Prove that `(2sqrt3 + 3) sintheta +2sqrt3 cos theta` lies between `-(2sqrt3 +sqrt15) and (2sqrt3+ sqrt15)`,

Explanation

Given,`(23+3)sintheta+2sqrt3costheta`

let `a=2sqrt3+3,b=2sqrt3`

`sqrt(a^2+b^2)`
=`sqrt((2sqrt3+3)^2+(2sqrt3)^2)`
`=sqrt(12+9+12sqrt3+12)=sqrt(33+12sqrt3)`

Dividing we get,

`((2sqrt3+3)/(sqrt(33+12sqrt3))sintheta+(2sqrt3)/(sqrt(33+12sqrt3))costheta)`
let` tanphi=a/b,`

So,`sinphi=a/(sqrta^2+b^2),cosphi=b/(sqrta^2+b^2)`

...