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The maximum values of 3 costheta+5sin(th...

The maximum values of `3 costheta+5sin(theta-(pi)/(6))` for any real value of `theta` is:

A

`sqrt(79)/(2)`

B

`sqrt(34)`

C

`sqrt(31)`

D

`sqrt(19)`

Text Solution

Verified by Experts

The correct Answer is:
D
Given expression 3 cos `theta+5sin(theta-(pi)/(6))`
`=3cos theta+5(sin thetacos.(pi)/(6)-sin.(pi)/(6)costheta)`
`=3cos theta+5((sqrt(3))/(2)sin theta-(1)/(2)cos theta)`
`=3cos theta-(5)/(2)cos theta+(5sqrt(3))/(2)sin theta`
`=(1)/(2)cos theta+(5sqrt(3))/(2)sin theta`
`therefore` The maximum value of a cos `theta` + b sin `theta` is `sqrt(a^(2)+b^(2))`
So, maximum value of `(1)/(2)` cos `theta+(5sqrt(3))/(2)sin theta` is
`sqrt(((1)/(2))^(2)+((5sqrt(3))/(2))^(2))=sqrt((1)/(4)+(75)/(4))=sqrt((76)/(4))=sqrt(19)`.
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