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Question Number 9525 by Joel575 last updated on 12/Dec/16

If x_1 , x_2 , x_3 , ..., x_(2009 ) ∈ R Find the minimum value from (cos x_1 )(sin x_2 ) + (cos x_2 )(sin x_3 ) + ... + (cos x_(2008) )(sin x_(2009) ) + (cos x_(2009) )(sin x_1 )

$$\mathrm{If}\:{x}_{\mathrm{1}} ,\:{x}_{\mathrm{2}} ,\:{x}_{\mathrm{3}} ,\:...,\:{x}_{\mathrm{2009}\:} \in\:\mathbb{R} \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{from} \\ $$$$\left(\mathrm{cos}\:{x}_{\mathrm{1}} \right)\left(\mathrm{sin}\:{x}_{\mathrm{2}} \right)\:+\:\left(\mathrm{cos}\:{x}_{\mathrm{2}} \right)\left(\mathrm{sin}\:{x}_{\mathrm{3}} \right)\:+\:...\:+\:\left(\mathrm{cos}\:{x}_{\mathrm{2008}} \right)\left(\mathrm{sin}\:{x}_{\mathrm{2009}} \right)\:+\:\left(\mathrm{cos}\:{x}_{\mathrm{2009}} \right)\left(\mathrm{sin}\:{x}_{\mathrm{1}} \right) \\ $$

Answered by mrW last updated on 15/Dec/16

S=(cos x_1 )(sin x_2 ) + (cos x_2 )(sin x_3 ) + ... + (cos x_(2008) )(sin x_(2009) ) + (cos x_(2009) )(sin x_1 ) k=1,2,3,∙∙∙,2009 following value combinations are possible and result to the minimum value for S: (i): cos x_k =((√2)/2), sin x_k =−((√2)/2) (ii): cos x_k =−((√2)/2), sin x_k =((√2)/2) we have min. S=2009×(−((√2)/2)×((√2)/2))=−1004.5

$$\mathrm{S}=\left(\mathrm{cos}\:{x}_{\mathrm{1}} \right)\left(\mathrm{sin}\:{x}_{\mathrm{2}} \right)\:+\:\left(\mathrm{cos}\:{x}_{\mathrm{2}} \right)\left(\mathrm{sin}\:{x}_{\mathrm{3}} \right)\:+\:...\:+\:\left(\mathrm{cos}\:{x}_{\mathrm{2008}} \right)\left(\mathrm{sin}\:{x}_{\mathrm{2009}} \right)\:+\:\left(\mathrm{cos}\:{x}_{\mathrm{2009}} \right)\left(\mathrm{sin}\:{x}_{\mathrm{1}} \right) \\ $$$$\mathrm{k}=\mathrm{1},\mathrm{2},\mathrm{3},\centerdot\centerdot\centerdot,\mathrm{2009} \\ $$$$\mathrm{following}\:\mathrm{value}\:\mathrm{combinations}\:\mathrm{are}\:\mathrm{possible}\:\mathrm{and}\:\mathrm{result}\:\mathrm{to} \\ $$$$\mathrm{the}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{for}\:\mathrm{S}: \\ $$$$\left(\mathrm{i}\right):\:\mathrm{cos}\:\mathrm{x}_{\mathrm{k}} =\frac{\sqrt{\mathrm{2}}}{\mathrm{2}},\:\mathrm{sin}\:\mathrm{x}_{\mathrm{k}} =−\frac{\sqrt{\mathrm{2}}}{\mathrm{2}} \\ $$$$\left(\mathrm{ii}\right):\:\mathrm{cos}\:\mathrm{x}_{\mathrm{k}} =−\frac{\sqrt{\mathrm{2}}}{\mathrm{2}},\:\mathrm{sin}\:\mathrm{x}_{\mathrm{k}} =\frac{\sqrt{\mathrm{2}}}{\mathrm{2}} \\ $$$$\mathrm{we}\:\mathrm{have} \\ $$$$\mathrm{min}.\:\mathrm{S}=\mathrm{2009}×\left(−\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}×\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\right)=−\mathrm{1004}.\mathrm{5} \\ $$

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