Question

Prove that: `-4le5costheta+3cos(theta+(pi)/(3))+3le10`, for all values of `theta`,

Answer 1

We have `, 5 cos theta+3 cos(theta+(pi)/(3))=5costheta+3costhetacos(pi)/(3)-3sinthetasin(pi)/(3)=(13)/(2)cos theta-(3sqrt(3))/(2)sin theta`
Since, `=sqrt(((13)/(3))^(2)+(-(3s qrt(3))/(2))^(2))le(13)/(2)costheta-(3sqrt(3))/(2)sin theta lesqrt(((13)/(2))^(2)+(-(3sqrt(3))/(2))^(2))`
`rArr-7le5cos theta+3 cos(theta+(pi)/(3))le7` for all `theta`,
`rArr-7+3le5costheta+3cos(theta+(pi)/(3))+3le7+3` for all `theta`.
`rArr -4le5cos theta+3cos(theta+(pi)/(3))+3le10` for all `theta`,