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Prove that: -4le5costheta+3cos(theta+(pi...

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

Text Solution

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We have, `5costheta+3costheta(theta+pi/3)=5costheta+3costhetacospi/3-3sinthetasinpi/3=13/2costheta-(3sqrt(3))/(2)sintheta`
Since `-sqrt((13/2)^(2)+(-3sqrt(3)/2)^(2)) le13/2 costheta-(3sqrt(3))/(2)sintheta le sqrt((13/2)^(2)+(-3sqrt(3))/(2)^(2))`
`rArr -7 le 13/2 costheta-(3sqrt(3))/(2)sintheta le7`
`rArr -7 le5costheta+3cos(theta+pi/3)le7` for all `theta`.
`rArr -7+3 le5costheta+3cos(theta+pi/3)+3 ge7 +3` for all `theta`
`rArr -4 le5 costheta+3cos(theta+pi/3)+3 le10` for all `theta`
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