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Question

Prove that 5cosθ+3cos(θ+π/3)+3 lies between 4 and 10.

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Solution

The given expression is
5cosθ+3(cosθcos60osinθsin60o)+3=12[13cosθ33sinθ]+3
Put 13=rcosα,33=rsinα
r=169+27=196=14
Given expression
=r2[cos(θ+α)]+3=7cos(θ+α)+3
Hence max. and min. value of expression are 7+3 and 7+3 i.e., 10 and 4 respectively

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