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DrunkenOldFool
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- Feb 6, 2012
- 20
If $\cos \alpha +\cos \beta + \cos \gamma=0$ and $\cos 3 \alpha +\cos 3\beta +\cos 3\gamma = \lambda \cos \alpha \cos \beta \cos \gamma$. What is the value of $\lambda$?
\[\begin{align*}\cos 3 \alpha +\cos 3\beta +\cos 3\gamma &= (4\cos^3 \alpha -3\cos \alpha)+(4\cos^3 \beta -3\cos \beta)+(4\cos^3 \gamma -3\cos \gamma) \\ &= 4(\cos^3 \alpha +\cos^3 \beta +\cos^3 \gamma)-3(\underbrace{\cos \alpha +\cos \beta + \cos \gamma}_{=0}) \\ &= 4(\cos^3 \alpha +\cos^3 \beta +\cos^3 \gamma)\end{align*}\]If $\cos \alpha +\cos \beta + \cos \gamma=0$ and $\cos 3 \alpha +\cos 3\beta +\cos 3\gamma = \lambda \cos \alpha \cos \beta \cos \gamma$. What is the value of $\lambda$?