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

Options

A. `sqrt(79)/(2)`
B. `sqrt(34)`
C. `sqrt(31)`
D. `sqrt(19)`

Answer

D

Explanation

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)`.