Find the maximum and minimum values of the following expressions. (i) `3costheta+5sin(theta-pi/6)` (ii) `4sintheta-3costheta+7`

Explanation

Maximum value of `asintheta+bcos theta ` is `sqrt(a^2+b^2).`

Minimum value of `asintheta+bcos theta ` is `-sqrt(a^2+b^2).`


`(i) 3costheta+5sin(theta-pi/6) = 3costheta+5(sinthetacos(pi/6)-costhetasin(pi/6))`

`=3costheta+5(sqrt3/2sintheta-1/2costheta)`

`=1/2costheta +(5sqrt3)/2sintheta`

So, maximum value of `1/2costheta +(5sqrt3)/2sintheta = sqrt((1/2)^2+((5sqrt3)/2)^2) = sqrt(76/4) = sqrt19`

Minimum value of `1/2costheta +(5sqrt3)/2sintheta = -sqrt((1/2)^2+((5sqrt3)/2)^2) = -sqrt(76/4) = -sqrt19`

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