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Prove that 5costheta+3cos(theta+pi/3)+3 ...

Prove that `5costheta+3cos(theta+pi/3)+3` lies between `-4a n d10.`

Text Solution

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We have `5 cos theta+3 cos (theta+(pi)/(3))+3`
`=5 cos theta+3 cos theta cos"" (pi)/(3)-3 sin theta sin"" (pi)/(3)+3`
`(13)/(2)cos theta-(3sqrt(3))/(2)sin theta+3`.
Now `-sqrt(((13)/(2))^(2)+((3sqrt(3))/(2))^(2))le(13)/(2)cos theta-(3sqrt(3))/(2)sin theta`
`lesqrt(((13)/(2))^(2)+((3sqrt(3))/(2))^(2))`
`rArr -7le(13)/(2)cos theta-(3sqrt(3))/(2)sin theta le 7`
`rArr -4le(13)/(2)cos theta-(3sqrt(3))/(2)sin theta+3 le10`
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