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Question

Write the interval in which the values of 5cosθ+3cos(θ+π3)+3 lie.

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Solution

Let f(θ)=5cosθ+3cos(θ+π3)+3then
f(θ)=5cosθ+3cos(θ+π3)+3
=5cosθ+3(cosθcosπ3sinθsinπ3)+3
=5cosθ+32cosθ332sinθ+3
=132cosθ332sinθ+3 ...(i)
Now,
(132)2+(332)2132cosθ332sinθ(132)2+(332)2
7132cosθ332sinθ+37+3
4132cosθ332sinθ+310
45cosθ3cos(θ+π3)+310 [Using 1] Hence, the required interval is [-4,10].

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