If sin alpha + sin beta + sin gamma = 0 and c | vedclass

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(B) Given: $\cos \alpha + \cos \beta + \cos \gamma = 0$ $(i)$ and $\sin \alpha + \sin \beta + \sin \gamma = 0$ $(ii)$.Let $a = \cos \alpha + i\sin \al

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$3/2$

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(B) Given: $\cos \alpha + \cos \beta + \cos \gamma = 0$ $(i)$ and $\sin \alpha + \sin \beta + \sin \gamma = 0$ $(ii)$.Let $a = \cos \alpha + i\sin \alpha$,$b = \cos \beta + i\sin \beta$,and $c = \cos \gamma + i\sin \gamma$.Then $a + b + c = (\cos \alpha + \cos \beta + \cos \gamma) + i(\sin \alpha + \sin \beta + \sin \gamma) = 0 + i(0) = 0$ $(iii)$.Also,$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = (\cos \alpha - i\sin \alpha) + (\cos \beta - i\sin \beta) + (\cos \gamma - i\sin \gamma) = 0 - i(0) = 0$.This implies $\frac{ab + bc + ca}{abc} = 0$,so $ab + bc + ca = 0$ $(iv)$.Squaring $(iii)$,we get $a^2 + b^2 + c^2 + 2(ab + bc + ca) = 0$.Using $(iv)$,$a^2 + b^2 + c^2 = 0$.Substituting the exponential forms: $(\cos 2\alpha + i\sin 2\alpha) + (\cos 2\beta + i\sin 2\beta) + (\cos 2\gamma + i\sin 2\gamma) = 0$.Equating real parts: $\cos 2\alpha + \cos 2\beta + \cos 2\gamma = 0$.Using the identity $\cos 2	heta = 1 - 2\sin^2 	heta$:$(1 - 2\sin^2 \alpha) + (1 - 2\sin^2 \beta) + (1 - 2\sin^2 \gamma) = 0$.$3 - 2(\sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma) = 0$.Therefore,$\sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma = 3/2$.

If $\sin \alpha + \sin \beta + \sin \gamma = 0$ and $\cos \alpha + \cos \beta + \cos \gamma = 0,$ then the value of $\sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma$ is

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