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-

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

If x_{1}, x_{2}, x_{3}, \dots, x_{2009} \in R

Find the minimum value from

(\cos x_{1}) (\sin x_{2}) + (\cos x_{2}) (\sin x_{3}) + \dots + (\cos x_{2008}) (\sin x_{2009}) + (\cos x_{2009}) (\sin x_{1})

Answered by mrW last updated on 15/Dec/16

S = (\cos x_{1}) (\sin x_{2}) + (\cos x_{2}) (\sin x_{3}) + \dots + (\cos x_{2008}) (\sin x_{2009}) + (\cos x_{2009}) (\sin x_{1})

k = 1, 2, 3, \cdot \cdot \cdot, 2009

following value combinations are possible and result to

the minimum value for S :

(i) : \cos x_{k} = \frac{\sqrt{2}}{2}, \sin x_{k} = - \frac{\sqrt{2}}{2}

(ii) : \cos x_{k} = - \frac{\sqrt{2}}{2}, \sin x_{k} = \frac{\sqrt{2}}{2}

we have

\min . S = 2009 \times (- \frac{\sqrt{2}}{2} \times \frac{\sqrt{2}}{2}) = - 1004.5

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