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If alpha in [-2pi, 2pi] and cos.(alpha)/...

If `alpha in [-2pi, 2pi]` and `cos.(alpha)/(2)+sin.(alpha)/(2)=sqrt(2)(cos 36^(@)-sin18^(@))`, then a value of `alpha`

A

`(7pi)/(6)`

B

`(pi)/(6)`

C

`-(5pi)/(6)`

D

`(pi)/(6)`

Text Solution

Verified by Experts

The correct Answer is:
A, D
`cos.(alpha)/(2)+sin.(alpha)/(2)=sqrt(2)(cos 36^(@)-sin 18^(@))`
`rArr cos((alpha)/(2)-(pi)/(4))=(sqrt(5)+1)/(4)-(sqrt(5)-1)/(4)`
`rArr cos((alpha)/(2)-(pi)/(4))=(1)/(2)`
`rArr (alpha)/(2)-(pi)/(4)=2n pi pm(pi)/(3), n in Z`
`therefore alpha = 4n pi+(pi)/(2)pm (2pi)/(3)`
`therefore alpha = 4n pi+(7pi)/(6)+(7pi)/(6), 4n pi-(pi)/(6)`
For `n=0, alpha=(7pi)/(6), (pi)/(6)`.
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