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Let A and B denote the statements A : ...

Let `A and B` denote the statements
`A : cos alpha + cos beta + cos gamma =0`
`B : sin alpha + siin beta + sin gamma = 0`
If `cos(beta - gamma) + cos (gamma -alpha) + cos (alpha -beta) = - (3)/(2)`,
then

A

`A` is true and `B` is false.

B

`A` is false and `B` is true.

C

Both `A and B` are true.

D

Both `A and B` are false.

Text Solution

Verified by Experts

The correct Answer is:
C
` cos (beta -gamma) +cos (gamma - alpha ) + cos (alphea - beta) =-(3)/(2)`
`rArr 2 [cos (beta -gamma) +cos (gamma -alpha ) + cos (alpha - beta)] + 3 =0`
`rArr 2 [cos (beta -gamma ) + cos (gamma -alpha) + cos (alpha -beta)] + sin ^(2) alpha + cos ^(2) alpha + sin ^(2) beta + cos ^(2) beta + sin ^(2) gamma + cos ^(2) gamma =0`
`rArr (sin alpha + sin beta + sin gamma)^(2) + ( cos alpha + cos beta + cos gamma)^(2) =0`
`rArr cos alpha + cos beta + cos gamma =0`
and `sin alpha + sin beta + sin gamma =0`
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