1.

Find the range of `f(theta) = 5 costheta + 3 cos(theta + pi/3) + 3 `

Answer» `f(theta) = 5costheta+3cos(theta+pi/3)+3`
`=>f(theta) = 5costheta + 3(costhetacospi/3-sinthetasinpi/3)+3`
`=>f(theta) = 5costheta + 3(costheta(1/2)-sintheta(sqrt3/2))+3`
`=>f(theta) = 5costheta + 3/2costheta-(3sqrt3)/2sintheta+3`
`=>f(theta) = 13/2costheta-(3sqrt3)/2sintheta+3`
We know, for a function `f(x) = acosx+bsinx+c`,
`f(x)_max = c + sqrt(a^2+b^2)`
`f(x)_min = c - sqrt(a^2+b^2)`
Here, `a = 13/2, b = (3sqrt3)/2 and c = 3`
`:. f(x)_max = 3 + sqrt((13/2)^2+((3sqrt3)/2)^2) = 3+sqrt49 = 3+7 = 10`
`:. f(x)_min = 3 - sqrt((13/2)^2+((3sqrt3)/2)^2) = 3-sqrt49 = 3-7 = -4`
So, range of `f(theta)` is between `-4` and `10`.
`f(theta) in [4,10]`.


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