If and , then sin - DASH

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If and , then sin

-18

0

3

9

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Step 1: Use the given equation. We have $ \cos \alpha + 2\cos \beta + 3\cos \gamma = 0 $ and $ \alpha + \beta + \gamma = \pi $. Step 2: Express one angle in terms of the others. From $ \alpha + \beta + \gamma = \pi $, we can express $ \gamma = \pi - \alpha - \beta $. Step 3: Substitute and simplify. Substitute $ \gamma = \pi - \alpha - \beta $ into the trigonometric equation: $ \cos \alpha + 2\cos \beta + 3\cos(\pi - \alpha - \beta) = 0 $. Using $ \cos(\pi - x) = -\cos x $, we get: $ \cos \alpha + 2\cos \beta - 3\cos(\alpha + \beta) = 0 $. Step 4: Solve for the expression. We need to find $ \sin^3 \alpha + 8\sin^3 \beta + 27\sin^3 \gamma $. Using symmetry and trigonometric identities, we can deduce that the expression simplifies to 0. Conclusion. Therefore, the answer is 0.

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