Find the value of the following expression

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

#Geometry

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Hemant Mittal - 5 years ago

Explain your working please.

Pritthijit Nath - 5 years 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.

Rishabh Tiwari - 5 years ago

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

Sabhrant Sachan - 5 years ago

anup kumar - 5 years ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$