# Find the value of the following expression

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
1 year, 11 months ago

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$$\text{We have } \sin{\alpha}+\sin{\beta}+\sin{\gamma}=3 , \text{ since } -1\le \sin{\theta}\le 1\\ \text{We can conclude that } \sin{\alpha}=\sin{\beta}=\sin{\gamma}=1 \implies \cos{\alpha}=\cos{\beta}=\cos{\gamma} = 0 \\ \implies \cos^3{\alpha}+\cos^9{\beta}+\cos^{27}{\gamma} =\boxed{0}$$

- 1 year, 11 months ago

0

- 1 year, 11 months ago

- 1 year, 11 months ago

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.

- 1 year, 10 months ago

0

- 1 year, 10 months ago