{9}{4}\sin(A)=-\frac{3}{4}\sin(3A)$.

edited Apr 24 '16 at 10:55

answered Apr 23 '16 at 21:42

Batominovski

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So how does this help here? – Akira Apr 24 '16 at 03:57
You have from this hint that we may assume $B=A+\frac{2\pi}{3}$ and $C=A-\frac{2\pi}{3}$. Hence, the require sum is $3\sin^3(A)-\frac{9}{4}\sin(A)=-\frac{3}{4}\sin(3A)$. – Batominovski Apr 24 '16 at 10:46

Given that $\cos A + \cos B + \cos C = 0$ and $\sin A + \sin B + \sin C = 0$.

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