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If \(\sin\alpha + \sin\beta + \sin \gamma = 3 \), fidn the value of \(\cos^3 \alpha + \cos^9 \beta + \cos^{27} \gamma \).

Note by Pritthijit Nath 6 months, 3 weeks ago

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\(\text{We have } \sin{\alpha}+\sin{\beta}+\sin{\gamma}=3 , \text{ since } -1\le \sin{\theta}\le 1\\ \text{We can conclude that } \sin{\alpha}=\sin{\beta}=\sin{\gamma}=1 \implies \cos{\alpha}=\cos{\beta}=\cos{\gamma} = 0 \\ \implies \cos^3{\alpha}+\cos^9{\beta}+\cos^{27}{\gamma} =\boxed{0}\) – Sambhrant Sachan · 6 months, 2 weeks ago

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0 – Hemant Mittal · 6 months, 3 weeks ago

@Hemant Mittal – Explain your working please. – Pritthijit Nath · 6 months, 2 weeks ago

@Pritthijit Nath – Max . Value of sum of three sines = 3 , since (-1 < sin¤ < +1)

Equality is there in both.,

Therefore we conclude that maximum value (=3) occurs when, all three sines are equal & hence 1.

Therefore cosines will be all 0 , Nd hence the given sum will be equal to zero. – Rishabh Tiwari · 6 months, 2 weeks ago

0 – Anup Kumar · 6 months, 1 week ago

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TopNewest\(\text{We have } \sin{\alpha}+\sin{\beta}+\sin{\gamma}=3 , \text{ since } -1\le \sin{\theta}\le 1\\ \text{We can conclude that } \sin{\alpha}=\sin{\beta}=\sin{\gamma}=1 \implies \cos{\alpha}=\cos{\beta}=\cos{\gamma} = 0 \\ \implies \cos^3{\alpha}+\cos^9{\beta}+\cos^{27}{\gamma} =\boxed{0}\) – Sambhrant Sachan · 6 months, 2 weeks ago

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0 – Hemant Mittal · 6 months, 3 weeks ago

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– Pritthijit Nath · 6 months, 2 weeks ago

Explain your working please.Log in to reply

Equality is there in both.,

Therefore we conclude that maximum value (=3) occurs when, all three sines are equal & hence 1.

Therefore cosines will be all 0 , Nd hence the given sum will be equal to zero. – Rishabh Tiwari · 6 months, 2 weeks ago

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0 – Anup Kumar · 6 months, 1 week ago

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