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Let alpha, beta, gamma be the measures o...

Let `alpha, beta, gamma` be the measures of angle such that `sin alpha+sin beta+sin gamma ge 2`. Then the possible value of `cos alpha + cos beta+cos gamma` is

A

3

B

`2.5`

C

`2.4`

D

2

Text Solution

Verified by Experts

The correct Answer is:
D
`sin alpha + sin beta + sin gamma ge 2`
or `Sigma sin alpha ge 2`
Squaring both sides, we get
`Sigma sin^(2)alpha + 2 Sigma sin alpha sin beta ge 4`
`rArr 3-Sigma cos^(2)alpha + 2Sigma sin alpha sin beta ge 4`
`rArr 2 Sigma sin alpha sin beta ge 1 1+Sigma cos^(2) alpha`
`rArr 2 Sigma (sin alpha sin beta + cos alpha cos beta) ge 1+(Sigma cos alpha)^(2)` (adding `2Sigma cos alpha cos beta` on both sides)
`rArr (cos alpha + cos beta + cos gamma)^(2)+1le 2[cos (alpha-beta)+cos(beta-gamma)+cos(gamma-alpha)]le 6`
`rArr (cos alpha + cos beta + cos gamma)^(2) le 5`
`rArr (cos alpha + cos beta + cos gamma le sqrt(5)`
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